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Counterpart of Advanced TS discrete controller without matrix inversion
4th 4th IFAC IFAC International International Conference Conference on on Intelligent Control and Automation Sciences Intelligent Control and Automationon Sciences 4th IFAC International Conference 4th IFAC International Conference on June 1-3, 2016. Reims, France Available online at www.sciencedirect.com June 1-3, 2016. Reims, France Intelligent Control Control and and Automation Automation Sciences Sciences Intelligent June 1-3, 2016. Reims, France June 1-3, 2016. Reims, France
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IFAC-PapersOnLine 49-5 (2016) 182–187 Counterpart Counterpart of of Advanced Advanced TS TS discrete discrete controller controller without without matrix matrix inversion inversion Counterpart of Advanced TS discrete controller without matrix Counterpart of Advanced TS discrete controller without matrix inversion inversion Thomas Laurain*, Jimmy Lauber*, Reinaldo Palhares**
Thomas Laurain*, Jimmy Lauber*, Reinaldo Palhares** Thomas Laurain*, Laurain*, Jimmy Jimmy Lauber*, Lauber*, Reinaldo Reinaldo Palhares** Palhares** Thomas *LAMIH *LAMIH UMR/CNRS UMR/CNRS 8201, 8201, University University of of Valenciennes, Valenciennes, France France (e-mail: {thomas.laurain, jimmy.lauber}@univ-valenciennes.fr). *LAMIH{thomas.laurain, UMR/CNRS 8201, 8201, University of of Valenciennes, Valenciennes, France France (e-mail: jimmy.lauber}@univ-valenciennes.fr). *LAMIH UMR/CNRS University **Department Engineering, University (e-mail: jimmy.lauber}@univ-valenciennes.fr). **Department of of Electronics Electronics Engineering, Federal Federal University of of Minas Minas Gerais, Gerais, Belo Belo Horizonte, Horizonte, Brazil Brazil (e-mail: {thomas.laurain, {thomas.laurain, jimmy.lauber}@univ-valenciennes.fr). (e-mail:[email protected]) **Department of Electronics Engineering, Federal University of Minas Gerais, Belo (e-mail:[email protected]) **Department of Electronics Engineering, Federal University of Minas Gerais, Belo Horizonte, Horizonte, Brazil Brazil (e-mail:[email protected]) (e-mail:[email protected])
Abstract: Abstract: This This paper paper aims aims to to present present aa systematic systematic methodology methodology for for designing designing aa Counterpart Counterpart of of an an Advanced Takagi-Sugeno (CATS) discrete controller. Advance controllers for nonlinear systems under Abstract: This paper aims to present a systematic methodology for designing a Counterpart of Advanced Takagi-Sugeno (CATS) discrete controller. methodology Advance controllers for nonlinear systems under Abstract: This paper aims to present a systematic for designing a Counterpart of an an Takagi-Sugeno representation have been for using efficient laws as nonAdvanced (CATS) controller. Advance for systems Takagi-Sugeno representation have discrete been designed designed for years years usingcontrollers efficient control control laws such such as the theunder nonAdvanced Takagi-Sugeno Takagi-Sugeno (CATS) discrete controller. Advance controllers for nonlinear nonlinear systems under PDC Compensation) controller. terms of that controllers is Takagi-Sugeno representation have for efficient laws as PDC (Parallel (Parallel Distributed Distributed Compensation) controller. Inyears termsusing of stabilization, stabilization, that kind kind ofsuch controllers is aa Takagi-Sugeno representation have been been designed designed forIn years using efficient control control lawsof such as the the nonnonpowerful tool which allows outperforming the classical PDC results using non quadratic Lyapunov PDC (Parallel Compensation) controller. In of of is powerful tool Distributed which allows outperforming the classical PDC results usingthat nonkind quadratic Lyapunov PDC (Parallel Distributed Compensation) controller. In terms terms of stabilization, stabilization, that kind of controllers controllers is aa function. spite these control strategy presents major inconvenient from powerful tool the classical results using quadratic Lyapunov function. However, However, in allows spite of ofoutperforming these advantages, advantages, this controlPDC strategy presents major inconvenient from powerful tool which whichin allows outperforming the this classical PDC results usingaa non non quadratic Lyapunov real implementation point of the nonlinear matrix inversion at sample In function. However, in spite spite of these these advantages, thisaa control control strategy presents major inconvenient from real time time However, implementation point of view: view: the use use of of nonlinear matrixpresents inversion at each eachinconvenient sample time. time.from In function. in of advantages, this strategy aa major order to solve this problem, our paper presents a CATS controller design methodology to obtain an real time implementation point of view: the use of a nonlinear matrix inversion at each sample time. In ordertime to solve this problem, paper the presents controller methodology to obtain real implementation pointour of view: use ofa aCATS nonlinear matrix design inversion at each sample time. an In equivalent to non-PDC controller without matrix inversion. Through aa given procedure associated order to this problem, our presents aa CATS controller design methodology to obtain equivalent to the the matrix inversion. Through given procedure order to solve solve thisnon-PDC problem,controller our paper paperwithout presents CATS controller design methodology to associated obtain an an with a stability analysis, not only the efficiency is proved but also its validity. Finally, some simulation equivalent to the non-PDC controller without matrix inversion. Through a given procedure associated with a stability analysis, notcontroller only the efficiency is proved but alsoThrough its validity. Finally, some simulation equivalent to the non-PDC without matrix inversion. a given procedure associated results emphasize the originality and the of proposed CATS with aa will stability analysis, only efficiency is but also Finally, results will emphasize thenot originality the usefulness usefulness of the the CATS controller. controller. with stability analysis, not only the theand efficiency is proved proved butproposed also its its validity. validity. Finally, some some simulation simulation results will emphasize the originality and the usefulness of the proposed CATS controller. results emphasize the originality and the usefulness of the proposed CATS controller. © 2016,will IFAC (International Federation of Automatic Control) Hosting bynon-PDC Elsevier Ltd. All rightsequivalence; reserved. Keywords: Takagi-Sugeno representation; discrete-time system; controller Keywords: Takagi-Sugeno representation; discrete-time system; non-PDC controller equivalence; advanced law. Keywords: Takagi-Sugeno representation; advanced control law. Keywords:control Takagi-Sugeno representation; discrete-time discrete-time system; system; non-PDC non-PDC controller controller equivalence; equivalence; advanced advanced control control law. law.
