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Structured observer-based controller for delayed systems with two unstable poles and minimum phase zeros.
15th IFAC Workshop on Time Delay Systems 15th IFAC Workshop on Time Delay Systems 15th IFAC Workshop on Delay Systems Sinaia, Romania, September 2019 15th IFAC Workshop on Time Time9-11, Delay Systems Sinaia, Romania, September 9-11, 2019 Available online at www.sciencedirect.com Sinaia, Romania, September 9-11, 2019 15th IFAC Workshop on Time Delay Systems Sinaia, Romania, September 9-11, 2019 Sinaia, Romania, September 9-11, 2019
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IFAC PapersOnLine 52-18 (2019) 162–167
Structured observer-based controller for Structured Structured observer-based observer-based controller controller for for Structured observer-based controller for delayed systems with two unstable poles Structured observer-based controller for delayed systems with two unstable poles delayed systems with two unstable poles delayed systems with two unstable poles and minimum zeros. delayed systems withphase two unstable poles and minimum phase zeros. and minimum phase zeros. and minimum phase zeros. and aminimum phase ∗∗ zeros. ∗ Carlos Daniel V´ zquez ∗∗∗ David Novella ∗∗ ∗∗ Basilio del Muro ∗ ∗
Carlos Daniel V´ a ∗ David Novella ∗∗ Basilio del Muro∗∗ ∗∗zquez ∗ David ∗∗ ∗∗ Basilio M´ Carlos Daniel V´ a zquez Novella delarquez Muro∗∗∗ ∗∗zquez ∗∗ Olivier Sename Luc Dugard Juan Francisco ∗∗ ∗∗ Carlos Daniel V´ a Novella del Muro Olivier Sename Luc Dugard Juan Francisco a rquez ∗ David ∗∗ ∗∗ Basilio M´ ∗∗ Carlos Daniel V´ a David Novella Basilio del Muro∗∗∗ ∗∗zquez ∗∗ Olivier Sename Luc Dugard Juan Francisco M´ a rquez Olivier Sename ∗∗ Luc Dugard ∗∗ Juan Francisco M´ arquez ∗ Olivier Sename Dugard Juan Francisco M´ arquez ∗ ∗ Escuela Superior Luc de Ingenier´ ıa Mec´ a nica yy El´ eectrica, Unidad ∗ ıa Mec´ a nica El´ ctrica, Unidad ∗ Escuela Superior de Ingenier´ ∗ Escuela Instituto Superior de ıa Mec´ a eectrica, Unidad Culhuacan, eecnico Nacional,Santa Ana 1000, M´ Superior Polit´ de Ingenier´ Ingenier´ Mec´ anica nica yy El´ El´ ctrica, Unidad Culhuacan, Polit´ cnico ıa Nacional,Santa Ana 1000, M´eexico xico ∗ Escuela Instituto Escuela Superior de Ingenier´ ıa Mec´ a nica y El´ e ctrica, Unidad Culhuacan, Instituto Polit´ e cnico Nacional,Santa Ana 1000, D.F., 04430, M´ e xico (e-mail: carlos daniel [email protected])(e-mail: Culhuacan, Instituto Polit´ecnico Nacional,Santa Ana 1000, M´ M´eexico xico daniel [email protected])(e-mail: D.F., 04430, M´ e xico (e-mail: carlos Culhuacan, Polit´ecnico Nacional,Santa Ana 1000, M´exico D.F., 04430, M´ carlos daniel [email protected])(e-mail: [email protected])(e-mail: [email protected]). D.F., 04430, Instituto M´eexico xico (e-mail: (e-mail: carlos daniel [email protected])(e-mail: [email protected])(e-mail: [email protected]). ∗∗ D.F., 04430, M´exicoAlpes, (e-mail: carlosGrenoble daniel [email protected])(e-mail: [email protected])(e-mail: [email protected]). ∗∗ Univ. Grenoble CNRS, INP, GIPSA-lab, 38000 ∗∗ [email protected])(e-mail: [email protected]). Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 ∗∗ Univ. [email protected])(e-mail: [email protected]). ∗∗ Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France (e-mail: [email protected])(e-mail: Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France (e-mail: [email protected])(e-mail: ∗∗ Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France (e-mail: [email protected])(e-mail: [email protected])(e-mail: Grenoble, France (e-mail: [email protected])(e-mail:38000 [email protected])(e-mail: Grenoble, France (e-mail: [email protected])(e-mail: [email protected])(e-mail: [email protected]) [email protected])(e-mail: [email protected]) [email protected])(e-mail: [email protected]) [email protected]) [email protected]) Abstract: This paper analyses the problem control stabilization of aa particular class Abstract: This paper analyses the problem of of control and and stabilization of particular class of of Abstract: This paper analyses the of and stabilization of particular of Linear Invariant systems. The under consideration two unstableclass poles, Abstract: paper (LTI) analyses the problem problem of control control and stabilizationhas of aatwo particular class of Linear Time TimeThis Invariant (LTI) systems. The system system under consideration has unstable poles, Abstract: This paper analyses the problem of control and stabilization of a particular class of Linear Time Invariant (LTI) systems. The system under consideration has two unstable poles, n real stable poles, m minimum phase zeros plus time delay. An observer based controller Linear systems. Thezeros system has twobased unstable poles, n real Time stableInvariant poles, m(LTI) minimum phase plusunder time consideration delay. An observer controller Linear Time Invariant systems. The system under has two unstable poles, n real realfour stable poles, m(LTI) minimum phase zeros plus time consideration delay. An observer based controller with tunable gains is proposed as a control strategy in order to ensure a stable behaviour n stable poles, m minimum phase zeros plus time delay. An observer based controller with four tunable gains is proposedphase as aa control strategy in order to ensure aa stable behaviour n real stable poles, m minimum zeros plus timeexistence delay. observer basedscheme controller with four tunable gains is as strategy in order to ensure behaviour of the closed-loop system. Sufficient for the of the proposed are with four tunable gains is proposed proposed asconditions a control control strategy in orderAn to ensure a stable stable behaviour of the closed-loop system. Sufficient conditions for the existence of the proposed scheme are with four tunable gains is proposed as a control strategy in order to ensure a stable behaviour of the closed-loop system. Sufficient conditions for the existence of the proposed scheme are obtained in terms of the upper limit of time delay size and the poles and zeros position. The of the closed-loop system. Sufficient for size the and existence of the scheme The are obtained in terms of the upper limit conditions of time delay the poles andproposed zeros position. of the closed-loop system. Sufficient conditions for the existence of the proposed scheme are obtained in terms of the upper limit of time delay size and the poles and zeros position. The optimization method. The proposed controller parameters are tuned using a non-smooth H ∞ obtained inparameters terms of the upper limit ofa time delay size and the poles and zeros position. The non-smooth optimization method. The proposed controller are tuned using H ∞ obtained inparameters terms of the limit delay model size and the poles and zeros The position. The optimization method. The proposed controller parameters are upper tuned usingofaa time non-smooth H∞ ∞ control strategy is applied to an unstable linearized of a continuously stirred tank reactor optimization method. proposed controller are tuned using non-smooth H ∞ control strategy is applied to an unstable linearized model aa continuously stirred tank reactor optimization method. proposed controller parameters are the tuned using a non-smooth H∞ of control strategy is applied to unstable linearized model of stirred tank reactor (CSTR) in order show effectiveness of the proposed design scheme. Numerical results are control strategy isto applied to an an unstable linearized model of a continuously continuously stirredThe tank reactor (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are control strategy is applied to an unstable linearized model of a continuously stirred tank reactor (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are presented. (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are presented. (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are presented. presented. Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. presented. Keywords: Keywords: Stability Stability and and stabilization, stabilization, Output Output feedback feedback control control under under I/O-delays, I/O-delays, Control Control Keywords: Stability design. Keywords: Stability and and stabilization, stabilization, Output Output feedback feedback control control under under I/O-delays, I/O-delays, Control Control design. Keywords: Stability and stabilization, Output feedback control under I/O-delays, Control design. design. design. 1. INTRODUCTION INTRODUCTION analyse aa high order system simple It 1. analyse order system by by simple controllers. controllers. It is is 1. analyse aa high high order by controllers. is important to highlight highlight that the the mentioned works It only 1. INTRODUCTION INTRODUCTION analyse high order system system by simple simple controllers. It is important to that mentioned works only 1. INTRODUCTION analyse a high order system by simple controllers. It is important to highlight that the mentioned works only consider the problem when only one minimum phase zero One way to define time-delay in the context of control important to highlight that the mentioned works only consider the problem when only one minimum phase zero One way to define time-delay in the context of control important toproblem highlight that theone mentioned works only consider the problem when only one minimum phase zero One way to define time-delay in the context of control and one unstable pole are involved in the original plant. system theory is the time interval from the application of consider the when only minimum phase zero One way to define in from the context of control system theory is thetime-delay time interval the application of and one unstable pole are involved the original plant. consider the problem when only onein minimum phase zero one unstable pole are involved in the original plant. One way to define time-delay in from the context of process control system theory is the time interval the application of control signal to any observable change in the and one cases, unstable pole are necessary involved into the original plant. theory is to theany time interval from the in application of and aasystem control signal observable change the process In real it is also guarantee typical and one unstable pole are involved in the original plant. In real cases, it is also necessary to guarantee typical system theory is the time interval from the application of avariable control signal to any observable change in the process (Wang et al., 1999). Such aachange phenomenon can be be design a control(Wang signal et to al., any1999). observable in the process In real real requirements cases, it it is is also also necessary to guarantee typical variable Such phenomenon can such as speed of response, control In cases, necessary to guarantee typical a control signal et to any1999). observable the process variable al., Such phenomenon can be such as speed of response, control found