A Polytopic Observer Design Approach for Landing Control of a Quadrotor UAV

Proceedings of the 20th World Congress Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress Proceedings of the 20th9-14, World Congress Control The International Federation Toulouse, France, July 2017 The International Federation of of Automatic Automatic Control Available online at www.sciencedirect.com The International Federation of Automatic Control Toulouse, Toulouse, France, France, July July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017

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A Polytopic Observer Approach for A Polytopic Observer Design Design Approach for A Polytopic Observer Design Approach for A Landing PolytopicControl Observer Design Approach for of a Quadrotor UAV Landing Control of a Quadrotor UAV Landing Control of a Quadrotor UAV Landing Control of a Quadrotor UAV † ∗ ∗ Souad Bezzaoucha ∗ Holger Voos ∗ Mohamed Darouach † ∗ Holger Voos ∗ Mohamed Darouach † Souad Bezzaoucha Souad Bezzaoucha ∗ Holger Voos ∗ Mohamed Darouach † Souad Bezzaoucha Holger Voos Mohamed Darouach ∗ Interdisciplinary Centre for Security, Reliability and Trust (SnT), ∗ ∗ Interdisciplinary Centre for Security, Reliability and Trust (SnT), Centre for Reliability and Trust Automatic Control Research Group, University ∗ Interdisciplinary Interdisciplinary Centre for Security, Security, Reliability of andLuxembourg, Trust (SnT), (SnT), Automatic Control Research Group, University of Luxembourg, Automatic Control Research Group, University of Luxembourg, Campus Kirchberg, 6 rue Coudenhove-Kalergi L-1359, Luxembourg; Automatic Control Research Group, University of Luxembourg, Campus Kirchberg, 6 rue Coudenhove-Kalergi L-1359, Luxembourg; Campus Kirchberg, 6 L-1359, e-mails: souad.bezzaoucha, [email protected] Campus Kirchberg, 6 rue rue Coudenhove-Kalergi Coudenhove-Kalergi L-1359, Luxembourg; Luxembourg; e-mails: souad.bezzaoucha, [email protected] † e-mails: souad.bezzaoucha, [email protected] Research Center for Automatic Control of Nancy (CRAN), † e-mails: souad.bezzaoucha, [email protected] † Research Center for Automatic Control of Nancy (CRAN), Center Automatic Control (CRAN), Universit´ e de Lorraine, de Longwy, 186 of rueNancy de Lorraine, 54400 † Research Research Center for forIUT Automatic Control of Nancy (CRAN), Universit´ ee de Lorraine, IUT de Longwy, 186 rue de Lorraine, 54400 Universit´ de Lorraine, IUT de Longwy, 186 rue de Lorraine, Cosnes et Romain, France; Universit´e de Lorraine, IUT et deRomain, Longwy, France; 186 rue de Lorraine, 54400 54400 Cosnes Cosnes e-mail: [email protected] Cosnes et et Romain, Romain, France; France; e-mail: [email protected] e-mail: e-mail: [email protected] [email protected]