1. 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. In In past past years, years, facing facing nonlinearity nonlinearity in in dynamical dynamical systems have become less and less a huge obstacle. Instead In past years, facing nonlinearity in systems have become less facing and lessnonlinearity a huge obstacle. Instead of of In past years, in dynamical dynamical the standard linearization that appeared in industry, some systems have become less and less a huge obstacle. Instead of the standard linearization thatless appeared in industry, some systems have become less and a huge obstacle. Instead of powerful tools have been developed. One of them is the the standard linearization that appeared in industry, some powerful toolslinearization have been that developed. One them issome the the standard appeared in of industry, Takagi-Sugeno (TS) been representation (Takagi and Sugeno, powerful developed. One them is Takagi-Sugeno (TS) representation powerful tools tools have have been developed.(Takagi One of of and themSugeno, is the the 1985) which can be expressed as aa polytopic model based on Takagi-Sugeno (TS) representation (Takagi and Sugeno, 1985) which can be expressed as polytopic model based on Takagi-Sugeno (TS) representation (Takagi and Sugeno, aa1985) combination of several linear models linked together with which expressed aa polytopic model based on combination ofbe several linearas linked together 1985) which can can be expressed asmodels polytopic model basedwith on terms. from nonlinear system based on aanonlinear combination several linked together nonlinear terms.of Obtained from aamodels nonlinear system basedwith on combination ofObtained several linear linear models linked together with aanonlinear sector approach, TS has terms. Obtained Obtained from aa nonlinear nonlinear system based based on sector nonlinearity nonlinearity approach, TS representation representation has the the nonlinear terms. from system on of being an exact equivalence of the nonlinear aadvantage sector nonlinearity approach, a TS representation has of being an exact equivalence of the nonlinear aadvantage sector nonlinearity approach, a TS representation has the the system on defined set. advantage being an system on aaof advantage ofdefined being set. an exact exact equivalence equivalence of of the the nonlinear nonlinear system on a defined set. system on a defined set. The The stability stability and and stabilization stabilization of of closed-loop closed-loop systems systems is is mainly studiedand using Lyapunov of functions. In (Guerra (Guerra and The stability stabilization closed-loop is mainly studied using Lyapunov functions. In The stability and stabilization of closed-loop systems systemsand is Vermeiren, 2004), the authors present aa In first case, the mainly studied using Lyapunov functions. (Guerra and Vermeiren, 2004), the authors present first case, the mainly studied using Lyapunov functions. In (Guerra and quadratic secondly, the Vermeiren, 2004), authors present aa first the quadratic Lyapunov function, and, secondly, the nonnonVermeiren,Lyapunov 2004), the thefunction, authors and, present first case, case, the quadratic one also introduced in discrete-time. Non-quadratic Lyapunov function, and, secondly, the nonquadratic one also introduced in discrete-time. Non-quadratic Lyapunov function, and, secondly, the nonquadratic Lyapunov functions have studied continuousquadratic one one also introduced introduced inbeen discrete-time. Non-quadratic Lyapunov functions have also alsoin been studied for for continuousquadratic also discrete-time. Non-quadratic time in (Guerra et al., 2012). Lyapunov functions have also been studied for time in (Guerra et al.,have 2012). Lyapunov functions also been studied for continuouscontinuoustime in (Guerra et al., 2012). time in (Guerra et al., 2012). In In order order to to control control nonlinear nonlinear systems, systems, lots lots of of methods methods have have been developed. Among them, the Parallel Distributed In order to control nonlinear systems, lots of have been developed. the lots Parallel Distributed In order to control Among nonlinearthem, systems, of methods methods have Compensation (PDC) controller developed in (Wang (Wang et al., al., been developed. Among them, the Parallel Distributed Compensation (PDC) controller developed in et been developed. Among them, the Parallel Distributed 1996) allows including nonlinearity inside aainclassic state or Compensation (PDC) controller developed (Wang et 1996) allows including nonlinearity inside classic state or Compensation (PDC) controller developed in (Wang et al., al., output feedback controller. Used in plenty of works, it can be 1996) including nonlinearity inside classic or output feedback controller. Used in plenty can be 1996) allows allows including nonlinearity insideofaa works, classicitstate state or however generalized in powerful output controller. Used in of works, it however generalized in aa more more powerful tool called non-PDC output feedback feedback controller. Used in plenty plentytool of called works,non-PDC it can can be be controller presented, for in and however more tool non-PDC controller presented, in foraaexample, example, in (Guerra (Guerra and Vermeiren, Vermeiren, however generalized generalized in more powerful powerful tool called called non-PDC 2004). As developed along this paper, the combination of controller presented, for example, in (Guerra (Guerra and Vermeiren, Vermeiren, 2004). Aspresented, developedfor along this paper, the combination of controller example, in and non-quadratic Lyapunov function and non-PDC controller 2004). As developed along this paper, the combination of non-quadratic Lyapunov function and non-PDC controller 2004). As developed along this paper, the combination of gives less conservative results. non-quadratic Lyapunov function and non-PDC controller gives less conservative results. non-quadratic Lyapunov function and non-PDC controller gives gives less less conservative conservative results. results.
For For many many years, years, the the non-PDC non-PDC controller controller has has been been used used through plenty of works and papers. (Ding et al., For many many years, non-PDC controller used through plenty works and papers. (Dinghas al., 2006) 2006) For years,of the the non-PDC controller haset been been used propose in their work an to previous results of through plenty works and et 2006) propose their of work an extension extension to the the(Ding previous results of through in plenty of works and papers. papers. (Ding et al., al., 2006) (Guerra and Vermeiren, 2004). (Bouarar et al., 2009) have propose in their work an extension to the previous results of (Guerra in andtheir Vermeiren, (Bouarar et al., 2009) have propose work an 2004). extension to the previous results of studied the particular case of static output feedback for (Guerra and Vermeiren, 2004). (Bouarar et al., 2009) have studied the of (Bouarar static output for (Guerra and particular Vermeiren,case 2004). et al.,feedback 2009) have descriptors systems using aa non-PDC control law. One studied the particular case of output feedback for descriptors using non-PDC One year year studied the systems particular case of static static control output law. feedback for after, (Bouarar (Bouarar et using al., 2010) 2010) focus control on robustness robustness with descriptors systems a non-PDC law. One year after, et al., focus on with descriptors systems using a non-PDC control law. One year H conditions. conditions. (Mozelli et al., al., 2010) 2010) based their work workwith on after, (Bouarar et focus on H et their on after, (Bouarar (Mozelli et al., al., 2010) 2010) focus based on robustness robustness with H conditions. (Mozelli et al., 2010) based their work on descriptors systems, Finsler’s lemma and so on to get new H conditions. (Mozelli et al.,lemma 2010)and based their on descriptors systems, Finsler’s so on to work get new relaxations and less conservativeness thanks to a non-PDC descriptors systems, Finsler’s lemma and so on to get new relaxations systems, and less Finsler’s conservativeness thanks to atonon-PDC descriptors lemma and so on get new controller. (Xie et 2013) aa new relaxations and conservativeness thanks to controller. et al., al., 2013) construct construct new relaxations (Xie and less less conservativeness thanks to aa non-PDC non-PDC control law structure for discrete-time systems. In the controller. (Xie et al., 2013) construct a new control law(Xie structure for 2013) discrete-time systems. the last last controller. et al., construct a newInnon-PDC non-PDC papers of (Lendek et for al., discrete-time 2013; Lendek Lendeksystems. et al., al., In 2015), the control law structure papers of (Lendek et al., 2013; et 2015), the control law structure for discrete-time systems. In the the last last theory of TS controllers has been extended, using periodic papers of (Lendek et al., 2013; Lendek et al., 2015), theory of TS controllers has been extended, using periodic papers of (Lendek et al., 2013; Lendek et al., 2015), the the Lyapunov functions and delayed theory controllers been extended, periodic Lyapunov functions and has delayed non-quadratic Lyapunov theory of of TS TS controllers has been non-quadratic extended, using usingLyapunov periodic functions. the works of et and Lyapunov functions and delayed non-quadratic Lyapunov functions. In the recent recent works of (Laurain (Laurain et al., al., 2015a) 2015a) and Lyapunov In functions and delayed non-quadratic Lyapunov (Laurain et al., 2015b), the non-PDC structure is used for functions.etIn Inal., the 2015b), recent works works of (Laurain (Laurain et al., al., is 2015a) and (Laurain the non-PDC structure used and for functions. the recent of et 2015a) periodic observers instead of control. (Laurain et al., 2015b), the non-PDC structure is used periodic observers insteadthe of control. (Laurain et al., 2015b), non-PDC structure is used for for periodic observers instead of control. periodic observers instead of control. In In both both cases cases discrete discrete and and continuous continuous time, time, the the non-PDC non-PDC controller, associated with a non-quadratic Lyapunov In both cases discrete and continuous time, non-PDC controller, associated withcontinuous a non-quadratic In both cases discrete and time, the the Lyapunov non-PDC function, implies a matrix inversion each time you reconsider controller, associated with a non-quadratic Lyapunov function, a matrixwith inversion each time you reconsider controller,implies associated a non-quadratic Lyapunov the control law, each clock time of embedded function, implies matrix inversion you the control law, aa i.e., i.e., each clock each time time of the the embedded function, implies matrix inversion each time you reconsider reconsider computer that calculates the control input. For matrices of the i.e., clock time of the computer thatlaw, calculates the control of big big the control control law, i.e., each each clock input. time For of matrices the embedded embedded size, such as ten squares and ten columns, this matrix computer the input. For of size, suchthat as calculates ten squares and ten columns, this matrix computer that calculates the control control input. For matrices matrices of big big inversion sampling time can be consuming, taking size, suchevery as ten ten squares and ten columns, this matrix matrix inversion every sampling timeand can ten be time time consuming, taking size, such as squares columns, this into the limited capacities of some inversion every sampling time can taking into account account limited some real-time real-time inversion every the sampling timecapacities can be be time timeofconsuming, consuming, taking embedded hardware. Even if non-PDC controllers offer into account the limited capacities of some real-time embedded hardware. Even capacities if non-PDC offer into account the limited of controllers some real-time useful and powerful results, they can be difficult to embedded hardware. Even if non-PDC controllers useful and hardware. powerful Even results, they can controllers be difficultoffer to embedded if non-PDC offer implement this limitation. useful and powerful results, they can be difficult implement limitation.results, they can be difficult to useful andthispowerful to implement implement this this limitation. limitation.