in (Wang several engineering processes, for in example: mavariable (Wang et al., 1999). Such aachange phenomenon canmabe design In real requirements cases, it is also necessary to guarantee typical design requirements such as speed of response, control found in several engineering processes, for example: bandwidth, disturbance rejection, and robustness unmoddesign requirements such as speed of response, control variable etengineering al., 1999). Such a biological phenomenon canmabe bandwidth, found in several processes, for disturbance rejection, and robustness unmodterial and information transmission, embedded found and in (Wang several engineering processes, for example: example: madesign requirements such as speed of response, control bandwidth, disturbance rejection, and robustness unmodterial information transmission, biological embedded eled dynamics. To meet these requirements, H theory is disturbance rejection, and robustness unmodeled dynamics. To meet these requirements, H∞ is found in biochemical several engineering processes, for example: terial information transmission, biological embedded ∞ theory systems, and chemical processes (Wu et maal., abandwidth, ∞ terial and and information transmission, biological embedded bandwidth, disturbance rejection, and robustness unmodeled dynamics. To meet these requirements, H theory is systems, biochemical and chemical processes (Wu et al., ∞ powerful technique used to design robust controllers for eled dynamics. To meet these requirements, H theory is ∞ terial and information transmission, biological embedded systems, biochemical and chemical processes (Wu et al., a powerful technique used to design robust controllers for 2015). systems, biochemical and chemical processes (Wu et al., alinear dynamics. To meet these requirements, H∞variations, theoryfor is aeled powerful technique used to design design robust controllers controllers for 2015). systems under uncertainties, parameter powerful technique used to robust systems, biochemical and chemical processes (Wu et al., 2015). linear systems under uncertainties, parameter variations, 2015). a powerful technique used to design robust controllers for linear systems under uncertainties, parameter variations, and disturbances. There exist different different systems systems with with two two unstable unstable poles poles and and linear systems under uncertainties, parameter variations, 2015). exist There and disturbances. linear systems under uncertainties, parameter variations, and disturbances. There exist different systems with two unstable poles and minimum phase zeros with time-delay. In chemical indusand disturbances. There exist different systems with two unstable poles and minimum phase zeros with time-delay. In chemical indus- The objective of this paper is two-fold: First we propose and disturbances. There exist different systems with two unstable poles and minimum phase zeros with time-delay. In chemical indusThe objective of this paper is two-fold: First we propose try, common example is the the CSTR (Bequette, (Bequette, 2003). In The minimum phaseexample zeros with time-delay. In chemical indusobjective of this paper is two-fold: First we propose try, aa common is CSTR 2003). In sufficient conditions in order to stabilize a specific class The objective of this paper is two-fold: First we propose minimum phase zeros with time-delay. In chemical industry, a common example is the CSTR (Bequette, 2003). In sufficient conditions in order to stabilize a specific class the military field, the dynamical model of a ballistic missile try, military a common example is the CSTR 2003). In The objective of this paper is two-fold: First we propose sufficient conditions in order to stabilize a specific class the field, the dynamical model(Bequette, of a ballistic missile of delayed systems with two real unstable poles, n stable sufficient conditions in order to stabilize a specific class of delayed systems with two real unstable poles, n stable try, military asimilar common example is the CSTR (Bequette, 2003). In sufficient the field, the dynamical model of aa ballistic missile has characteristics (Blakelock, 1991). Then, the the military field, the dynamical model of ballistic missile conditions in order to stabilize a specific class of delayed systems with two real unstable poles, n stable has similar characteristics (Blakelock, 1991). Then, the poles and m minimum phase zeros (m < n + 2), using of delayed systems with two real unstable poles, n stable the military field, the dynamical model of a ballistic missile has similar characteristics (Blakelock, 1991). Then, the poles and m minimum phase zeros (m < n + 2), using problem of stabilization stabilization of this this class of of 1991). systemsThen, becomes has similar characteristics (Blakelock, the poles of delayed systems with phase two controller real unstable poles, ntunable stable and m minimum zeros (m < n + 2), using problem of of class systems becomes a structured observer-based with four poles and m minimum phase zeros (m < n + 2), using has similar characteristics (Blakelock, the problem of this class of systems becomes aa structured controller tunable an interesting topic where whereof explicit stabilizability results for poles problem of stabilization stabilization ofexplicit this class of 1991). systemsThen, becomes and maobserver-based minimum phase zeros (mwith < of nfour + 2), using structured observer-based controller with four tunable an interesting topic stabilizability results for Then systematic design procedure the control agains. structured observer-based controller with four tunable problem of stabilization of this class of systems becomes an interesting topic where explicit stabilizability results for gains. Then a systematic design procedure of the control many unstable processes are not available yet. an interesting whereare explicit stabilizability agains. structured observer-based controller with four tunable Then a systematic design procedure of the control many unstabletopic processes not available yet. results for gain is proposed in the H framework. Since the controller gains.is Then a systematic procedure control proposed in the H∞ framework. Sinceof thethe controller an interesting topic whereare explicit stabilizability for gain many unstable processes not available yet. ∞design ∞ many unstable available yet. results a the systematic procedure of the control gain is proposed in the framework. Since the controller ∞design is structured, smooth H proposed in Few works haveprocesses addressedare thenot problem of delayed delayed unstable gains. ∞ synthesis gain is Then proposed in non the H H Since the controller ∞ framework. manyworks unstable processes are not available yet. unstable Few have addressed the problem of is structured, the non smooth H synthesis proposed in ∞ ∞ synthesis gain is proposed in non the H framework. Since the controller is structured, the smooth H proposed in ∞has Few works have addressed the problem of delayed unstable ∞ (Apkarian and Noll, 2006) been considered, which is an systems with zero dynamics. In (Kwak et al., 2000), the is structured, the non smooth H synthesis proposed in Few works have addressed the In problem delayed unstable ∞ considered, which is an systems with zero dynamics. (Kwakofet al., 2000), the (Apkarian and Noll, 2006) has been is structured, the non smooth H synthesis proposed in (Apkarian and Noll, 2006) has been considered, which is an Few works have addressed the problem of delayed unstable ∞ systems with zero dynamics. In (Kwak et al., 2000), the central design method (Doyle et al., extension of the H authors face the problem of stabilization for first and ∞ (Apkarian and Noll, 2006) has been considered, which is an systems with zero dynamics. In (Kwak et al., 2000), the central design method (Doyle et al., extension of the H authors face the problem of stabilization for first and (Apkarian ∞ ∞ and Noll, 2006) has been considered, which is an central design method (Doyle et al., extension of the H systems with zero dynamics. In (Kwak et al., 2000), the authors face the problem of stabilization for first and ∞ 1989). One of the main advantages of H structured desecond order systems using ofaa P P controller. for An extended extended design of method (Doyle et deal., authors order face systems the problem stabilization first and extension ∞ central 1989). Oneofofthe theHmain advantages H∞ second using controller. An ∞ structured ∞ designand method (Doyle et al., the H 1989). One the main advantages of H structured deauthors face systems the problem ofaa P stabilization first and extension ∞ central second using controller. An sign method to choose the order the of the result isorder given in (Lee (Lee and Wang, 2010), wherefor theextended authors 1989). Oneofof ofis the main advantages of H∞ structured desecondis order systems using P controller. An extended ∞structure result given in and Wang, 2010), where the authors sign method is to choose the order and the structure of the 1989). One ofis the main advantages of the H∞is structured design method to choose the order and structure of the second order systems using a P controller. An extended result is given in (Lee and Wang, 2010), where the authors desired controller. An illustrative example considered sign method is to choose the order and the structure of the result is given in (Lee and Wang, 2010), where the authors desired controller. An illustrative example is considered to to sign method isinterest to choose the order and the is structure of the desired controller. An illustrative example considered to This work was the of Sciresult given (Lee and by Wang, 2010), where the authors highlight the this work, in particular compared desired controller. An of illustrative example is considered to This is work wasinsupported supported by the Secretary Secretary of Education, Education, Scihighlight the interest of this work, in particular compared This work was supported by the Secretary of Education, Sciences, Technology and Innovation of Mexico City, under the grant desired controller. An illustrative example is considered to highlight the interest of this work, in particular compared with a full order H controller. ThisTechnology work was supported by the Secretary of under Education, Sci∞ of highlight the interest this work, in particular compared ences, and Innovation of Mexico City, the grant with a full order H controller. ∞ ences, and of City, the SECITI/079/2017. ThisTechnology work was supported by the Secretary of under Education, Scihighlight interest this work, in particular compared with aa full fullthe order H∞ controller. ences, Technology and Innovation Innovation of Mexico Mexico City, under the grant grant ∞ of SECITI/079/2017. with order H controller. ∞ SECITI/079/2017. ences, Technology and Innovation of Mexico City, under the grant with a full order H∞ controller. SECITI/079/2017.