Abstract: In this paper, a constructive procedure to design functional unknown input observer Abstract: In this paper, atime constructive design functional unknown input observer Abstract: In paper, constructive procedure to design unknown input for nonlinear continuous systems procedure under theto Takagi-Sugeno framework (also Abstract: In this this paper, a atime constructive procedure toPolytopic design functional functional unknown input observer observer for nonlinear continuous systems under the Polytopic Takagi-Sugeno framework (also for nonlinear continuous time systems under the Polytopic Takagi-Sugeno framework (also known as multiple models systems) is proposed. Applying the Lyapunov theory, Linear Matrix for nonlinear continuous time systems under the Polytopic Takagi-Sugeno framework (also known as multiple models systems) is proposed. Applying the Lyapunov theory, Linear Matrix known as multiple models systems) is proposed. Applying the Lyapunov theory, Linear Matrix Inequalities (LMI)s conditions are deduced which are solved for feasibility to obtain observer known as multiple models systems) is proposed. Applying the for Lyapunov theory, Linearobserver Matrix Inequalities (LMI)s conditions are deduced which are solved feasibility to obtain Inequalities (LMI)s deduced which are for observer design matrices. To conditions reject the are effect of unknown classical approach to of obtain decoupling the Inequalities (LMI)s conditions are deduced which input, are solved solved for feasibility feasibility to obtain observer design matrices. To reject the effect of unknown input, classical approach of decoupling the design matrices. To reject the effect of unknown input, classical approach of decoupling the unknown input for the linear case is used. A comparative study between single and Polytopic design matrices. Tothe reject thecase effect of unknown input, classical approach of decoupling the unknown input for linear is used. A comparative study between single and Polytopic unknown input for the linear case is used. A comparative study between single and Polytopic Lyapunov function is made in order to prove the relaxation effect of the Multiple functions. A unknown input for the linear case is used. A comparative study between single and Polytopic Lyapunov function is made in order to prove the relaxation effect of the Multiple functions. A Lyapunov function is made in order to prove the relaxation effect of the Multiple functions. A solver based solution is then proposed. Lyapunov function is ismade inproposed. order to prove the relaxation effect of the Multiple functions. A solver solution then solver based solution is proposed. It will based be shown through applicative example (a Quadrotor Aerial Robots Landing) that even solver based solution is then then proposed. It will be shown through applicative example (a Quadrotor Aerial Robots Landing) that even It will be shown through applicative example (a Aerial Landing) that even if the proposed LMIs solver based solution look conservative, an adequate choice the It will proposed be shownLMIs through applicative examplemay (a Quadrotor Quadrotor Aerial Robots Robots Landing) thatof even if the solver based solution may look conservative, an adequate choice of the if the proposed LMIs solver based solution may look conservative, an adequate choice of solver makes it suitable for the application of the proposed approach. if the proposed LMIs solver based solutionofmay look conservative, an adequate choice of the the solver makes it suitable for the application the proposed approach. solver makes it suitable for the application of the proposed approach. solver suitable forFederation the application of theControl) proposed approach. © 2017,makes IFAC it (International of Automatic Hosting by Elsevier Ltd. All rights reserved. Keywords: Functional Observer, Unknown Inputs, Polytopic approach, Takagi-Sugeno models, Keywords: continuous Functional time Observer, Unknown Inputs, Inputs, Polytopic approach, approach, Takagi-Sugeno models, models, Keywords: Functional Observer, Unknown Nonlinear systems. Keywords: Functional time Observer, Unknown Inputs, Polytopic Polytopic approach, Takagi-Sugeno Takagi-Sugeno models, Nonlinear continuous continuous systems. Nonlinear time systems. Nonlinear continuous time systems. 1. INTRODUCTION can be established for nonlinear systems (Chadli et al. 1. INTRODUCTION INTRODUCTION can be be established established for [2006]), nonlinear systems (Chadli (Chadli et al. al. 1. can for nonlinear systems et [2002]), (Guerra et al. (Kruszewski et al. [2008]) 1. INTRODUCTION can be established for [2006]), nonlinear systems (Chadli et al. [2002]), (Guerra et al. (Kruszewski et al. [2008]) (Guerra et [2006]), al. using tools borrowed the (Kruszewski linear theory et (Luenberger The polytopic T-S approach has proved its effectiveness [2002]), [2002]), (Guerra et al. al.from [2006]), al. [2008]) [2008]) using tools tools borrowed from the (Kruszewski linear theory theory et(Luenberger (Luenberger using borrowed from the linear The polytopic T-S approach has proved proved its advantage effectiveness [1971]), (Darouach et al. [1994]). The polytopic T-S approach has its effectiveness in the study of nonlinear systems. The main of using tools borrowed from the linear theory (Luenberger [1971]), (Darouach et al. [1994]). The polytopic T-S approach has proved its effectiveness (Darouach al. in the therepresentation study of of nonlinear systems. The main main advantage of [1971]), In this paper, the