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systems such as Internal Combustion (IC) engines, an efficient control needs powerful controllers. As a conclusion for the motivating example, some nonlinear and complex systems can be controlled only by advanced control laws such as non-PDC controllers, even if this implies matrices inversion and high computational cost. Consequently, there is a need for an equivalent control law that avoids the matrix inversion.
As a conclusion of the state-of-the-art, non-PDC structure have been hugely used in the literature, in order to improve controllers or observers efficiency, develop new relaxations and reduce conservativeness for an important family of systems (continuous-time, discrete-time, robust systems, disturbed-by-noise systems…). However, in spite of its efficiency, non-PDC structure presents the inconvenient of the matrix inversion. This paper proposes a method for finding an equivalency to a non-PDC control law for a discrete-time system, called CATS controller.
2.3 Non-quadratic Lyapunov function and non-PDC controller conditions
It is organized following the structure: Some preliminaries are presented to introduce the context, including a motivating example. The main contribution is detailed in Section 3 with the new control law, the choice of the Lyapunov function and the stability analysis. Finally, Section 4 emphasizes the efficiency of the proposed methodology by presenting some simulation results.
For this paper, we base this work on the conditions for a new non-quadratic Lyapunov function and a non-PDC controller such as the ones developed in (Guerra and Vermeiren, 2004): 1 u t Fz H z x T T 1 V t x H z Pz H z x
2. PRELIMINARIES
Pi ijk Ai H j Bi Fj
2.1 Notation
i 1
i 1
Az hi z t Ai , Azz hi z t h j z t Aij
iik 0 , i, k 1,
, r
2 k ii ijk kji 0 , i, j, k 1, r 1
r
Azz
() T H k H k Pk
(3)
In order to make the system (1) stable, the previous quantity must check the following conditions:
Along this paper, the following notations are used: r
(2)
Applying the Lyapunov stability method with such a controller and the previously detailed system, the authors define the following quantity:
Keeping in mind that our purpose is real applications, we will mainly focus on discrete-time because the embedded computer that calculates the control law is triggered at every sample time.
r
183
hi z t hj z t 1 Aij , i 1
(4)
, r, i j
(5)
This leads to a Linear Matrix Inequality (LMI) problem such as defined in (Boyd et al., 1994). This problem can be solved by the LMI Toolbox of Matlab, which can provide a solution and, consequently, the gains of the non-PDC controller.
r
Azz hi z t h j z t 1 Aij i 1
2.2 Motivating example 3. MAIN CONTRIBUTION
For this paper, we base this work on the previous study of the literature, (Guerra and Vermeiren, 2004). In their paper, they propose an example that has been considered in many other papers as a good academic example for nonlinear research, such as (Nguyen et al., 2015).
3.1 Control law approximation with multiple sums In this subsection, we present the main contribution: the procedure to develop an equivalent control law to the nonPDC controller, based on a Counterpart of this Advanced Takagi-Sugeno (CATS) controller. As detailed in the preliminaries, we base the original contribution on non-PDC controller for discrete time (Guerra and Vermeiren, 2004).
The nonlinear system is based on the following matrices:
1 5 A1 , B1 2 1 0.5 1 5 A2 , B2 2 1 0.5
(1)
By this way, the first step of the approximation process is to solve the LMI problem defined in (3) in order to get the gains of the non-PDC controller. To explain the principle, let us consider the following non-PDC control law taken from (2):
For a given , it is proved that a quadratic Lyapunov function and a PDC controller cannot stabilize this system which is not so complex (only 2 rules and square matrices of size 2). However, for a given , the use of a non-PDC control law is mandatory. Moreover, for highly complex real
u k Fz H z1 x k
(6)
The objective is to find an equivalent controller which involves non inversion of weighted matrices. To this purpose, 183
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1 , 2 1 1 ,
the idea is to approximate the H z1 matrix as new nonlinear multiple sums matrix Gz ... z . A hypothesis is that the more sums you add, the more accurate the approximation is. Let us consider the new CATS controller with two sums for example:
u k Gzz x k
2
H
Gzz
T
H
1 z
i
T z
j
k
(13)
0
ijk
i 1 j 1 k 1
Remark 2: The previous result can be extended with a CATS controller with, for example, five sums:
(7)
u k Gzzzzz x k
(14)
Then, conditions (12) become:
Gzz I
(8)
1 ,
After matrices manipulations, (8) can be written as:
I H
2
The main idea is then to minimize a criterion such as: 1 z
2
GzzT I H zGzz H zT H z 0
i 1 , i \ 1 p , p 1
,1 ,
r 1
, r 1 i , i 1
(9) r
r
r
r
r
r
Using Schur complement, (9) leads to:
i
j
k
l
p ijklmp 0
m
(15)
i 1 j 1 k 1 l 1 m 1 p 1
0
H zT H z zzz I H z Gzz
I
(10)
With the quantity:
H iT H j ijklmp I H j Giklmp
In classical approaches to obtain LMI, relaxations are used to remove the nonlinear hi z t functions from the conditions. Our problem defined by (10) is to search for the Gij matrices
the inverted matrix H z1 . We can now apply the Lyapunov stability method by searching some candidate Lyapunov functions stabilizing the system. Let us keep the double sum for the CATS controller and let us write the system:
a desired step . Let us define the set:
x k 1 Az x k Bz Fz u k u k Gzz x k
(11)
The inequality (10) can be solved using the following relaxation where denotes the values taken by the membership functions among the set :
1 ,
i 1 , i \ 1 p , p 1
r
r
r
i
j
k
ijk
,1 ,
, r 1 i ,
0
(12)
(16)
By solving (12), the first step is achieved: obtaining the matrix Gzz (for the example with two sums) that approximate
The weighted functions hi z t are tabled from 0 to 1 with
,1
I
3.2 Candidate Lyapunov function
that minimize . Generally, only the vertices of the polytopes and some cross-terms between the vertices are considered. Here, to obtain an accurate approximation, we need to include more information from the membership functions.
0, ,2 ,
(17)
The candidate Lyapunov function is chosen as a nonquadratic one based on the same as in (Guerra and Vermeiren, 2004): V xT GzzT PzGzz x
r 1 i 1
(18)
The Lyapunov stability method ensures that the closed-loop system is stable if:
V 0
(19)
i 1 j 1 k 1
By replacing (18) into (19), it is deduced that:
If the conditions (12) are verified, then a minimum for can be searched (using the “mincx” function of LMI Toolbox of Matlab). This value represents the degree of accuracy of the approximation.