SECITI/079/2017. 2405-8963 © Copyright © 2019. The Authors. Published by Elsevier Ltd. 284 All rights reserved. Copyright 2019 Copyright 2019 IFAC IFAC 284 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 284 Copyright © 2019 IFAC 284 10.1016/j.ifacol.2019.12.224 Copyright © 2019 IFAC 284
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The following result is related to the stability of the closedloop transfer function (4). This result is an extended version of the conditions given in (V´azquez-Rosas et al., 2017). Lemma 1. Consider the delayed system (2) and the proportional output feedback control (3), the associated closed-loop is stable if : m
τ<
n
1 1 1 + − , a bl i=1 ci
(5)
l=1
and Fig. 1. Proposed control strategy . (reduced-order) The remainder of the paper is organized as follows: In Section 2, we present the problem statement and some preliminary results on stabilization of time-delay systems with zero dynamics. Section 3 provides the proposed control strategy. The standard formulation of structured H∞ synthesis and H∞ control design procedure are given in Section 4. An example and numerical simulation are shown in Section 5. Finally, we provide some conclusions in Section 6. 2. PROBLEM STATEMENT Let us consider the observer based controller scheme shown in Fig. 1 and the class of a Single-Input Single-Output (SISO) systems with delay in the direct path, two unstable poles, m minimum phase zeros and n real stable poles given in the form:
G(s)e
−τ s
=
m
b) l=1 (s + nl e−τ s (s − a1 )(s − a2 ) i=1 (s + ci )
(1)
where a1 , a2 , m, n > 0,m < n + 2 and a1 ≥ a2 . G(s) is the delay-free transfer function and τ > 0 is the time-delay. The objective is to provide a control strategy (i.e to find parameters k1 , k2 , g1 and g2 ) in order to stabilize the closed-loop system shown in Fig. 1. 2.1 Preliminary Results This section presents a preliminary result used later to obtain the main results of the paper. Consider the highorder unstable system given by: m (s + bl ) Y (s) l=1 n = H(s) = e−τ s U (s) (s − a) i=1 (s + ci )
(2)
where a, m, τ > 0 and n 0. A proportional control strategy based on an output feedback of the form: U (s) = kp [R(s) − Y (s)]
(3)
yields the closed-loop transfer function: Y (s) kp H(s) = R(s) 1 + kp H(s)
m m (ω 2 + b2l ) bl l=1 l=1 n > ∀ω > 0 n 2 2 a i=1 cl (ω + a ) i=1 (ω 2 + c2i )
(6)
We will demonstrate that the Nyquist plot of the controller system encircles once the critical point in counter clockwise, guaranteeing the closed-loop stability according to the Nyquist stability criteria. An analysis in frequency domain is used. Proof. To obtain this encirclement, it is required to have two intersections with the real negative axis, one intersection should occur in (−∞, −1) and the second one should be appear in (−1, 0) . That is: MH (ωc1 ) > 1, φH (ωc1 ) = −π ,MH (ωc2 ) < 1, φH (ωc2 ) = −π, where MH (ω) denote the magnitude of H(ω) given by:
MH (ω) = kp
m
(ω 2 +
(ω 2 + b2l ) l=1 n a2 ) i=1 (ω 2 + c2i )
(7)
ωc1 , ωc2 are non-negative crossover frequencies at which the phase angle is −π and ωc1 < ωc2 , φH (ω) denote the phase of H(ω) given by: m n ω ω ω φH (ω) = −(π −tan−1 )+ tan−1 − tan−1 −ωτ a bl i=1 ci l=1 (8)
Let us assume that the conditions of Lemma 1 are satisfied. From (8) we can see that the phase trajectory begins at φH (0) = −π, therefore ωc1 = 0 . A growing of φH (ω) is H (ω) > 0. In this required since φH (0) = −π, this implies dφdω n m dφH (ω) 1 1 way, we evaluate: dω |ω→0 = a + l=1 bl − i=1 c1i − τ > 0. Thus, it is clear that under condition (5), the phase dφH (ω) > 0 for a ω > 0. dω The value of the magnitude MH (ω) with ω = 0 is m bl . Then, in order to archive the closedMH (0) = a l=1 n c i=1
l
loop stabilization, assuming condition (6) we assure that the MH (ω) is a decreasing function for ω ≈ 0 obtaining a correct direction of the Nyquist trajectory. 3. OBSERVER BASED-CONTROLLER
(4) 285
To handle the problem of stabilization of the time delay system (1), an observer based control strategy is proposed.
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The principal idea is to get a non-delayed estimation of the internal variables of the system to be used as control signals for the real process. Regardless of the order of the system, the control scheme involves only four tunable gains to obtain a stable behaviour of the closed-loop system. Theorem 1. Consider the observer based controller scheme shown in Fig. 1. There exist constants g1 , g2 , k1 and k2 such that the closed-loop system is stable if: m n 1 1 1 1 + − − τ< a1 bl i=1 ci β
(9)
l=1
and m m 2 b2l ) bl l=1 (ω + l=1 n > n a1 β i=1 cl (ω 2 + a21 )(ω 2 + β 2 ) i=1 (ω 2 + c2i ) (10) ∀ω > 0 where β is a positive real constant. Proof. A similar observer-based controller was analysed in (Novella-Rodr´ıguez et al., 2014), where it is demonstrated that the separation principle holds for the control scheme shown in Fig. 1. Then, the observer and the controller can be designed in an independent way. To simplify the analysis, we state stability conditions for the controller and the observer separately. Let us consider the controller scheme, identified from Fig. 1; the following result can be stated: Lemma 2. Consider the delayed system (1) and the control scheme shown in Fig. 1 considering the controller only. There exist constants k1 and k2 such that the closed-loop system is stable if : m
n
1 1 1 1 + − − τ< a2 bl i=1 ci β
(11)
l=1
and m m 2 b2 ) bl l=1 (ω + l=1 n nl > 2 2 2 2 a2 β i=1 cl (ω + a2 )(ω + β ) i=1 (ω 2 + c2i ) (12) ∀ω > 0 where β is a positive real constant.
Proof. Consider the delayed system (1) and the state feedback controller shown in Fig. 1, with a constant gain k1 > a1 . The closed loop transfer function of the system can be written as follows: m (s + bl )e−τ s Y (s) nl=1 m = R(s) (s + β)(s − a2 ) i=1 (s + ci ) + l=1 (s + bl )e−τ s (13)
with β = k1 − a1 . Note that β is a free parameter function of k1 , with β > 0, the system only has one unstable pole and n + 1 stable poles. If there exist β such that it satisfies the conditions (11) and (12) and taking account 286
Fig. 2. Closed-loop scheme with the weighting transfer functions. the Lemma 1, we can assure a stable behaviour of the closed-loop using the gain g2 as a P controller. Let us consider the static output injection scheme shown in Fig. 1, we can formulate the next result: Lemma 3. Consider the delayed system (1) and the observer scheme shown in Fig 1. There exist constants g1 and g2 such that the closed-loop system is stable if the conditions (9) and (10) are true. Proof. The proof can be easily derived from a dual procedure of the previous result. Reminding the stability conditions stated previously in Lemma 2 and Lemma 3 and considering a1 ≥ a2 , we can conclude that the stability conditions given for the observer are more restrictive than the stability conditions of the controller, namely the condition (9) and (10) holds, there exist gains k1 , k2 , g1 and g2 such that the closed-loop system shown in Fig. 1 is stable. 4. H∞ CONTROL DESIGN PROCEDURE 4.1 Background on H∞ control Mixed sensitivity optimization is a useful design tool that allows simultaneous design for performance and robustness. Fig. 2 shows the generalized plant for H∞ mixed sensitivity problem where G(s) is the open loop plant and K(s) is the controller that combines all tunable control elements. Each element is assumed to be linear time invariant, We , Wu , Wd are weights for specify the system performance, d is the disturbance input, u is the control input, y is the measured output, e1 and e2 are regulated outputs and r is the reference input. The transfer matrix from r and d is given by: e1 We S We SGWd r = d Wu KS Wu T Wd e2
(14)
where S = (1 + GK)−1 is the sensitivity function and T = KGS is the complementary sensitivity function. The main result of the H∞ standard problem is: for γ as small as possible, find a stabilizing controller K(s) such that: We S We SGWd Wu KS Wu T Wd < γ ∞
(15)
Our objective is to use the optimum H∞ controller theory in order to find an optimal fixed-structure controller, so to find the tunable gains shown in Fig. 1, as explained in (Apkarian and Noll, 2006).
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where ωb is the lowest allowable bandwidth of the system, chosen close to the minimum required bandwidth. Ms is the maximum allowable peak for the frequency response of the sensitivity function. The weighting function Wu should be a high-pass filter in order to guarantee the stability of the controlled system under diverse operating conditions. The weighting function Wu is defined as follows: ω
Fig. 3. Closed-loop with the weighting transfer function of the alternative observer scheme representation. 4.2 Proposed design procedure Let us point out that the controller K(s) cannot be analytically designed if the time delay involved in the process is treated strictly (Zhang, 1998). Consider a Pad´e approximation in the form: e−τ s ≈ Gp (s) =
1 − p1 s + p2 s2 + ... + (−1)q pq sq 1 + p1 s + p2 s2 + ... + pq sq
(16)
where q is the approximation order to be chosen. The coefficients pi depend on q and τ and are determined from a Taylor series expansion of the transcendental function. When an approximation of the time delay is introduced, the plant (1) becomes: G(s)e−τ s ≈ G(s)Gp (s)
(17)
With the rational transfer function (17), a controller using H∞ theory may now be calculated. Augmented plant structure. The Fig. 3 show an alternative configuration of the observer scheme where GO represents a subsystem that contains the next elements of the system (17): transfer function of mthe stable poles and minimum (s+bl ) phase zeros given by: nl=1 (s+c ) , the transfer function i=1
Wu (s) =
1 s + Mhu εu s + ωuh
(19)
where εu > 0 must be chosen as a small constant in order to ensure good rejection of measurement errors. The pulsation ωh limits the bandwidth and must be chosen sufficiently far from the desired grid frequency for closed loop control. Mu represent the effort of the controller. The principal effect of the weighting function Wd is to ensure the rejection of disturbances, it is chosen here as a constant function. Controller using H∞ structured design. The controller is synthesized by H∞ structured design method to get a feasible solution until the cost function γ is minimized. To apply the H∞ structured design, in addition to provide the augmented plant and the required structure of the controller, it is necessary to provide an initial controller as a starting point. The initial controllers values can be selected using the gains where the system (1) is stable using the control strategy shown in Fig. 1. The following procedure is proposed to obtain the gains g1 and g2 such that the closed-loop of the observer scheme is stable. The value of the proportional gain g1 can be stated as follows. Consider that of the Theorem 1 are mthe conditions n true and τ = a12 + l=1 b1l − i=1 c1i − β1 . Thus exist g1 1 such that β > g1 −a > 0. We can obtain: 2
i
g1 >
m
1 n − i=1
+ a2
(20)
of the Pad´e-aproximation is Gp , the transfer function of 1 the unstable pole a1 is given by: (s−a . The subsystem 1) KO contains the tunable gains g1 , g2 and the transfer 1 . A similar function of the unstable pole a2 given by (s−a 2) configuration of the controller scheme can be easily derived from a dual procedure of the previous structure.