et functional observers design with unin study systems. The advantage of such is that it provides a natural, simple [1971]), (Darouach et al. [1994]). [1994]). In this this inputs paper, the nonlinear functional observers design with with ununin therepresentation study of nonlinear nonlinear systems. The main advantage of In paper, the functional observers design such is that it provides a natural, simple known for systems is addressed. such representation that it provides aa natural, and effective design is approach to complement othersimple non- In this inputs paper, for the nonlinear functionalsystems observers design withMost unknown is addressed. Most such representation is that it provides natural, simple inputs for nonlinear systems is addressed. Most and effective effective design approach to complement complement other non- known of the works devoted to these observers are for linear and design approach to other nonlinear techniques that require special and rather involved known inputs for nonlinear systems is addressed. Most of the works devoted to these observers are for linear and design approach to complement other non- systems the these observers are lineareffective techniques that require special and rather involved where devoted necessaryto sufficient conditions forlinear their linear techniques and rather knowledge. In fact,that therequire study ofspecial generic nonlinear systems of of the works works devoted toand these observers are for for systems where necessary and sufficient conditions forlinear their linear techniques that require special and rather involved involved systems where necessary and sufficient conditions for their knowledge. In fact, the study of generic nonlinear systems existence were given (see (Darouach [2000]), (Watson and knowledge. fact, the study of nonlinear systems can result inIn complex models togeneric be dealt with, requiring systems where necessary and sufficient conditions for their existence were given (see (Darouach [2000]), (Watson and knowledge. In fact, the study of generic nonlinear systems existence were given (see (Darouach [2000]), (Watson and can result in complex models to be dealt with, requiring Grigoriadis [1998]), (Nagpal et al. [1987]), (Ezzine et al. can result in complex models to be dealt with, requiring heavy mathematical tools and does not systematically lead existence were given (Nagpal (see (Darouach [2000]),(Ezzine (Watsonetand Grigoriadis [1998]), et al. [1987]), al. can result in complex models to be dealt with, requiring Grigoriadis [1998]), (Nagpal et al. [1987]), (Ezzine et al. heavy mathematical tools and does not systematically lead [2011]) and (N’Doye et al. [2013])). The proposed work heavy mathematical tools and does not systematically lead to unified results. Consequently, many of nonlinear Grigoriadis [1998]), (Nagpal et al. [1987]), (Ezzine et al. [2011]) and (N’Doye et al. al. approaches [2013])). The proposed work heavy mathematical tools and does not classes systematically lead [2011]) and (N’Doye et [2013])). The proposed work to unified results. Consequently, many classes of nonlinear is based on these classical of decoupling the to unified results. Consequently, classes of nonlinear systems are studied with specificmany assumptions. [2011]) and (N’Doye et al. approaches [2013])). The proposed work is based on these classical of decoupling the to unified results. Consequently, many classes of nonlinear based these classical approaches decoupling systems are studied studied with specific specific assumptions. unknown input the linear case andof adapted to the systems are with assumptions. The polytopic Takagi-Sugeno (T-S) models or Multiple is is based on on thesefor approaches decoupling unknown input forclassical thethelinear linear case framework. andof adapted adapted to the the systems are studied with specific assumptions. unknown input for the case and to The polytopic Takagi-Sugeno (T-S) models or Multiple nonlinear one through polytopic Applying The polytopic Takagi-Sugeno (T-S) models or Multiple Models (MM) decomposition is one appealing alternative unknown input for the linear case and adapted to the the nonlinear one through the polytopic framework. Applying The polytopic Takagi-Sugeno (T-S) models or Multiple nonlinear one through the polytopic framework. Applying Models (MM) (MM) decomposition is systems one appealing appealing alternative the Lyapunov theory, the Linear Matrixframework. InequalitiesApplying (LMI)s Models decomposition is one alternative solution to deal with nonlinear and to obtain the nonlinear one through polytopic the Lyapunov theory, Linear Matrix Inequalities (LMI)s Models is systems one appealing alternative Lyapunov theory, Inequalities (LMI)s solution(MM) to representation dealdecomposition with nonlinear nonlinear and tolinear obtainstate the the conditions are deduced which Matrix are solved for feasibility to solution to deal systems obtain the equivalent by a compact set ofto the Lyapunov theory, Linear Linear Inequalities (LMI)s conditions are deduced deduced which Matrix are solved solved for feasibility feasibility to solution to representation deal with with nonlinear systems and and tolinear obtainstate the conditions are which are for to equivalent by a compact set of obtain observer design matrices. equivalent representation by aa compact set of linear state space models with nonlinear weighting functions satisfying conditions are deduced which are solved for feasibility to obtain observer design matrices. equivalent representation