GzzT Pz Gzz
Az Bz Fz Gzz
T
GzT z Pz Gz z Az Bz Fz Gzz 0
(20)
Using Schur complement, the following inequality can be obtained from (20), where denotes the values taken by the membership functions at the t moment and for the membership functions at the t 1 among the set :
Remark 1: Considering the motivating example (1) with two rules, inequality (12) becomes:
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1 ,
i 1 , i \ 1 p , p 1
,1 ,
1 ,
, i \ 1 p , p 1
,1 ,
r
r
r
i 1
r
r
r
r
r
i
j
k
l
m
F1 2.4917 14.4205 F2 6.5470 13.2318
r 1
, r 1 i ,
22.0636 H1 1.9517 31.6556 H2 4.1046
i 1
r 1
, r 1 i ,
p q s ijklmpqs 0
185
i 1
0.1605 60.1749
(23)
8.0931 43.3994
Then, the problem (13) can be solved with a membership step of 0.5 and using the LMI Toolbox of Matlab in order to obtain the gains of our CATS controller:
(21)
i 1 j 1 k 1 l 1 m 1 p 1 q 1 s 1
With the following quantity defined by:
ijklmpqs
GijT Pk Glm PpGqs Ai Bi Fj Glm Pp
0.0453 0.0001 G11 0.0015 0.0166 2.6432 0.0923 G12 0.5763 0.4428 2.7158 0.0865 G21 0.5806 0.4805 0.0324 0.0060 G22 0.0031 0.0236
(22)
Then, in order to pay the same attention to accuracy, the same method is realized with the membership functions. They are tabled from 0 to 1 with a given step . Consequently, if the conditions (21) are verified for the quantity defined in (22), then there exists possible Lyapunov matrices Pi obtained with the “feasp” function of LMI Toolbox.
(24)
According to the conditions, these gains approximate the non-PDC controller gains H z with a certain accuracy that
Remark 3: Because the membership functions are tabled with values among the set , finding Lyapunov matrices resolving (21) do not ensure the stability of the closed-loop system.
has been maximized. For our example, 8.3756 1012 . These results can be validated by simulation using a sampling time of T 0.5s . However, in a first time, let us demonstrate why the cross-terms are needed and why the vertices are not sufficient. Considering a membership set 0,1 , the
3.3 Stability analysis
CATS gains are obtained and the simulation can be realized, comparing the results of the non-PDC controller and the CATS controller on the same system (1). Figure 1 depicts the two different commands generated by the non-PDC (blue) and the CATS (green) controller.
After getting the non-PDC controller gain composed of Fz and H z by solving (3) negative, after obtaining the CATS controller gain Gz ... z by solving (12) and after having found the Lyapunov function Pz by solving (21), it is possible now to validate the stability of the closed-loop system. In order to solve this problem, let us use the Proposition 1 of (Sala and Ariño, 2007) for multi-dimensional fuzzy summations (MDFS) which allow removing the nonlinear weighted terms from the conditions, leading to verify a set of matrix inequalities that have to be definite negative (i.e., because all the terms of the expression are known, this is equivalent to check if the real part of their eigenvalues are negative).
0.02 0
command
-0.02 -0.04 -0.06 -0.08 non-PDC controller CATS controller
-0.1
4. SIMULATION RESULTS
-0.12 0
For simulation results, let us use the previously detailed motivating example (1) from (Guerra and Vermeiren, 2004) with 1.5 . As presented along this paper, the first step of the methodology is to get the gains from the conditions (4) and (5). The following gains are obtained:
1
2
3
4
5 time (s)
6
7
8
9
10
Fig. 1. Difference between the command generated by the non-PDC and the two-sums CATS controller with a membership step of 1 . Considering only the vertices leads to a difference between the two control laws. That is because there is a need for taking into account the cross-terms. With a membership step of 0.5 in order to get the CATS controller already
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Now, let us study the impact of increasing the number of sums of the proposed CATS controller. However, the more sums you add, the more degrees of freedom you have, and, consequently, the more points taken by the membership functions you need. Increasing the number of sums without increasing the membership step cannot ensure that the conditions (12) are verified and cannot guarantee a correct accuracy for the cross-terms. By consequent, the CATS controllers with more sums and not enough points in the membership set cannot stabilize the system.
presented in (24), the error can be displayed. Even if the purpose is to compare the commands, the error on the state is more significant because of the nonlinearity of the studied system: A small error between the control laws can lead to a huge error on the state. Figure 2 depicts the error between the state of a system controlled by a non-PDC and the two-sums CATS controller, calculated in percent.
error on the first state (%)
0 error on x1 -0.5
Let us consider a step 0.1 . Four sums are needed to get an error on the state nearly equal to zero even in transient phases, i.e., the closed-loop system can be controlled and there exists a CATS controller that is totally equivalent to the non-PDC controller (6) such as:
-1
-1.5
-2 0
1
2
3
4
5 time (s)
6
7
8
9
u k Gzzzz x k
10
5. CONCLUSION
Fig. 2. Error in percent between the first state of a system piloted by a non-PDC and the two-sums CATS controller with a membership step of 0.5 .
This paper has presented an original methodology to replace non-PDC controllers. These control laws are necessary for plenty of researches in advanced Takagi-Sugeno topics, including command, observation, relaxations and so on. The main inconvenient of such controllers remains in the matrix inversion that leads to high computational cost for an embedded computer in a real application. Along this paper, a Counterpart for this Advanced Takagi-Sugeno (CATS) controller has been designed based on nonlinear matrices with several sums and LMI formulation. In order to improve the accuracy of the approximation, instead using a relaxation to remove the membership functions, they are taken into account among a special set divided with a particular step. This leads to two tuneable parameters: the number of sums involved in the new controller and the step of the membership functions gridding. As usual in control theory, a compromise has to be realized between getting a really accurate equivalent control law and generating higher computational cost with too many sums and a too big membership functions step.
As it is presented, the error is very small (less than 2%), which validates that the presented CATS controller (7) is an equivalence to the non-PDC controller (6) with an accuracy that allows only 2% of error in the transient phase. Based on our hypothesis, two parameters can improve this accuracy: The number of sums, as it has been introduced in Remark 2, and the step . Let us consider a smaller membership step . Figure 3 depicts the error on the state between the non-PDC and the two-sums CATS controller with the membership functions included into the set:
0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1
(25)
0 error on x1
error on the first state (%)
-0.2
(26)
-0.4 -0.6
ACKNOWLEDGMENT
-0.8
This research is sponsored by the International Campus on Safety and Intermodality in Transportation, the Nord-Pas-deCalais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the French National Center for Scientific Research (CNRS).
-1 -1.2 -1.4 0
1
2
3
4
5 time (s)
6
7
8
9
10
Fig. 3. Error in percent between the first state of a system piloted by a non-PDC and the two-sums CATS controller with membership taken in .
REFERENCES Bouarar, T., Guelton, K., Manamanni, N. (2010) Robust fuzzy Lyapunov stabilization for uncertain and disturbed Takagi–Sugeno descriptors. ISA Transactions. 49(4), 447– 461. Bouarar, T., Guelton, K., Manamanni, N. (2009) Static output feedback controller design for Takagi-Sugeno systems-a fuzzy Lyapunov LMI approach. In IEEE Conference on
The error during the transient phase is always inferior to 1.2%, which validates our hypothesis: The more complete the set where the membership functions are taken is, the more accurate the approximation of the non-PDC controller is.