We can select a g1 :
The representation shown in Fig. 3 keeps the same dynamic behaviour of the original proposal shown in Fig. 1 and represent a one way to build the augmented system (nominal system + weighting functions).
with εg1 being a positive real constant such that (5) and (6) holds.
Weighting functions selection. The weighting functions We , Wu , Wd are selected taking into account the basic requirement of mixed-sensitivity design (Lundstrom et al., 1991). The weighting function We represents the performance objective of the error sensitivity function S(s), it should work as a lowpass-filter in order to reduce the error sensitivity in the low frequency range for output disturbance rejection. We is defined in the next form (Skogestad and Postlethwaite, 2005): 1 s + ω b Ms We (s) = Ms s + ω b ε
(18) 287
g1 =
1 a1
1 a1
+
+
1 l=1 bl
m
1 l=1 bl
1 n − i=1
1 ci
1 ci
−τ
−τ
+ a2 + ε g 1
Now, from Lemma 1, there exist g2 such that the behaviour of the subsystem (13) is stable. We can find the values of g2 as follows: m
+ ωc2 ) n > g2 > (a1 + ωc2 )(g1 − a2 + ωc2 ) i=1 (ci + ωc2 ) m ) l=1 (b l n a1 (g1 − a2 ) i=2 (ci ) l=1 (bl
It is important to headline that if g1 is selected near to the bound stated in Ec (20), then the margin for the gain g2 will be reduced.
2019 IFAC TDS 166 Sinaia, Romania, September 9-11, 2019Carlos Daniel Vázquez et al. / IFAC PapersOnLine 52-18 (2019) 162–167
k1 >
1 a2
m
+ m
1 l=1 bl
1 n − i=1
1 ci
−τ
+ a1
Sensitivity function S 10
0
-10
-20
Singular Values (dB)
The indicated values ensure a stable behaviour of the closed-loop system and represent a stable start point of the initial controller in order to find the optimal values of the tunable gains. The values for k1 and k2 can be stated as follows:
-30
-40
-50
-60
Sensitivity function full order control Sensitivity function proposed method Template
-70
2 c) l=1 (bl + ω > k2 > n (a2 + ωc2 )(k1 − a1 + ωc2 ) i=1 (ci + ωc2 ) m ) l=1 (b l n a2 (k1 − a1 ) i=2 (ci )
-80 10 -5
10 -4
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
Frequency (rad/s)
Fig. 4. Frequency response : Sensitivity function. Comparison between different calculated controllers Controller*Sensitivity KS
5. EXAMPLE
20
10
0
Singular Values (dB)
The proposed method is applied to a linearizing model of CSTR with zero dynamics by numerical simulation. We illustrate the design of the controller through the following example.
Controller sensitivity full order control Controller sensitivity proposed method Template
-10
-20
-30
Let us consider the example proposed in (Novella-Rodr´ıguez et al., 2014) . In this example, the flow rate is the manipulated variable and the temperature of the CSTR is the controlled variable. The linearization of the equation assuming a measurement delay of 54 min and taking account a one hour as the time scale of the model gives the transfer Fig. 5. Frequency response : Controller sensitivity function. Comparison between different calculated confunction model as: trollers proposed control strategy and a classical H∞ full order −4.7475(s + 3.129) e−0.9s (21) control design method. The simulation scheme is the one G(s) = (s − 0.5545)(s − 0.09395)(s + 6.394) given in Fig. 1. -40
-50
-60 10 -5
10 -4
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
Frequency (rad/s)
The stability conditions given in Theorem 1 taking into account β = 1 are satisfied, due to:
Fig. 4 shows the shape of the sensitivity function S , which also corresponds to the closed-loop transfer function and Fig. 5 shows the shape of the controller sensitivity. This indicates that the selected weighting functions for controller design using a H∞ structured design have achieved the goal of command tracking.
1 1 1 1 0.9 < − + − 0.5545 6.394 3.129 1 and 3.129 > (0.5545)(1)(6.394)
(ω 2
5.1 Frequency response analysis
5.2 Time response analysis
(ω 2 + 3.1292 ) ∀ω + 0.55452 )(ω 2 + 1)(ω 2 + 6.3942 )
therefore there exists an observer based structure as shown in the Fig. 1 with proportional gains g1 , g2 , k1 and k2 such that the resulting closed-loop system is stable. The weighting function We (s) is selected with the parameters: Ms = 1.2, ωb = 0.1, ε = 0.0103. The weighting function Wu (s) is calculated with the parameters: Mu = 10, ωh = 10, ε = 0.0001. And Wd (s) = 0.001. Following the methodology shown in section 4 and using a Pad´e-approximation with order q = 9, the calculated gains for the example 1 are shown in Table 1. The optimal H∞ performance level γ is γ = 2.54. Additionally, we compute a full-order H∞ controller using the same weighting functions and the same order of Pad´eapproximation in order to make a comparison between the 288
The step response of the closed-loop system is shown in Fig. 6. Blue lines indicates the simulation of the closed-loop system using the scheme given in Fig. 1 as a control strategy. The black line indicates the full order H∞ controller, which is an eighth order controller. The results show that the methodology used in Section 4 with the reduced-order observer based controller is significantly faster and provides a smaller overshoot than the full order H∞ controller. The error introduced by the rational approximation will not cause instability in the proposed method. The simulations are carried out using c Matlab /Simulink. Table 1. calculated gains for the example 1 Gains g1 k1 g2 k2
Values 816.7051 101.6702 −544.9466 −64.8597
2019 IFAC TDS Sinaia, Romania, September 9-11, 2019Carlos Daniel Vázquez et al. / IFAC PapersOnLine 52-18 (2019) 162–167
Step Response
3 Nominal delay full order controller Nominal delay proposed method Uncertain delay proposed method
2.5 2
Output y(t)
1.5 1
0.5 0 -0.5 0
10
20
30
40
50
Time (seconds)
60
70
80
90
100
Fig. 6. Closed-loop behavior of the example 1 with a delay uncertainty of 5 percent Due to the use of a delayed observer in this work, the uncertainties in the time-delay of the plant may cause oscillations or even instability. Fig. 5 also shows the effect of delay mismatches between the plant nominal delay and the observer delay. The dashed red line shows the response of the system when the time-varying uncertain delay is introduced into the process. The time-delay in the plant can be considered as τ (t) = 0.9 + δ(t), where δ(t) is a pseudo-random binary signal scaled to a magnitude of 0.05. 6. CONCLUSION The paper described the control design of an observer based controller scheme that guarantees the stabilization of delayed systems with two unstable poles, n stable poles and m minimum phase zeros using only 4 tunable gains. The conditions that guarantee the stability of the closedloop system are derived in terms of the upper limit of time delay size and the poles and zeros position. Based on the H∞ control theory the controller gains are tuned using a non-smooth H∞ optimization method. Some uncertainty in the knowledge of the delay value has been considered in the numerical simulations which demonstrates the effectiveness of the method. The test demonstrates that the reduced-order observer based controller is faster and presents a minor overshoot when compared to the full order controller for the example used. REFERENCES Apkarian, P. and Noll, D. (2006). Nonsmooth optimization for multidisk h synthesis. European Journal of Control, 12(3), 229 – 244. Bequette, B.W. (2003). Process Control. Modeling, Design and Simulation. Prentice Hall International. Blakelock, J. (1991). Automatic control of aircraft and missiles. Wiley-interscience publicationl. Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989). State-space solutions to standard h2 and h∞ control problems. IEEE Transactions on Automatic Control, 34, 831–847. Kwak, H.J., Sung, S.W., and Lee, I.B. (2000). Stabilizability conditions and controller design for unstable processes. Chemical Engineering Research and Design, 78(4), 549 – 556. Lee, S.C. and Wang, Q.G. (2010). Stabilization conditions for a class of unstable delay processes of higher order. 289
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Journal of the Taiwan Institute of Chemical Engineers, 41(4), 440 – 445. Lundstrom, P., Skogestad, S., and Wang, Z.Q. (1991). Performance weight selection for h-infinity and -control methods. Transactions of the Institute of Measurement and Control, 13, 241–252. Novella-Rodr´ıguez, D., del Muro-Cu´ellar, B., and Sename, O. (2014). Observer based scheme for the control of high order systems with two unstable poles plus time delay. Asia-Pacific Journal of Chemical Engineering, 9. Skogestad, S. and Postlethwaite, I. (2005). Multivariable feedback control: Analysis and Design. John Wiley. V´azquez-Rosas, C., del Muro-Cu´ellar, B., and M´arquez Rubio, J. (2017). Stability conditions for unstable high order systems with minimum-phase zeros plus time delay. Wang, Q.G., Lee, T.H., and k.Tan, K. (1999). FiniteSpectrum Assignment for Time-Delay Systems. Springer-Verlag, London. Wu, L., Lam, H.K., Zhao, Y., and Shu, Z. (2015). Timedelay systems and their applications in engineering 2014. Mathematical Problems in Engineering, 2015, 1–3. Zhang, W.D. (1998). Analytical design methods for process control. In Postdoctoral research report. Shanghai Jiaotong University.