by compact set of linear state matrices. space models with nonlinear weighting functions satisfying A key observer point fordesign the synthesis observer is the search space models with nonlinear weighting functions satisfying the convex sum property (Tanaka and Wang [2001]), (Tuan obtain obtain observer matrices.of A key key point fordesign the synthesis of observer observer is the the search space models with nonlinear weighting functions satisfying A point for the synthesis of is search theal. convex sum property (Tanaka and Wang Wang [2001]), (Tuan for an adequate Lyapunov function. A usual approach the convex sum property (Tanaka and [2001]), (Tuan et [2001]), and (Yoneyama [2006]). Herein, the well A key point for the synthesis of observer is the search for ancall adequate Lyapunov function. A usual usual approach the convex sum and property (Tanaka[2006]). and Wang [2001]), (Tuan for an adequate Lyapunov function. A approach et al. [2001]), (Yoneyama Herein, the well is to for a single quadratic Lyapunov function. This et al. [2001]), and (Yoneyama [2006]). Herein, known sector nonlinearity transformation (SNT) the maywell be for ancall adequate Lyapunov function. A usual approach is to for a single quadratic Lyapunov function. This et al. [2001]), and (Yoneyama [2006]). Herein, the well is to call for a single quadratic Lyapunov function. This known sector nonlinearity transformation (SNT) may be approach suffers from conservatism since it does not take known nonlinearity transformation may be used as sector a systematic and analytical method(SNT) to rewrite and is to call for a single quadratic Lyapunov function. This approach suffers from conservatism since it does not take known sector nonlinearity transformation (SNT) may be approach suffers from conservatism since it does not take used as a systematic and analytical method to rewrite and into account the time varying of the polytopic systems. used as aa systematic and to thus transform a nonlinear systemmethod into a polytopic T-S approach suffers from since it does systems. not take intosome account the time conservatism varying ofproblem the polytopic used as systematic and analytical analytical method to rewrite rewrite and and into account the time varying of the polytopic systems. thus transform a nonlinear system into a polytopic T-S In cases, it may cause the to be infeasible, thus transform aa nonlinear system aa polytopic T-S form without any loss of information (Tanaka and Wang into account the time cause varying the polytopic systems. In some some cases, it may may theofproblem problem to be be infeasible, thus nonlinear system into into polytopic T-S In it the to form transform without any loss of ofetinformation information (Tanaka and Wang Wang meaning that quadratic stabilization cannot beinfeasible, achieved. form without any loss (Tanaka and [2001]) and (Kawamoto al. [1992]). In some cases, cases, it may cause cause the problem to be infeasible, meaning that quadratic stabilization cannot be achieved. form without any loss of information (Tanaka and Wang that quadratic cannot achieved. [2001]) and (Kawamoto et al. al. [1992]). A significant can be obtained bybe considering [2001]) (Kawamoto et A T-S and model can be understood as polytopic systems, meaning meaning thatimprovement quadratic stabilization stabilization cannot achieved. A significant significant improvement can be be obtained obtained bybe considering [2001]) and (Kawamoto et al. [1992]). [1992]). A improvement can by considering A T-S model can be understood as polytopic systems, time-varying or polytopic Lyapunov Functions which inA T-S model can be understood as polytopic systems, where the blending between the subsystems is time varying A significant improvement can be obtained by considering time-varying or polytopic Lyapunov Functions which inA T-Sthe model can between be understood as polytopic systems, time-varying or polytopic Lyapunov Functions which inwhere blending the subsystems subsystems is time time varying corporate the system dynamics (Sename et al. [2013]). where the blending between the is varying according to the so called weighting functions. Thanks to time-varying or polytopic Lyapunov Functions which incorporate the system dynamics (Sename et al. [2013]). where the blending the subsystems is time varying the system dynamics (Sename al. according to the the sobetween called weighting functions. Thanks to corporate In this paper, solutions are presented. A classical single according to so called weighting functions. to the convexity of the weighting functions and to Thanks the linearcorporate the two system dynamics (Sename et et al. [2013]). [2013]). In this paper, two solutions are presented. A classical single according to the so called weighting functions. Thanks to this two are A single the of convexity of the the weighting functions and to to the results linear- In Lyapunov function and a polytopic Lyapunov functions, the convexity of and the linearity the subsystems definingfunctions the vertices, some In this paper, paper, two solutions solutions are presented. presented. A classical classical single Lyapunov function and aa polytopic polytopic Lyapunov functions, the convexity of the weighting weighting and to the results linear- Lyapunov function and Lyapunov functions, ity of of the subsystems subsystems definingfunctions the vertices, vertices, some ity the defining the some results Lyapunov function and a polytopic Lyapunov functions, ity of the subsystems defining the vertices, some results