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Decision and Control / Chinese Control Conference (CDC/CCC). IEEE, pp. 4150–4155. Ding, B., Sun, H., Yang, P. (2006) Further studies on LMIbased relaxed stabilization conditions for nonlinear systems in Takagi–Sugeno’s form. Automatica. 42(3), 503–508. Guerra, T.M., Bernal, M., Guelton, K., Labiod, S. (2012) Non-quadratic local stabilization for continuous-time Takagi– Sugeno models. Fuzzy Sets and Systems. 201, 40–54. Guerra, T.M., Vermeiren, L. (2004) LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno’s form. Automatica. 40(5), 823–829. Laurain, T., Lauber, J., Palhares, R. (2015) Periodic TakagiSugeno Observers for Individual Cylinder Spark Imbalance in Idle Speed Control Context. In IFAC International Conference on Informatics in Control, Automation and Robotics (ICINCO). IFAC. Laurain, T., Lauber, J., Palhares, R.M. (2015) Observer design to control individual cylinder spark advance for idle speed management of a SI engine. In IEEE International Conference on Industrial Electronics and Applications (ICIEA). IEEE. Lendek, Z., Guerra, T.M., Lauber, J. (2015) Controller Design for TS Models Using Delayed Nonquadratic Lyapunov Functions. IEEE Transactions on Cybernetics. 45(3), 439–450. Lendek, Z., Lauber, J., Guerra, T.-M. (2013) Periodic Lyapunov functions for periodic TS systems. Systems & Control Letters. 62(4), 303–310. Mozelli, L.A., Palhares, R.M., Mendes, E.M.A.M. (2010) Equivalent techniques, extra comparisons and less conservative control design for Takagi–Sugeno (TS) fuzzy systems. IET Control Theory & Applications. 4(12), 2813– 2822. Nguyen, A., Laurain, T., Lauber, J., Sentouh, C., Popieul, J.C. (2015) Non-quadratic approach for control design of constrained Takagi-Sugeno fuzzy systems subject to persistent disturbances. In IEEE International Conference on Fuzzy Systems (FUZZ-IEEE). IEEE, pp. 1–6. Sala, A., Ariño, C. (2007) Asymptotically necessary and sufficient conditions for stability and performance in fuzzy control: Applications of Polya’s theorem. Fuzzy Sets and Systems. 158(24), 2671–2686. Takagi, T., Sugeno, M. (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics. (1), 116– 132. Wang, H.O., Tanaka, K., Griffin, M.F. (1996) An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Transactions on Fuzzy Systems. 4(1), 14–23. Xie, X., Ma, H., Zhao, Y., Ding, D.-W., Wang, Y. (2013) Control Synthesis of Discrete-Time T-S Fuzzy Systems Based on a Novel Non-PDC Control Scheme. IEEE Transactions on Fuzzy Systems. 21(1), 147–157.
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IFAC-PapersOnLine 49-5 (2016) 182–187 Counterpart Counterpart of of Advanced Advanced TS TS discrete discrete controller controller without without matrix matrix inversion inversion Counterpart of Advanced TS discrete controller without matrix Counterpart of Advanced TS discrete controller without matrix inversion inversion Thomas Laurain*, Jimmy Lauber*, Reinaldo Palhares**
Thomas Laurain*, Jimmy Lauber*, Reinaldo Palhares** Thomas Laurain*, Laurain*, Jimmy Jimmy Lauber*, Lauber*, Reinaldo Reinaldo Palhares** Palhares** Thomas *LAMIH *LAMIH UMR/CNRS UMR/CNRS 8201, 8201, University University of of Valenciennes, Valenciennes, France France (e-mail: {thomas.laurain, jimmy.lauber}@univ-valenciennes.fr). *LAMIH{thomas.laurain, UMR/CNRS 8201, 8201, University of of Valenciennes, Valenciennes, France France (e-mail: jimmy.lauber}@univ-valenciennes.fr). *LAMIH UMR/CNRS University **Department Engineering, University (e-mail: jimmy.lauber}@univ-valenciennes.fr). **Department of of Electronics Electronics Engineering, Federal Federal University of of Minas Minas Gerais, Gerais, Belo Belo Horizonte, Horizonte, Brazil Brazil (e-mail: {thomas.laurain, {thomas.laurain, jimmy.lauber}@univ-valenciennes.fr). (e-mail:[email protected]) **Department of Electronics Engineering, Federal University of Minas Gerais, Belo (e-mail:[email protected]) **Department of Electronics Engineering, Federal University of Minas Gerais, Belo Horizonte, Horizonte, Brazil Brazil (e-mail:[email protected]) (e-mail:[email protected])
Abstract: Abstract: This This paper paper aims aims to to present present aa systematic systematic methodology methodology for for designing designing aa Counterpart Counterpart of of an an Advanced Takagi-Sugeno (CATS) discrete controller. Advance controllers for nonlinear systems under Abstract: This paper aims to present a systematic methodology for designing a Counterpart of Advanced Takagi-Sugeno (CATS) discrete controller. methodology Advance controllers for nonlinear systems under Abstract: This paper aims to present a systematic for designing a Counterpart of an an Takagi-Sugeno representation have been for using efficient laws as nonAdvanced (CATS) controller. Advance for systems Takagi-Sugeno representation have discrete been designed designed for years years usingcontrollers efficient control control laws such such as the theunder nonAdvanced Takagi-Sugeno Takagi-Sugeno (CATS) discrete controller. Advance controllers for nonlinear nonlinear systems under PDC Compensation) controller. terms of that controllers is Takagi-Sugeno representation have for efficient laws as PDC (Parallel (Parallel Distributed Distributed Compensation) controller. Inyears termsusing of stabilization, stabilization, that kind kind ofsuch controllers is aa Takagi-Sugeno representation have been been designed designed forIn years using efficient control control lawsof such as the the nonnonpowerful tool which allows outperforming the classical PDC results using non quadratic Lyapunov PDC (Parallel Compensation) controller. In of of is powerful tool Distributed which allows outperforming the classical PDC results usingthat nonkind quadratic Lyapunov PDC (Parallel Distributed Compensation) controller. In terms terms of stabilization, stabilization, that kind of controllers controllers is aa function. spite these control strategy presents major inconvenient from powerful tool the classical results using quadratic Lyapunov function. However, However, in allows spite of ofoutperforming these advantages, advantages, this controlPDC strategy presents major inconvenient from powerful tool which whichin allows outperforming the this classical PDC results usingaa non non quadratic Lyapunov real implementation point of the nonlinear matrix inversion at sample In function. However, in spite spite of these these advantages, thisaa control control strategy presents major inconvenient from real time time However, implementation point of view: view: the use use of of nonlinear matrixpresents inversion at each eachinconvenient sample time. time.from In function. in of advantages, this strategy aa major order to solve this problem, our paper presents a CATS controller design methodology to obtain an real time implementation point of view: the use of a nonlinear matrix inversion at each sample time. In ordertime to solve this problem, paper the presents controller methodology to obtain real implementation pointour of view: use ofa aCATS nonlinear matrix design inversion at each sample time. an In equivalent to non-PDC controller without matrix inversion. Through aa given procedure associated order to this problem, our presents aa CATS controller design methodology to obtain equivalent to the the matrix inversion. Through given procedure order to solve solve thisnon-PDC problem,controller our paper paperwithout presents CATS controller design methodology to associated obtain an an with a stability analysis, not only the efficiency is proved but also its validity. Finally, some simulation equivalent to the non-PDC controller without matrix inversion. Through a given procedure associated with a stability analysis, notcontroller only the efficiency is proved but alsoThrough its validity. Finally, some simulation equivalent to the non-PDC without matrix inversion. a given procedure associated results emphasize the originality and the of proposed CATS with aa will stability analysis, only efficiency is but also Finally, results will emphasize thenot originality the usefulness usefulness of the the CATS controller. controller. with stability analysis, not only the theand efficiency is proved proved butproposed also its its validity. validity. Finally, some some simulation simulation results will emphasize the originality and the usefulness of the proposed CATS controller. results emphasize the originality and the usefulness of the proposed CATS controller. © 2016,will IFAC (International Federation of Automatic Control) Hosting bynon-PDC Elsevier Ltd. All rightsequivalence; reserved. Keywords: Takagi-Sugeno representation; discrete-time system; controller Keywords: Takagi-Sugeno representation; discrete-time system; non-PDC controller equivalence; advanced law. Keywords: Takagi-Sugeno representation; advanced control law. Keywords:control Takagi-Sugeno representation; discrete-time discrete-time system; system; non-PDC non-PDC controller controller equivalence; equivalence; advanced advanced control control law. law.