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IFAC PapersOnLine 52-18 (2019) 162–167
Structured observer-based controller for Structured Structured observer-based observer-based controller controller for for Structured observer-based controller for delayed systems with two unstable poles Structured observer-based controller for delayed systems with two unstable poles delayed systems with two unstable poles delayed systems with two unstable poles and minimum zeros. delayed systems withphase two unstable poles and minimum phase zeros. and minimum phase zeros. and minimum phase zeros. and aminimum phase ∗∗ zeros. ∗ Carlos Daniel V´ zquez ∗∗∗ David Novella ∗∗ ∗∗ Basilio del Muro ∗ ∗
Carlos Daniel V´ a ∗ David Novella ∗∗ Basilio del Muro∗∗ ∗∗zquez ∗ David ∗∗ ∗∗ Basilio M´ Carlos Daniel V´ a zquez Novella delarquez Muro∗∗∗ ∗∗zquez ∗∗ Olivier Sename Luc Dugard Juan Francisco ∗∗ ∗∗ Carlos Daniel V´ a Novella del Muro Olivier Sename Luc Dugard Juan Francisco a rquez ∗ David ∗∗ ∗∗ Basilio M´ ∗∗ Carlos Daniel V´ a David Novella Basilio del Muro∗∗∗ ∗∗zquez ∗∗ Olivier Sename Luc Dugard Juan Francisco M´ a rquez Olivier Sename ∗∗ Luc Dugard ∗∗ Juan Francisco M´ arquez ∗ Olivier Sename Dugard Juan Francisco M´ arquez ∗ ∗ Escuela Superior Luc de Ingenier´ ıa Mec´ a nica yy El´ eectrica, Unidad ∗ ıa Mec´ a nica El´ ctrica, Unidad ∗ Escuela Superior de Ingenier´ ∗ Escuela Instituto Superior de ıa Mec´ a eectrica, Unidad Culhuacan, eecnico Nacional,Santa Ana 1000, M´ Superior Polit´ de Ingenier´ Ingenier´ Mec´ anica nica yy El´ El´ ctrica, Unidad Culhuacan, Polit´ cnico ıa Nacional,Santa Ana 1000, M´eexico xico ∗ Escuela Instituto Escuela Superior de Ingenier´ ıa Mec´ a nica y El´ e ctrica, Unidad Culhuacan, Instituto Polit´ e cnico Nacional,Santa Ana 1000, D.F., 04430, M´ e xico (e-mail: carlos daniel [email protected])(e-mail: Culhuacan, Instituto Polit´ecnico Nacional,Santa Ana 1000, M´ M´eexico xico daniel [email protected])(e-mail: D.F., 04430, M´ e xico (e-mail: carlos Culhuacan, Polit´ecnico Nacional,Santa Ana 1000, M´exico D.F., 04430, M´ carlos daniel [email protected])(e-mail: [email protected])(e-mail: [email protected]). D.F., 04430, Instituto M´eexico xico (e-mail: (e-mail: carlos daniel [email protected])(e-mail: [email protected])(e-mail: [email protected]). ∗∗ D.F., 04430, M´exicoAlpes, (e-mail: carlosGrenoble daniel [email protected])(e-mail: [email protected])(e-mail: [email protected]). ∗∗ Univ. Grenoble CNRS, INP, GIPSA-lab, 38000 ∗∗ [email protected])(e-mail: [email protected]). Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 ∗∗ Univ. [email protected])(e-mail: [email protected]). ∗∗ Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France (e-mail: [email protected])(e-mail: Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble, France (e-mail: [email protected])(e-mail: ∗∗ Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, Grenoble, France (e-mail: [email protected])(e-mail: [email protected])(e-mail: Grenoble, France (e-mail: [email protected])(e-mail:38000 [email protected])(e-mail: Grenoble, France (e-mail: [email protected])(e-mail: [email protected])(e-mail: [email protected]) [email protected])(e-mail: [email protected]) [email protected])(e-mail: [email protected]) [email protected]) [email protected]) Abstract: This paper analyses the problem control stabilization of aa particular class Abstract: This paper analyses the problem of of control and and stabilization of particular class of of Abstract: This paper analyses the of and stabilization of particular of Linear Invariant systems. The under consideration two unstableclass poles, Abstract: paper (LTI) analyses the problem problem of control control and stabilizationhas of aatwo particular class of Linear Time TimeThis Invariant (LTI) systems. The system system under consideration has unstable poles, Abstract: This paper analyses the problem of control and stabilization of a particular class of Linear Time Invariant (LTI) systems. The system under consideration has two unstable poles, n real stable poles, m minimum phase zeros plus time delay. An observer based controller Linear systems. Thezeros system has twobased unstable poles, n real Time stableInvariant poles, m(LTI) minimum phase plusunder time consideration delay. An observer controller Linear Time Invariant systems. The system under has two unstable poles, n real realfour stable poles, m(LTI) minimum phase zeros plus time consideration delay. An observer based controller with tunable gains is proposed as a control strategy in order to ensure a stable behaviour n stable poles, m minimum phase zeros plus time delay. An observer based controller with four tunable gains is proposedphase as aa control strategy in order to ensure aa stable behaviour n real stable poles, m minimum zeros plus timeexistence delay. observer basedscheme controller with four tunable gains is as strategy in order to ensure behaviour of the closed-loop system. Sufficient for the of the proposed are with four tunable gains is proposed proposed asconditions a control control strategy in orderAn to ensure a stable stable behaviour of the closed-loop system. Sufficient conditions for the existence of the proposed scheme are with four tunable gains is proposed as a control strategy in order to ensure a stable behaviour of the closed-loop system. Sufficient conditions for the existence of the proposed scheme are obtained in terms of the upper limit of time delay size and the poles and zeros position. The of the closed-loop system. Sufficient for size the and existence of the scheme The are obtained in terms of the upper limit conditions of time delay the poles andproposed zeros position. of the closed-loop system. Sufficient conditions for the existence of the proposed scheme are obtained in terms of the upper limit of time delay size and the poles and zeros position. The optimization method. The proposed controller parameters are tuned using a non-smooth H ∞ obtained inparameters terms of the upper limit ofa time delay size and the poles and zeros position. The non-smooth optimization method. The proposed controller are tuned using H ∞ obtained inparameters terms of the limit delay model size and the poles and zeros The position. The optimization method. The proposed controller parameters are upper tuned usingofaa time non-smooth H∞ ∞ control strategy is applied to an unstable linearized of a continuously stirred tank reactor optimization method. proposed controller are tuned using non-smooth H ∞ control strategy is applied to an unstable linearized model aa continuously stirred tank reactor optimization method. proposed controller parameters are the tuned using a non-smooth H∞ of control strategy is applied to unstable linearized model of stirred tank reactor (CSTR) in order show effectiveness of the proposed design scheme. Numerical results are control strategy isto applied to an an unstable linearized model of a continuously continuously stirredThe tank reactor (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are control strategy is applied to an unstable linearized model of a continuously stirred tank reactor (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are presented. (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are presented. (CSTR) in order to show the effectiveness of the proposed design scheme. Numerical results are presented. presented. Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. presented. Keywords: Keywords: Stability Stability and and stabilization, stabilization, Output Output feedback feedback control control under under I/O-delays, I/O-delays, Control Control Keywords: Stability design. Keywords: Stability and and stabilization, stabilization, Output Output feedback feedback control control under under I/O-delays, I/O-delays, Control Control design. Keywords: Stability and stabilization, Output feedback control under I/O-delays, Control design. design. design. 1. INTRODUCTION INTRODUCTION analyse aa high order system simple It 1. analyse order system by by simple controllers. controllers. It is is 1. analyse aa high high order by controllers. is important to highlight highlight that the the mentioned works It only 1. INTRODUCTION INTRODUCTION analyse high order system system by simple simple controllers. It is important to that mentioned works only 1. INTRODUCTION analyse a high order system by simple controllers. It is important to highlight that the mentioned works only consider the problem when only one minimum phase zero One way to define time-delay in the context of control important to highlight that the mentioned works only consider the problem when only one minimum phase zero One way to define time-delay in the context of control important toproblem highlight that theone mentioned works only consider the problem when only one minimum phase zero One way to define time-delay in the context of control and one unstable pole are involved in the original plant. system theory is the time interval from the application of consider the when only minimum phase zero One way to define in from the context of control system theory is thetime-delay time interval the application of and one unstable pole are involved the original plant. consider the problem when only onein minimum phase zero one unstable pole are involved in the original plant. One way to define time-delay in from the context of process control system theory is the time interval the application of control signal to any observable change in the and one cases, unstable pole are necessary involved into the original plant. theory is to theany time interval from the in application of and aasystem control signal observable change the process In real it is also guarantee typical and one unstable pole are involved in the original plant. In real cases, it is also necessary to guarantee typical system theory is the time interval from the application of avariable control signal to any observable change in the process (Wang et al., 1999). Such aachange phenomenon can be be design a control(Wang signal et to al., any1999). observable in the process In real real requirements cases, it it is is also also necessary to guarantee typical variable Such phenomenon can such as speed of response, control In cases, necessary to guarantee typical a control signal et to any1999). observable the process variable al., Such phenomenon can be such as speed of response, control found in (Wang several engineering processes, for in example: mavariable (Wang et al., 1999). Such aachange phenomenon canmabe design In real requirements cases, it is also necessary to guarantee typical design requirements such as speed of response, control found in several