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Proceedings of the 20th IFAC World Congress 9754 Souad Bezzaoucha et al. / IFAC PapersOnLine 50-1 (2017) 9753–9759 Toulouse, France, July 9-14, 2017

where the design conditions are still cast in terms of LMIs but the the switching speed of each linear subsystems (exactly speaking, to the lower bounds of time derivatives of membership functions) too (see (Tanaka et al. [2007]) for more details). The second approach is introduced to overcome the conservativeness and to obtain more relaxed constraints. In other words, to improve the numerical results in terms of convergence by introducing new types of polytopic Lyapunov function. The paper is organized as follows. In section II, a functional observer for nonlinear polytopic systems with unknown inputs is addressed. Sufficient conditions for its existence and a step-by-step design algorithms are given. The single and multiple Lyapunov functions case are considered. Simulation results of the application of the proposed approaches to a quadrotor landing on a moving platform are given in section III with a comparison study in terms of performance. Conclusions are detailed in section IV. 2. FUNCTIONAL OBSERVERS DESIGN FOR NONLINEAR POLYTOPIC SYSTEMS WITH UNKNOWN INPUTS In this section, sufficient conditions for the existence of a polytopic functional observer with unknown inputs are given. A constructive procedure for its design is proposed. Let us consider the following polytopic model subject to unknown inputs where each sub-model contributes to the global behavior of the nonlinear system through a weighting function µi (ξ(t)). The polytopic structure is given by:  M     ˙ = µi (ξ(t))(Ai x(t) + Bi u(t) + Fi d(t))  x(t) (1) i=1  y(t) = Cx(t)    z(t) = Lx(t) where x(t) ∈ Rn is the system state, y(t) ∈ Rp and u(t) ∈ Rm represent respectively the system output and input. d(t) ∈ Rq is the unknown input vector and z(t) ∈ Rr where r ≤ n is the vector to be estimated. Ai , Bi , C, Fi and L are known constant matrices of appropriate dimensions. It is assumed, without loss of generality that rank(C) = p and rank(L) = r. The weighting functions µi (ξ(t)) verify the following convex sum property: r  µi (ξ(t)) = 1, 0 ≤ µi (ξ(t)) ≤ 1, i = 1, . . . , M, ∀t i=1