1. 1. INTRODUCTION INTRODUCTION 1. INTRODUCTION INTRODUCTION 1. In In past past years, years, facing facing nonlinearity nonlinearity in in dynamical dynamical systems have become less and less a huge obstacle. Instead In past years, facing nonlinearity in systems have become less facing and lessnonlinearity a huge obstacle. Instead of of In past years, in dynamical dynamical the standard linearization that appeared in industry, some systems have become less and less a huge obstacle. Instead of the standard linearization thatless appeared in industry, some systems have become less and a huge obstacle. Instead of powerful tools have been developed. One of them is the the standard linearization that appeared in industry, some powerful toolslinearization have been that developed. One them issome the the standard appeared in of industry, Takagi-Sugeno (TS) been representation (Takagi and Sugeno, powerful developed. One them is Takagi-Sugeno (TS) representation powerful tools tools have have been developed.(Takagi One of of and themSugeno, is the the 1985) which can be expressed as aa polytopic model based on Takagi-Sugeno (TS) representation (Takagi and Sugeno, 1985) which can be expressed as polytopic model based on Takagi-Sugeno (TS) representation (Takagi and Sugeno, aa1985) combination of several linear models linked together with which expressed aa polytopic model based on combination ofbe several linearas linked together 1985) which can can be expressed asmodels polytopic model basedwith on terms. from nonlinear system based on aanonlinear combination several linked together nonlinear terms.of Obtained from aamodels nonlinear system basedwith on combination ofObtained several linear linear models linked together with aanonlinear sector approach, TS has terms. Obtained Obtained from aa nonlinear nonlinear system based based on sector nonlinearity nonlinearity approach, TS representation representation has the the nonlinear terms. from system on of being an exact equivalence of the nonlinear aadvantage sector nonlinearity approach, a TS representation has of being an exact equivalence of the nonlinear aadvantage sector nonlinearity approach, a TS representation has the the system on defined set. advantage being an system on aaof advantage ofdefined being set. an exact exact equivalence equivalence of of the the nonlinear nonlinear system on a defined set. system on a defined set. The The stability stability and and stabilization stabilization of of closed-loop closed-loop systems systems is is mainly studiedand using Lyapunov of functions. In (Guerra (Guerra and The stability stabilization closed-loop is mainly studied using Lyapunov functions. In The stability and stabilization of closed-loop systems systemsand is Vermeiren, 2004), the authors present aa In first case, the mainly studied using Lyapunov functions. (Guerra and Vermeiren, 2004), the authors present first case, the mainly studied using Lyapunov functions. In (Guerra and quadratic secondly, the Vermeiren, 2004), authors present aa first the quadratic Lyapunov function, and, secondly, the nonnonVermeiren,Lyapunov 2004), the thefunction, authors and, present first case, case, the quadratic one also introduced in discrete-time. Non-quadratic Lyapunov function, and, secondly, the nonquadratic one also introduced in discrete-time. Non-quadratic Lyapunov function, and, secondly, the nonquadratic Lyapunov functions have studied continuousquadratic one one also introduced introduced inbeen discrete-time. Non-quadratic Lyapunov functions have also alsoin been studied for for continuousquadratic also discrete-time. Non-quadratic time in (Guerra et al., 2012). Lyapunov functions have also been studied for time in (Guerra et al.,have 2012). Lyapunov functions also been studied for continuouscontinuoustime in (Guerra et al., 2012). time in (Guerra et al., 2012). In In order order to to control control nonlinear nonlinear systems, systems, lots lots of of methods methods have have been developed. Among them, the Parallel Distributed In order to control nonlinear systems, lots of have been developed. the lots Parallel Distributed In order to control Among nonlinearthem, systems, of methods methods have Compensation (PDC) controller developed in (Wang (Wang et al., al., been developed. Among them, the Parallel Distributed Compensation (PDC) controller developed in et been developed. Among them, the Parallel Distributed 1996) allows including nonlinearity inside aainclassic state or Compensation (PDC) controller developed (Wang et 1996) allows including nonlinearity inside classic state or Compensation (PDC) controller developed in (Wang et al., al., output feedback controller. Used in plenty of works, it can be 1996) including nonlinearity inside classic or output feedback controller. Used in plenty can be 1996) allows allows including nonlinearity insideofaa works, classicitstate state or however generalized in powerful output controller. Used in of works, it however generalized in aa more more powerful tool called non-PDC output feedback feedback controller. Used in plenty plentytool of called works,non-PDC it can can be be controller presented, for in and however more tool non-PDC controller presented, in foraaexample, example, in (Guerra (Guerra and Vermeiren, Vermeiren, however generalized generalized in more powerful powerful tool called called non-PDC 2004). As developed along this paper, the combination of controller presented, for example, in (Guerra (Guerra and Vermeiren, Vermeiren, 2004). Aspresented, developedfor along this paper, the combination of controller example, in and non-quadratic Lyapunov function and non-PDC controller 2004). As developed along this paper, the combination of non-quadratic Lyapunov function and non-PDC controller 2004). As developed along this paper, the combination of gives less conservative results. non-quadratic Lyapunov function and non-PDC controller gives less conservative results. non-quadratic Lyapunov function and non-PDC controller gives gives less less conservative conservative results. results.
For For many many years, years, the the non-PDC non-PDC controller controller has has been been used used through plenty of works and papers. (Ding et al., For many many years, non-PDC controller used through plenty works and papers. (Dinghas al., 2006) 2006) For years,of the the non-PDC controller haset been been used propose in their work an to previous results of through plenty works and et 2006) propose their of work an extension extension to the the(Ding previous results of through in plenty of works and papers. papers. (Ding et al., al., 2006) (Guerra and Vermeiren, 2004). (Bouarar et al., 2009) have propose in their work an extension to the previous results of (Guerra in andtheir Vermeiren, (Bouarar et al., 2009) have propose work an 2004). extension to the previous results of studied the particular case of static output feedback for (Guerra and Vermeiren, 2004). (Bouarar et al., 2009) have studied the of (Bouarar static output for (Guerra and particular Vermeiren,case 2004). et al.,feedback 2009) have descriptors systems using aa non-PDC control law. One studied the particular case of output feedback for descriptors using non-PDC One year year studied the systems particular case of static static control output law. feedback for after, (Bouarar (Bouarar et using al., 2010) 2010) focus control on robustness robustness with descriptors systems a non-PDC law. One year after, et al., focus on with descriptors systems using a non-PDC control law. One year H conditions. conditions. (Mozelli et al., al., 2010) 2010) based their work workwith on after, (Bouarar et focus on H et their on after, (Bouarar (Mozelli et al., al., 2010) 2010) focus based on robustness robustness with H conditions. (Mozelli et al., 2010) based their work on descriptors systems, Finsler’s lemma and so on to get new H conditions. (Mozelli et al.,lemma 2010)and based their on descriptors systems, Finsler’s so on to work get new relaxations and less conservativeness thanks to a non-PDC descriptors systems, Finsler’s lemma and so on to get new relaxations systems, and less Finsler’s conservativeness thanks to atonon-PDC descriptors lemma and so on get new controller. (Xie et 2013) aa new relaxations and conservativeness thanks to controller. et al., al., 2013) construct construct new relaxations (Xie and less less conservativeness thanks to aa non-PDC non-PDC control law structure for discrete-time systems. In the controller. (Xie et al., 2013) construct a new control law(Xie structure for 2013) discrete-time systems. the last last controller. et al., construct a newInnon-PDC non-PDC papers of (Lendek et for al., discrete-time 2013; Lendek Lendeksystems. et al., al., In 2015), the control law structure papers of (Lendek et al., 2013; et 2015), the control law structure for discrete-time systems. In the the last last theory of TS controllers has been extended, using periodic papers of (Lendek et al., 2013; Lendek et al., 2015), theory of TS controllers has been extended, using periodic papers of (Lendek et al., 2013; Lendek et al., 2015), the the Lyapunov functions and delayed theory controllers been extended, periodic Lyapunov functions and has delayed non-quadratic Lyapunov theory of of TS TS controllers has been non-quadratic extended, using usingLyapunov periodic functions. the works of et and Lyapunov functions and delayed non-quadratic Lyapunov functions. In the recent recent works of (Laurain (Laurain et al., al., 2015a) 2015a) and Lyapunov In functions and delayed non-quadratic Lyapunov (Laurain et al., 2015b), the non-PDC structure is used for functions.etIn Inal., the 2015b), recent works works of (Laurain (Laurain et al., al., is 2015a) and (Laurain the non-PDC structure used and for functions. the recent of et 2015a) periodic observers instead of control. (Laurain et al., 2015b), the non-PDC structure is used periodic observers insteadthe of control. (Laurain et al., 2015b), non-PDC structure is used for for periodic observers instead of control. periodic observers instead of control. In In both both cases cases discrete discrete and and continuous continuous time, time, the the non-PDC non-PDC controller, associated with a non-quadratic Lyapunov In both cases discrete and continuous time, non-PDC controller, associated withcontinuous a non-quadratic In both cases discrete and time, the the Lyapunov non-PDC function, implies a matrix inversion each time you reconsider controller, associated with a non-quadratic Lyapunov function, a matrixwith inversion each time you reconsider controller,implies associated a non-quadratic Lyapunov the control law, each clock time of embedded function, implies matrix inversion you the control law, aa i.e., i.e., each clock each time time of the the embedded function, implies matrix inversion each time you reconsider reconsider computer that calculates the control input. For matrices of the i.e., clock time of the computer thatlaw, calculates the control of big big the control control law, i.e., each each clock input. time For of matrices the embedded embedded size, such as ten squares and ten columns, this matrix computer the input. For of size, suchthat as calculates ten squares and ten columns, this matrix computer that calculates the control control input. For matrices matrices of big big inversion sampling time can be consuming, taking size, suchevery as ten ten squares and ten columns, this matrix matrix inversion every sampling timeand can ten be time time consuming, taking size, such as squares columns, this into the limited capacities of some inversion every sampling time can taking into account account limited some real-time real-time inversion every the sampling timecapacities can be be time timeofconsuming, consuming, taking embedded hardware. Even if non-PDC controllers offer into account the limited capacities of some real-time embedded hardware. Even capacities if non-PDC offer into account the limited of controllers some real-time useful and powerful results, they can be difficult to embedded hardware. Even if non-PDC controllers useful and hardware. powerful Even results, they can controllers be difficultoffer to embedded if non-PDC offer implement this limitation. useful and powerful results, they can be difficult implement limitation.results, they can be difficult to useful andthispowerful to implement implement this this limitation. limitation.