engineering processes, for example: bandwidth, disturbance rejection, and robustness unmoddesign requirements such as speed of response, control variable etengineering al., 1999). Such a biological phenomenon canmabe bandwidth, found in several processes, for disturbance rejection, and robustness unmodterial and information transmission, embedded found and in (Wang several engineering processes, for example: example: madesign requirements such as speed of response, control bandwidth, disturbance rejection, and robustness unmodterial information transmission, biological embedded eled dynamics. To meet these requirements, H theory is disturbance rejection, and robustness unmodeled dynamics. To meet these requirements, H∞ is found in biochemical several engineering processes, for example: terial information transmission, biological embedded ∞ theory systems, and chemical processes (Wu et maal., abandwidth, ∞ terial and and information transmission, biological embedded bandwidth, disturbance rejection, and robustness unmodeled dynamics. To meet these requirements, H theory is systems, biochemical and chemical processes (Wu et al., ∞ powerful technique used to design robust controllers for eled dynamics. To meet these requirements, H theory is ∞ terial and information transmission, biological embedded systems, biochemical and chemical processes (Wu et al., a powerful technique used to design robust controllers for 2015). systems, biochemical and chemical processes (Wu et al., alinear dynamics. To meet these requirements, H∞variations, theoryfor is aeled powerful technique used to design design robust controllers controllers for 2015). systems under uncertainties, parameter powerful technique used to robust systems, biochemical and chemical processes (Wu et al., 2015). linear systems under uncertainties, parameter variations, 2015). a powerful technique used to design robust controllers for linear systems under uncertainties, parameter variations, and disturbances. There exist different different systems systems with with two two unstable unstable poles poles and and linear systems under uncertainties, parameter variations, 2015). exist There and disturbances. linear systems under uncertainties, parameter variations, and disturbances. There exist different systems with two unstable poles and minimum phase zeros with time-delay. In chemical indusand disturbances. There exist different systems with two unstable poles and minimum phase zeros with time-delay. In chemical indus- The objective of this paper is two-fold: First we propose and disturbances. There exist different systems with two unstable poles and minimum phase zeros with time-delay. In chemical indusThe objective of this paper is two-fold: First we propose try, common example is the the CSTR (Bequette, (Bequette, 2003). In The minimum phaseexample zeros with time-delay. In chemical indusobjective of this paper is two-fold: First we propose try, aa common is CSTR 2003). In sufficient conditions in order to stabilize a specific class The objective of this paper is two-fold: First we propose minimum phase zeros with time-delay. In chemical industry, a common example is the CSTR (Bequette, 2003). In sufficient conditions in order to stabilize a specific class the military field, the dynamical model of a ballistic missile try, military a common example is the CSTR 2003). In The objective of this paper is two-fold: First we propose sufficient conditions in order to stabilize a specific class the field, the dynamical model(Bequette, of a ballistic missile of delayed systems with two real unstable poles, n stable sufficient conditions in order to stabilize a specific class of delayed systems with two real unstable poles, n stable try, military asimilar common example is the CSTR (Bequette, 2003). In sufficient the field, the dynamical model of aa ballistic missile has characteristics (Blakelock, 1991). Then, the the military field, the dynamical model of ballistic missile conditions in order to stabilize a specific class of delayed systems with two real unstable poles, n stable has similar characteristics (Blakelock, 1991). Then, the poles and m minimum phase zeros (m < n + 2), using of delayed systems with two real unstable poles, n stable the military field, the dynamical model of a ballistic missile has similar characteristics (Blakelock, 1991). Then, the poles and m minimum phase zeros (m < n + 2), using problem of stabilization stabilization of this this class of of 1991). systemsThen, becomes has similar characteristics (Blakelock, the poles of delayed systems with phase two controller real unstable poles, ntunable stable and m minimum zeros (m < n + 2), using problem of of class systems becomes a structured observer-based with four poles and m minimum phase zeros (m < n + 2), using has similar characteristics (Blakelock, the problem of this class of systems becomes aa structured controller tunable an interesting topic where whereof explicit stabilizability results for poles problem of stabilization stabilization ofexplicit this class of 1991). systemsThen, becomes and maobserver-based minimum phase zeros (mwith < of nfour + 2), using structured observer-based controller with four tunable an interesting topic stabilizability results for Then systematic design procedure the control agains. structured observer-based controller with four tunable problem of stabilization of this class of systems becomes an interesting topic where explicit stabilizability results for gains. Then a systematic design procedure of the control many unstable processes are not available yet. an interesting whereare explicit stabilizability agains. structured observer-based controller with four tunable Then a systematic design procedure of the control many unstabletopic processes not available yet. results for gain is proposed in the H framework. Since the controller gains.is Then a systematic procedure control proposed in the H∞ framework. Sinceof thethe controller an interesting topic whereare explicit stabilizability for gain many unstable processes not available yet. ∞design ∞ many unstable available yet. results a the systematic procedure of the control gain is proposed in the framework. Since the controller ∞design is structured, smooth H proposed in Few works haveprocesses addressedare thenot problem of delayed delayed unstable gains. ∞ synthesis gain is Then proposed in non the H H Since the controller ∞ framework. manyworks unstable processes are not available yet. unstable Few have addressed the problem of is structured, the non smooth H synthesis proposed in ∞ ∞ synthesis gain is proposed in non the H framework. Since the controller is structured, the smooth H proposed in ∞has Few works have addressed the problem of delayed unstable ∞ (Apkarian and Noll, 2006) been considered, which is an systems with zero dynamics. In (Kwak et al., 2000), the is structured, the non smooth H synthesis proposed in Few works have addressed the In problem delayed unstable ∞ considered, which is an systems with zero dynamics. (Kwakofet al., 2000), the (Apkarian and Noll, 2006) has been is structured, the non smooth H synthesis proposed in (Apkarian and Noll, 2006) has been considered, which is an Few works have addressed the problem of delayed unstable ∞ systems with zero dynamics. In (Kwak et al., 2000), the central design method (Doyle et al., extension of the H authors face the problem of stabilization for first and ∞ (Apkarian and Noll, 2006) has been considered, which is an systems with zero dynamics. In (Kwak et al., 2000), the central design method (Doyle et al., extension of the H authors face the problem of stabilization for first and (Apkarian ∞ ∞ and Noll, 2006) has been considered, which is an central design method (Doyle et al., extension of the H systems with zero dynamics. In (Kwak et al., 2000), the authors face the problem of stabilization for first and ∞ 1989). One of the main advantages of H structured desecond order systems using ofaa P P controller. for An extended extended design of method (Doyle et deal., authors order face systems the problem stabilization first and extension ∞ central 1989). Oneofofthe theHmain advantages H∞ second using controller. An ∞ structured ∞ designand method (Doyle et al., the H 1989). One the main advantages of H structured deauthors face systems the problem ofaa P stabilization first and extension ∞ central second using controller. An sign method to choose the order the of the result isorder given in (Lee (Lee and Wang, 2010), wherefor theextended authors 1989). Oneofof ofis the main advantages of H∞ structured desecondis order systems using P controller. An extended ∞structure result given in and Wang, 2010), where the authors sign method is to choose the order and the structure of the 1989). One ofis the main advantages of the H∞is structured design method to choose the order and structure of the second order systems using a P controller. An extended result is given in (Lee and Wang, 2010), where the authors desired controller. An illustrative example considered sign method is to choose the order and the structure of the result is given in (Lee and Wang, 2010), where the authors desired controller. An illustrative example is considered to to sign method isinterest to choose the order and the is structure of the desired controller. An illustrative example considered to This work was the of Sciresult given (Lee and by Wang, 2010), where the authors highlight the this work, in particular compared desired controller. An of illustrative example is considered to This is work wasinsupported supported by the Secretary Secretary of Education, Education, Scihighlight the interest of this work, in particular compared This work was supported by the Secretary of Education, Sciences, Technology and Innovation of Mexico City, under the grant desired controller. An illustrative example is considered to highlight the interest of this work, in particular compared with a full order H controller. ThisTechnology work was supported by the Secretary of under Education, Sci∞ of highlight the interest this work, in particular compared ences, and Innovation of Mexico City, the grant with a full order H controller. ∞ ences, and of City, the SECITI/079/2017. ThisTechnology work was supported by the Secretary of under Education, Scihighlight interest this work, in particular compared with aa full fullthe order H∞ controller. ences, Technology and Innovation Innovation of Mexico Mexico City, under the grant grant ∞ of SECITI/079/2017. with order H controller. ∞ SECITI/079/2017. ences, Technology and Innovation of Mexico City, under the grant with a full order H∞ controller. SECITI/079/2017.
SECITI/079/2017. 2405-8963 © Copyright © 2019. The Authors. Published by Elsevier Ltd. 284 All rights reserved. Copyright 2019 Copyright 2019 IFAC IFAC 284 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 284 Copyright © 2019 IFAC 284 10.1016/j.ifacol.2019.12.224 Copyright © 2019 IFAC 284
2019 IFAC TDS Sinaia, Romania, September 9-11, 2019Carlos Daniel Vázquez et al. / IFAC PapersOnLine 52-18 (2019) 162–167
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The following result is related to the stability of the closedloop transfer function (4). This result is an extended version of the conditions given in (V´azquez-Rosas et al., 2017). Lemma 1. Consider the delayed system (2) and the proportional output feedback control (3), the associated closed-loop is stable if : m
τ<
n
1 1 1 + − , a bl i=1 ci
(5)
l=1
and Fig. 1. Proposed control strategy . (reduced-order) The remainder of the paper is organized as follows: In Section 2, we present the problem statement and some preliminary results on stabilization of time-delay systems with zero dynamics. Section 3 provides the proposed control strategy. The standard formulation of structured H∞ synthesis and H∞ control design procedure are given in Section 4. An example and numerical simulation are shown in Section 5. Finally, we provide some conclusions in Section 6. 2. PROBLEM STATEMENT Let us consider the observer based controller scheme shown in Fig. 1 and the class of a Single-Input Single-Output (SISO) systems with delay in the direct path, two unstable poles, m minimum phase zeros and n real stable poles given in the form:
G(s)e
−τ s
=
m
b) l=1 (s + nl e−τ s (s − a1 )(s − a2 ) i=1 (s + ci )
(1)
where a1 , a2 , m, n > 0,m < n + 2 and a1 ≥ a2 . G(s) is the delay-free transfer function and τ > 0 is the time-delay. The objective is to provide a control strategy (i.e to find parameters k1 , k2 , g1 and g2 ) in order to stabilize the closed-loop system shown in Fig. 1. 2.1 Preliminary Results This section presents a preliminary result used later to obtain the main results of the paper. Consider the highorder unstable system given by: m (s + bl ) Y (s) l=1 n = H(s) = e−τ s U (s) (s − a) i=1 (s + ci )
(2)
where a, m, τ > 0 and n 0. A proportional control strategy based on an output feedback of the form: U (s) = kp [R(s) − Y (s)]
(3)
yields the closed-loop transfer function: Y (s) kp H(s) = R(s) 1 + kp H(s)
m m (ω 2 + b2l ) bl l=1 l=1 n > ∀ω > 0 n 2 2 a i=1 cl (ω + a ) i=1 (ω 2 + c2i )
(6)
We will demonstrate that the Nyquist plot of the controller system encircles once the critical point in counter clockwise, guaranteeing the closed-loop stability according to the Nyquist stability criteria. An analysis in frequency domain is used. Proof. To obtain this encirclement, it is required to have two intersections with the real negative axis, one intersection should occur in (−∞, −1) and the second one should be appear in (−1, 0) . That is: MH (ωc1 ) > 1, φH (ωc1 ) = −π ,MH (ωc2 ) < 1, φH (ωc2 ) = −π, where MH (ω) denote the magnitude of H(ω) given by:
MH (ω) = kp
m
(ω 2 +
(ω 2 + b2l ) l=1 n a2 ) i=1 (ω 2 + c2i )
(7)
ωc1 , ωc2 are non-negative crossover frequencies at which the phase angle is −π and ωc1 < ωc2 , φH (ω) denote the phase of H(ω) given by: m n ω ω ω φH (ω) = −(π −tan−1 )+ tan−1 − tan−1 −ωτ a bl i=1 ci l=1 (8)
Let us assume that the conditions of Lemma 1 are satisfied. From (8) we can see that the phase trajectory begins at φH (0) = −π, therefore ωc1 = 0 . A growing of φH (ω) is H (ω) > 0. In this required since φH (0) = −π, this implies dφdω n m dφH (ω) 1 1 way, we evaluate: dω |ω→0 = a + l=1 bl − i=1 c1i − τ > 0. Thus, it is clear that under condition (5), the phase dφH (ω) > 0 for a ω > 0. dω The value of the magnitude MH (ω) with ω = 0 is m bl . Then, in order to archive the closedMH (0) = a l=1 n c i=1
l
loop stabilization, assuming condition (6) we assure that the MH (ω) is a decreasing function for ω ≈ 0 obtaining a correct direction of the Nyquist trajectory. 3. OBSERVER BASED-CONTROLLER
(4) 285
To handle the problem of stabilization of the time delay system (1), an observer based control strategy is proposed.
2019 IFAC TDS 164 Sinaia, Romania, September 9-11, 2019Carlos Daniel Vázquez et al. / IFAC PapersOnLine 52-18 (2019) 162–167
The principal idea is to get a non-delayed estimation of the internal variables of the system to be used as control signals for the real process. Regardless of the order of the system, the control scheme involves only four tunable gains to obtain a stable behaviour of the closed-loop system. Theorem 1. Consider the observer based controller scheme shown in Fig. 1. There exist constants g1 , g2 , k1 and k2 such that the closed-loop system is stable if: m n 1 1 1 1 + − − τ< a1 bl i=1 ci β
(9)
l=1
and m m 2 b2l ) bl l=1 (ω + l=1 n > n a1 β i=1 cl (ω 2 + a21 )(ω 2 + β 2 ) i=1 (ω 2 + c2i ) (10) ∀ω > 0 where β is a positive real constant. Proof. A similar observer-based controller was analysed in (Novella-Rodr´ıguez et al., 2014), where it is demonstrated that the separation principle holds for the control scheme shown in Fig. 1. Then, the observer and the controller can be designed in an independent way. To simplify the analysis, we state stability conditions for the controller and the observer separately. Let us consider the controller scheme, identified from Fig. 1; the following result can be stated: Lemma 2. Consider the delayed system (1) and the control scheme shown in Fig. 1 considering the controller only. There exist constants k1 and k2 such that the closed-loop system is stable if : m
n
1 1 1 1 + − − τ< a2 bl i=1 ci β
(11)
l=1
and m m 2 b2 ) bl l=1 (ω + l=1 n nl > 2 2 2 2 a2 β i=1 cl (ω + a2 )(ω + β ) i=1 (ω 2 + c2i ) (12) ∀ω > 0 where β is a positive real constant.
Proof. Consider the delayed system (1) and the state feedback controller shown in Fig. 1, with a constant gain k1 > a1 . The closed loop transfer function of the system can be written as follows: m (s + bl )e−τ s Y (s) nl=1 m = R(s) (s + β)(s − a2 ) i=1 (s + ci ) + l=1 (s + bl )e−τ s (13)
with β = k1 − a1 . Note that β is a free parameter function of k1 , with β > 0, the system only has one unstable pole and n + 1 stable poles. If there exist β such that it satisfies the conditions (11) and (12) and taking account 286
Fig. 2. Closed-loop scheme with the weighting transfer functions. the Lemma 1, we can assure a stable behaviour of the closed-loop using the gain g2 as a P controller. Let us consider the static output injection scheme shown in Fig. 1, we can formulate the next result: Lemma 3. Consider the delayed system (1) and the observer scheme shown in Fig 1. There exist constants g1 and g2 such that the closed-loop system is stable if the conditions (9) and (10) are true. Proof. The proof can be easily derived from a dual procedure of the previous result. Reminding the stability conditions stated previously in Lemma 2 and Lemma 3 and considering a1 ≥ a2 , we can conclude that the stability conditions given for the observer are more restrictive than the stability conditions of the controller, namely the condition (9) and (10) holds, there exist gains k1 , k2 , g1 and g2 such that the closed-loop system shown in Fig. 1 is stable. 4. H∞ CONTROL DESIGN PROCEDURE 4.1 Background on H∞ control Mixed sensitivity optimization is a useful design tool that allows simultaneous design for performance and robustness. Fig. 2 shows the generalized plant for H∞ mixed sensitivity problem where G(s) is the open loop plant and K(s) is the controller that combines all tunable control elements. Each element is assumed to be linear time invariant, We , Wu , Wd are weights for specify the system performance, d is the disturbance input, u is the control input, y is the measured output, e1 and e2 are regulated outputs and r is the reference input. The transfer matrix from r and d is given by: e1 We S We SGWd r = d Wu KS Wu T Wd e2
(14)
where S = (1 + GK)−1 is the sensitivity function and T = KGS is the complementary sensitivity function. The main result of the H∞ standard problem is: for γ as small as possible, find a stabilizing controller K(s) such that: We S We SGWd Wu KS Wu T Wd < γ ∞
(15)
Our objective is to use the optimum H∞ controller theory in order to find an optimal fixed-structure controller, so to find the tunable gains shown in Fig. 1, as explained in (Apkarian and Noll, 2006).
2019 IFAC TDS Sinaia, Romania, September 9-11, 2019Carlos Daniel Vázquez et al. / IFAC PapersOnLine 52-18 (2019) 162–167
165
where ωb is the lowest allowable bandwidth of the system, chosen close to the minimum required bandwidth. Ms is the maximum allowable peak for the frequency response of the sensitivity function. The weighting function Wu should be a high-pass filter in order to guarantee the stability of the controlled system under diverse operating conditions. The weighting function Wu is defined as follows: ω
Fig. 3. Closed-loop with the weighting transfer function of the alternative observer scheme representation. 4.2 Proposed design procedure Let us point out that the controller K(s) cannot be analytically designed if the time delay involved in the process is treated strictly (Zhang, 1998). Consider a Pad´e approximation in the form: e−τ s ≈ Gp (s) =
1 − p1 s + p2 s2 + ... + (−1)q pq sq 1 + p1 s + p2 s2 + ... + pq sq
(16)
where q is the approximation order to be chosen. The coefficients pi depend on q and τ and are determined from a Taylor series expansion of the transcendental function. When an approximation of the time delay is introduced, the plant (1) becomes: G(s)e−τ s ≈ G(s)Gp (s)
(17)
With the rational transfer function (17), a controller using H∞ theory may now be calculated. Augmented plant structure. The Fig. 3 show an alternative configuration of the observer scheme where GO represents a subsystem that contains the next elements of the system (17): transfer function of mthe stable poles and minimum (s+bl ) phase zeros given by: nl=1 (s+c ) , the transfer function i=1
Wu (s) =
1 s + Mhu εu s + ωuh
(19)
where εu > 0 must be chosen as a small constant in order to ensure good rejection of measurement errors. The pulsation ωh limits the bandwidth and must be chosen sufficiently far from the desired grid frequency for closed loop control. Mu represent the effort of the controller. The principal effect of the weighting function Wd is to ensure the rejection of disturbances, it is chosen here as a constant function. Controller using H∞ structured design. The controller is synthesized by H∞ structured design method to get a feasible solution until the cost function γ is minimized. To apply the H∞ structured design, in addition to provide the augmented plant and the required structure of the controller, it is necessary to provide an initial controller as a starting point. The initial controllers values can be selected using the gains where the system (1) is stable using the control strategy shown in Fig. 1. The following procedure is proposed to obtain the gains g1 and g2 such that the closed-loop of the observer scheme is stable. The value of the proportional gain g1 can be stated as follows. Consider that of the Theorem 1 are mthe conditions n true and τ = a12 + l=1 b1l − i=1 c1i − β1 . Thus exist g1 1 such that β > g1 −a > 0. We can obtain: 2
i
g1 >
m
1 n − i=1
+ a2
(20)
of the Pad´e-aproximation is Gp , the transfer function of 1 the unstable pole a1 is given by: (s−a . The subsystem 1) KO contains the tunable gains g1 , g2 and the transfer 1 . A similar function of the unstable pole a2 given by (s−a 2) configuration of the controller scheme can be easily derived from a dual procedure of the previous structure.