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systems such as Internal Combustion (IC) engines, an efficient control needs powerful controllers. As a conclusion for the motivating example, some nonlinear and complex systems can be controlled only by advanced control laws such as non-PDC controllers, even if this implies matrices inversion and high computational cost. Consequently, there is a need for an equivalent control law that avoids the matrix inversion.
As a conclusion of the state-of-the-art, non-PDC structure have been hugely used in the literature, in order to improve controllers or observers efficiency, develop new relaxations and reduce conservativeness for an important family of systems (continuous-time, discrete-time, robust systems, disturbed-by-noise systems…). However, in spite of its efficiency, non-PDC structure presents the inconvenient of the matrix inversion. This paper proposes a method for finding an equivalency to a non-PDC control law for a discrete-time system, called CATS controller.
2.3 Non-quadratic Lyapunov function and non-PDC controller conditions
It is organized following the structure: Some preliminaries are presented to introduce the context, including a motivating example. The main contribution is detailed in Section 3 with the new control law, the choice of the Lyapunov function and the stability analysis. Finally, Section 4 emphasizes the efficiency of the proposed methodology by presenting some simulation results.
For this paper, we base this work on the conditions for a new non-quadratic Lyapunov function and a non-PDC controller such as the ones developed in (Guerra and Vermeiren, 2004): 1 u t Fz H z x T T 1 V t x H z Pz H z x
2. PRELIMINARIES
Pi ijk Ai H j Bi Fj
2.1 Notation
i 1
i 1
Az hi z t Ai , Azz hi z t h j z t Aij
iik 0 , i, k 1,
, r
2 k ii ijk kji 0 , i, j, k 1, r 1
r
Azz
() T H k H k Pk
(3)
In order to make the system (1) stable, the previous quantity must check the following conditions:
Along this paper, the following notations are used: r
(2)
Applying the Lyapunov stability method with such a controller and the previously detailed system, the authors define the following quantity:
Keeping in mind that our purpose is real applications, we will mainly focus on discrete-time because the embedded computer that calculates the control law is triggered at every sample time.
r
183
hi z t hj z t 1 Aij , i 1
(4)
, r, i j
(5)
This leads to a Linear Matrix Inequality (LMI) problem such as defined in (Boyd et al., 1994). This problem can be solved by the LMI Toolbox of Matlab, which can provide a solution and, consequently, the gains of the non-PDC controller.
r
Azz hi z t h j z t 1 Aij i 1
2.2 Motivating example 3. MAIN CONTRIBUTION
For this paper, we base this work on the previous study of the literature, (Guerra and Vermeiren, 2004). In their paper, they propose an example that has been considered in many other papers as a good academic example for nonlinear research, such as (Nguyen et al., 2015).
3.1 Control law approximation with multiple sums In this subsection, we present the main contribution: the procedure to develop an equivalent control law to the nonPDC controller, based on a Counterpart of this Advanced Takagi-Sugeno (CATS) controller. As detailed in the preliminaries, we base the original contribution on non-PDC controller for discrete time (Guerra and Vermeiren, 2004).
The nonlinear system is based on the following matrices:
1 5 A1 , B1 2 1 0.5 1 5 A2 , B2 2 1 0.5
(1)
By this way, the first step of the approximation process is to solve the LMI problem defined in (3) in order to get the gains of the non-PDC controller. To explain the principle, let us consider the following non-PDC control law taken from (2):
For a given , it is proved that a quadratic Lyapunov function and a PDC controller cannot stabilize this system which is not so complex (only 2 rules and square matrices of size 2). However, for a given , the use of a non-PDC control law is mandatory. Moreover, for highly complex real
u k Fz H z1 x k
(6)
The objective is to find an equivalent controller which involves non inversion of weighted matrices. To this purpose, 183
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1 , 2 1 1 ,
the idea is to approximate the H z1 matrix as new nonlinear multiple sums matrix Gz ... z . A hypothesis is that the more sums you add, the more accurate the approximation is. Let us consider the new CATS controller with two sums for example:
u k Gzz x k
2
H
Gzz
T
H
1 z
i
T z
j
k
(13)
0
ijk
i 1 j 1 k 1
Remark 2: The previous result can be extended with a CATS controller with, for example, five sums:
(7)
u k Gzzzzz x k
(14)
Then, conditions (12) become:
Gzz I
(8)
1 ,
After matrices manipulations, (8) can be written as:
I H
2
The main idea is then to minimize a criterion such as: 1 z
2
GzzT I H zGzz H zT H z 0
i 1 , i \ 1 p , p 1
,1 ,
r 1
, r 1 i , i 1
(9) r
r
r
r
r
r
Using Schur complement, (9) leads to:
i
j
k
l
p ijklmp 0
m
(15)
i 1 j 1 k 1 l 1 m 1 p 1
0
H zT H z zzz I H z Gzz
I
(10)
With the quantity:
H iT H j ijklmp I H j Giklmp
In classical approaches to obtain LMI, relaxations are used to remove the nonlinear hi z t functions from the conditions. Our problem defined by (10) is to search for the Gij matrices
the inverted matrix H z1 . We can now apply the Lyapunov stability method by searching some candidate Lyapunov functions stabilizing the system. Let us keep the double sum for the CATS controller and let us write the system:
a desired step . Let us define the set:
x k 1 Az x k Bz Fz u k u k Gzz x k
(11)
The inequality (10) can be solved using the following relaxation where denotes the values taken by the membership functions among the set :
1 ,
i 1 , i \ 1 p , p 1
r
r
r
i
j
k
ijk
,1 ,
, r 1 i ,
0
(12)
(16)
By solving (12), the first step is achieved: obtaining the matrix Gzz (for the example with two sums) that approximate
The weighted functions hi z t are tabled from 0 to 1 with
,1
I
3.2 Candidate Lyapunov function
that minimize . Generally, only the vertices of the polytopes and some cross-terms between the vertices are considered. Here, to obtain an accurate approximation, we need to include more information from the membership functions.