We can select a g1 :
The representation shown in Fig. 3 keeps the same dynamic behaviour of the original proposal shown in Fig. 1 and represent a one way to build the augmented system (nominal system + weighting functions).
with εg1 being a positive real constant such that (5) and (6) holds.
Weighting functions selection. The weighting functions We , Wu , Wd are selected taking into account the basic requirement of mixed-sensitivity design (Lundstrom et al., 1991). The weighting function We represents the performance objective of the error sensitivity function S(s), it should work as a lowpass-filter in order to reduce the error sensitivity in the low frequency range for output disturbance rejection. We is defined in the next form (Skogestad and Postlethwaite, 2005): 1 s + ω b Ms We (s) = Ms s + ω b ε
(18) 287
g1 =
1 a1
1 a1
+
+
1 l=1 bl
m
1 l=1 bl
1 n − i=1
1 ci
1 ci
−τ
−τ
+ a2 + ε g 1
Now, from Lemma 1, there exist g2 such that the behaviour of the subsystem (13) is stable. We can find the values of g2 as follows: m
+ ωc2 ) n > g2 > (a1 + ωc2 )(g1 − a2 + ωc2 ) i=1 (ci + ωc2 ) m ) l=1 (b l n a1 (g1 − a2 ) i=2 (ci ) l=1 (bl
It is important to headline that if g1 is selected near to the bound stated in Ec (20), then the margin for the gain g2 will be reduced.
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k1 >
1 a2
m
+ m
1 l=1 bl
1 n − i=1
1 ci
−τ
+ a1
Sensitivity function S 10
0
-10
-20
Singular Values (dB)
The indicated values ensure a stable behaviour of the closed-loop system and represent a stable start point of the initial controller in order to find the optimal values of the tunable gains. The values for k1 and k2 can be stated as follows:
-30
-40
-50
-60
Sensitivity function full order control Sensitivity function proposed method Template
-70
2 c) l=1 (bl + ω > k2 > n (a2 + ωc2 )(k1 − a1 + ωc2 ) i=1 (ci + ωc2 ) m ) l=1 (b l n a2 (k1 − a1 ) i=2 (ci )
-80 10 -5
10 -4
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
Frequency (rad/s)
Fig. 4. Frequency response : Sensitivity function. Comparison between different calculated controllers Controller*Sensitivity KS
5. EXAMPLE
20
10
0
Singular Values (dB)
The proposed method is applied to a linearizing model of CSTR with zero dynamics by numerical simulation. We illustrate the design of the controller through the following example.
Controller sensitivity full order control Controller sensitivity proposed method Template
-10
-20
-30
Let us consider the example proposed in (Novella-Rodr´ıguez et al., 2014) . In this example, the flow rate is the manipulated variable and the temperature of the CSTR is the controlled variable. The linearization of the equation assuming a measurement delay of 54 min and taking account a one hour as the time scale of the model gives the transfer Fig. 5. Frequency response : Controller sensitivity function. Comparison between different calculated confunction model as: trollers proposed control strategy and a classical H∞ full order −4.7475(s + 3.129) e−0.9s (21) control design method. The simulation scheme is the one G(s) = (s − 0.5545)(s − 0.09395)(s + 6.394) given in Fig. 1. -40
-50
-60 10 -5
10 -4
10 -3
10 -2
10 -1
10 0
10 1
10 2
10 3
Frequency (rad/s)
The stability conditions given in Theorem 1 taking into account β = 1 are satisfied, due to:
Fig. 4 shows the shape of the sensitivity function S , which also corresponds to the closed-loop transfer function and Fig. 5 shows the shape of the controller sensitivity. This indicates that the selected weighting functions for controller design using a H∞ structured design have achieved the goal of command tracking.
1 1 1 1 0.9 < − + − 0.5545 6.394 3.129 1 and 3.129 > (0.5545)(1)(6.394)
(ω 2
5.1 Frequency response analysis
5.2 Time response analysis
(ω 2 + 3.1292 ) ∀ω + 0.55452 )(ω 2 + 1)(ω 2 + 6.3942 )
therefore there exists an observer based structure as shown in the Fig. 1 with proportional gains g1 , g2 , k1 and k2 such that the resulting closed-loop system is stable. The weighting function We (s) is selected with the parameters: Ms = 1.2, ωb = 0.1, ε = 0.0103. The weighting function Wu (s) is calculated with the parameters: Mu = 10, ωh = 10, ε = 0.0001. And Wd (s) = 0.001. Following the methodology shown in section 4 and using a Pad´e-approximation with order q = 9, the calculated gains for the example 1 are shown in Table 1. The optimal H∞ performance level γ is γ = 2.54. Additionally, we compute a full-order H∞ controller using the same weighting functions and the same order of Pad´eapproximation in order to make a comparison between the 288
The step response of the closed-loop system is shown in Fig. 6. Blue lines indicates the simulation of the closed-loop system using the scheme given in Fig. 1 as a control strategy. The black line indicates the full order H∞ controller, which is an eighth order controller. The results show that the methodology used in Section 4 with the reduced-order observer based controller is significantly faster and provides a smaller overshoot than the full order H∞ controller. The error introduced by the rational approximation will not cause instability in the proposed method. The simulations are carried out using c Matlab /Simulink. Table 1. calculated gains for the example 1 Gains g1 k1 g2 k2
Values 816.7051 101.6702 −544.9466 −64.8597
2019 IFAC TDS Sinaia, Romania, September 9-11, 2019Carlos Daniel Vázquez et al. / IFAC PapersOnLine 52-18 (2019) 162–167
Step Response
3 Nominal delay full order controller Nominal delay proposed method Uncertain delay proposed method
2.5 2
Output y(t)
1.5 1
0.5 0 -0.5 0
10
20
30
40
50
Time (seconds)
60
70
80
90
100
Fig. 6. Closed-loop behavior of the example 1 with a delay uncertainty of 5 percent Due to the use of a delayed observer in this work, the uncertainties in the time-delay of the plant may cause oscillations or even instability. Fig. 5 also shows the effect of delay mismatches between the plant nominal delay and the observer delay. The dashed red line shows the response of the system when the time-varying uncertain delay is introduced into the process. The time-delay in the plant can be considered as τ (t) = 0.9 + δ(t), where δ(t) is a pseudo-random binary signal scaled to a magnitude of 0.05. 6. CONCLUSION The paper described the control design of an observer based controller scheme that guarantees the stabilization of delayed systems with two unstable poles, n stable poles and m minimum phase zeros using only 4 tunable gains. The conditions that guarantee the stability of the closedloop system are derived in terms of the upper limit of time delay size and the poles and zeros position. Based on the H∞ control theory the controller gains are tuned using a non-smooth H∞ optimization method. Some uncertainty in the knowledge of the delay value has been considered in the numerical simulations which demonstrates the effectiveness of the method. The test demonstrates that the reduced-order observer based controller is faster and presents a minor overshoot when compared to the full order controller for the example used. REFERENCES Apkarian, P. and Noll, D. (2006). Nonsmooth optimization for multidisk h synthesis. European Journal of Control, 12(3), 229 – 244. Bequette, B.W. (2003). Process Control. Modeling, Design and Simulation. Prentice Hall International. Blakelock, J. (1991). Automatic control of aircraft and missiles. Wiley-interscience publicationl. Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989). State-space solutions to standard h2 and h∞ control problems. IEEE Transactions on Automatic Control, 34, 831–847. Kwak, H.J., Sung, S.W., and Lee, I.B. (2000). Stabilizability conditions and controller design for unstable processes. Chemical Engineering Research and Design, 78(4), 549 – 556. Lee, S.C. and Wang, Q.G. (2010). Stabilization conditions for a class of unstable delay processes of higher order. 289
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Journal of the Taiwan Institute of Chemical Engineers, 41(4), 440 – 445. Lundstrom, P., Skogestad, S., and Wang, Z.Q. (1991). Performance weight selection for h-infinity and -control methods. Transactions of the Institute of Measurement and Control, 13, 241–252. Novella-Rodr´ıguez, D., del Muro-Cu´ellar, B., and Sename, O. (2014). Observer based scheme for the control of high order systems with two unstable poles plus time delay. Asia-Pacific Journal of Chemical Engineering, 9. Skogestad, S. and Postlethwaite, I. (2005). Multivariable feedback control: Analysis and Design. John Wiley. V´azquez-Rosas, C., del Muro-Cu´ellar, B., and M´arquez Rubio, J. (2017). Stability conditions for unstable high order systems with minimum-phase zeros plus time delay. Wang, Q.G., Lee, T.H., and k.Tan, K. (1999). FiniteSpectrum Assignment for Time-Delay Systems. Springer-Verlag, London. Wu, L., Lam, H.K., Zhao, Y., and Shu, Z. (2015). Timedelay systems and their applications in engineering 2014. Mathematical Problems in Engineering, 2015, 1–3. Zhang, W.D. (1998). Analytical design methods for process control. In Postdoctoral research report. Shanghai Jiaotong University.