0, ,2 ,
(17)
The candidate Lyapunov function is chosen as a nonquadratic one based on the same as in (Guerra and Vermeiren, 2004): V xT GzzT PzGzz x
r 1 i 1
(18)
The Lyapunov stability method ensures that the closed-loop system is stable if:
V 0
(19)
i 1 j 1 k 1
By replacing (18) into (19), it is deduced that:
If the conditions (12) are verified, then a minimum for can be searched (using the “mincx” function of LMI Toolbox of Matlab). This value represents the degree of accuracy of the approximation.
GzzT Pz Gzz
Az Bz Fz Gzz
T
GzT z Pz Gz z Az Bz Fz Gzz 0
(20)
Using Schur complement, the following inequality can be obtained from (20), where denotes the values taken by the membership functions at the t moment and for the membership functions at the t 1 among the set :
Remark 1: Considering the motivating example (1) with two rules, inequality (12) becomes:
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1 ,
i 1 , i \ 1 p , p 1
,1 ,
1 ,
, i \ 1 p , p 1
,1 ,
r
r
r
i 1
r
r
r
r
r
i
j
k
l
m
F1 2.4917 14.4205 F2 6.5470 13.2318
r 1
, r 1 i ,
22.0636 H1 1.9517 31.6556 H2 4.1046
i 1
r 1
, r 1 i ,
p q s ijklmpqs 0
185
i 1
0.1605 60.1749
(23)
8.0931 43.3994
Then, the problem (13) can be solved with a membership step of 0.5 and using the LMI Toolbox of Matlab in order to obtain the gains of our CATS controller:
(21)
i 1 j 1 k 1 l 1 m 1 p 1 q 1 s 1
With the following quantity defined by:
ijklmpqs
GijT Pk Glm PpGqs Ai Bi Fj Glm Pp
0.0453 0.0001 G11 0.0015 0.0166 2.6432 0.0923 G12 0.5763 0.4428 2.7158 0.0865 G21 0.5806 0.4805 0.0324 0.0060 G22 0.0031 0.0236
(22)
Then, in order to pay the same attention to accuracy, the same method is realized with the membership functions. They are tabled from 0 to 1 with a given step . Consequently, if the conditions (21) are verified for the quantity defined in (22), then there exists possible Lyapunov matrices Pi obtained with the “feasp” function of LMI Toolbox.
(24)
According to the conditions, these gains approximate the non-PDC controller gains H z with a certain accuracy that
Remark 3: Because the membership functions are tabled with values among the set , finding Lyapunov matrices resolving (21) do not ensure the stability of the closed-loop system.
has been maximized. For our example, 8.3756 1012 . These results can be validated by simulation using a sampling time of T 0.5s . However, in a first time, let us demonstrate why the cross-terms are needed and why the vertices are not sufficient. Considering a membership set 0,1 , the
3.3 Stability analysis
CATS gains are obtained and the simulation can be realized, comparing the results of the non-PDC controller and the CATS controller on the same system (1). Figure 1 depicts the two different commands generated by the non-PDC (blue) and the CATS (green) controller.
After getting the non-PDC controller gain composed of Fz and H z by solving (3) negative, after obtaining the CATS controller gain Gz ... z by solving (12) and after having found the Lyapunov function Pz by solving (21), it is possible now to validate the stability of the closed-loop system. In order to solve this problem, let us use the Proposition 1 of (Sala and Ariño, 2007) for multi-dimensional fuzzy summations (MDFS) which allow removing the nonlinear weighted terms from the conditions, leading to verify a set of matrix inequalities that have to be definite negative (i.e., because all the terms of the expression are known, this is equivalent to check if the real part of their eigenvalues are negative).
0.02 0
command
-0.02 -0.04 -0.06 -0.08 non-PDC controller CATS controller
-0.1
4. SIMULATION RESULTS
-0.12 0
For simulation results, let us use the previously detailed motivating example (1) from (Guerra and Vermeiren, 2004) with 1.5 . As presented along this paper, the first step of the methodology is to get the gains from the conditions (4) and (5). The following gains are obtained:
1
2
3
4
5 time (s)
6
7
8
9
10
Fig. 1. Difference between the command generated by the non-PDC and the two-sums CATS controller with a membership step of 1 . Considering only the vertices leads to a difference between the two control laws. That is because there is a need for taking into account the cross-terms. With a membership step of 0.5 in order to get the CATS controller already
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Thomas Laurain et al. / IFAC-PapersOnLine 49-5 (2016) 182–187
Now, let us study the impact of increasing the number of sums of the proposed CATS controller. However, the more sums you add, the more degrees of freedom you have, and, consequently, the more points taken by the membership functions you need. Increasing the number of sums without increasing the membership step cannot ensure that the conditions (12) are verified and cannot guarantee a correct accuracy for the cross-terms. By consequent, the CATS controllers with more sums and not enough points in the membership set cannot stabilize the system.
presented in (24), the error can be displayed. Even if the purpose is to compare the commands, the error on the state is more significant because of the nonlinearity of the studied system: A small error between the control laws can lead to a huge error on the state. Figure 2 depicts the error between the state of a system controlled by a non-PDC and the two-sums CATS controller, calculated in percent.
error on the first state (%)
0 error on x1 -0.5
Let us consider a step 0.1 . Four sums are needed to get an error on the state nearly equal to zero even in transient phases, i.e., the closed-loop system can be controlled and there exists a CATS controller that is totally equivalent to the non-PDC controller (6) such as:
-1
-1.5
-2 0
1
2
3
4
5 time (s)
6
7
8
9
u k Gzzzz x k
10
5. CONCLUSION
Fig. 2. Error in percent between the first state of a system piloted by a non-PDC and the two-sums CATS controller with a membership step of 0.5 .
This paper has presented an original methodology to replace non-PDC controllers. These control laws are necessary for plenty of researches in advanced Takagi-Sugeno topics, including command, observation, relaxations and so on. The main inconvenient of such controllers remains in the matrix inversion that leads to high computational cost for an embedded computer in a real application. Along this paper, a Counterpart for this Advanced Takagi-Sugeno (CATS) controller has been designed based on nonlinear matrices with several sums and LMI formulation. In order to improve the accuracy of the approximation, instead using a relaxation to remove the membership functions, they are taken into account among a special set divided with a particular step. This leads to two tuneable parameters: the number of sums involved in the new controller and the step of the membership functions gridding. As usual in control theory, a compromise has to be realized between getting a really accurate equivalent control law and generating higher computational cost with too many sums and a too big membership functions step.
As it is presented, the error is very small (less than 2%), which validates that the presented CATS controller (7) is an equivalence to the non-PDC controller (6) with an accuracy that allows only 2% of error in the transient phase. Based on our hypothesis, two parameters can improve this accuracy: The number of sums, as it has been introduced in Remark 2, and the step . Let us consider a smaller membership step . Figure 3 depicts the error on the state between the non-PDC and the two-sums CATS controller with the membership functions included into the set:
0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1
(25)
0 error on x1
error on the first state (%)
-0.2
(26)
-0.4 -0.6
ACKNOWLEDGMENT
-0.8
This research is sponsored by the International Campus on Safety and Intermodality in Transportation, the Nord-Pas-deCalais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, and the French National Center for Scientific Research (CNRS).
-1 -1.2 -1.4 0
1
2
3
4
5 time (s)
6
7
8
9
10
Fig. 3. Error in percent between the first state of a system piloted by a non-PDC and the two-sums CATS controller with membership taken in .
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The error during the transient phase is always inferior to 1.2%, which validates our hypothesis: The more complete the set where the membership functions are taken is, the more accurate the approximation of the non-PDC controller is.
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