ξ(t) may depend on measurable premises variables (a part of the input u(t) or the output y(t)) or unmeasurable premises variables (as the system states, UIs). As a first contribution and for simplicity reasons, in this paper, the case of measurable premise variables is considered. In order to reconstruct the state function using measurable signals (i.e. inputs u(t) and output y(t)), we define a functional polytopic observer of the form:  M    η(t) ˙ = µi (ξ(t))(Ni η(t) + Ji y(t) + Hi u(t)) (2)  i=1  zˆ(t) = η(t) + Ey(t) where η ∈ Rr is the state vector of the observer and zˆ(t) ∈ Rr is the estimate of z(t). Ni , Ji , Hi and E are

unknown and constant matrices of appropriate dimension to be designed. The following proposition gives the conditions for the existence and stability of the functional polytopic observer (2). Proposition 1. The state zˆ(t) in (2) is an asymptotic estimate of z(t) (1) for any x(0), zˆ(0) and u(t) if for i, j = 1, . . . , M : e(t) ˙ =

M 

ˆ µi (ξ(t))N i e(t)

i=1

(3)

is asymptotically stable

and P Ai − Ni P − Ji C = 0 P Bi = Hi P Fi = 0 where P is defined by: P = L − EC Proof 1. Let us define the estimation error e(t) as: e(t) = z(t) − zˆ(t) = P x(t) − η(t) with P = L − EC. Its time derivative is then deduced: M  ˆ µi (ξ(t))(N e(t) ˙ = i e(t) + P Fi d(t)

(4a) (4b) (4c) (5)

(6)

i=1

+(P Ai − Ni P − Ji C)x(t) + (P Bi − Hi )u(t)) (7) Now, under conditions (4), the estimation error dynamics becomes: M  ˆ (8) µi (ξ(t))N e(t) ˙ = i e(t) i=1

Then we can see that if (3) is satisfied, zˆ(t) → z(t).

Now, the design of the functional polytopic observer is reduced to finding the gain matrices Ni , Pi , Ji , Hi and E such that proposition 1 is satisfied. From the definition of P , the conditions (4a) and (4c) are written as: Ni L + ECAi + Ki C = LAi (9a) ECFi = LFi (9b) with Ki = Ji − Ni E. Equations (9) can be written as: [ Ni Ki E ] Σ1i = Σ2i (10) where for i = 1, . . . , M   L 0 0  Σ1i =  C (11) CAi CFi and (12) Σ2i = [ LAi LFi ] The following lemma gives necessary and sufficient conditions for the existence of a solution to (10). Lemma 1. There exists a solution to (10) if and only if:   Σ1i rank = rank [Σ1i ] (13) Σ2i

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Proof 2. From the general solution of linear algebraic equations (Rao and Mitra [1971]), there exists a solution to (10) if and only if: Σ2i Σ+ 1i Σ1i = Σ2i

(14)

where Σ+ 2i denotes any generalized inverse of Σ2i satisfying + Σ2i Σ2i Σ2i = Σ2i and (13) are satisfied for i = 1, . . . , M . Equation (14) is also equivalent to:   Σ1i rank = rank [Σ1i ] Σ2i which corresponds to condition (13).

In the sequel of the paper we assume that (14) is satisfied. In this case, the general solution of (10) is given by: Σ2i Σ+ 1i

Σ1i Σ+ 1i )

[ N i Ki E ] = − Zi (I − (15) where Zi is an arbitrary matrix of appropriate dimension that will be determined in the sequel using LMI approach. Remark 1. Solving (15) may leads to several solutions for E (one different solution for each sub-model i). In this paper, we are looking for a single solution, common to all the sub-models. This common value will be given by combining (15) and the inequality constraints given in the above devlopment (solver based numerical solution). From (15), Ni is given by: N i = Ai − Z i B i where     I + I + Ai = Σ2i Σ1i , Bi = (I − Σ1i Σ1i ) 0 0

(16) (17)

Under conditions (4), the estimation error dynamics is given by (8). Now the design problem is reduced to find the arbitrary matrix Zi such that condition (3) of proposition 1 is satisfied. Based on the Lyapunov theory, two solutions are proposed; the first one based on a classical single Lyapunov matrix and the second one base on multiple polytopic Lyapunov matrices. It is important to highlight that the second approach introduces more relaxation in the synthesis problem that the first one. 2.1 Single Lyapunov matrice

The following lemma gives the necessary and sufficient conditions to ensure the asymptotic stability of the estimation error dynamics. Lemma 2. Under conditions (13), there exists an asymptotic stable polytopic functional observer of the form (2) for the system (1), if there exists a symmetric positive matrix X = X T > 0 solution of the inequalities: XNi + XNiT < 0 (18) for i = 1, . . . , M Proof 3. The proof of this lemma is straightforwardly obtained from the application of the Lyapunov theory (Tanaka and Wang [2001]). Then, the following theorem is proposed: Theorem 1. Under conditions (13), there exists an asymptotic stable polytopic functional observer of the form (2) for the system (1), if there exists a symmetric positive

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matrix X = X T > 0, matrices Wi , Ri and S, solution of the following inequality: (19) XAi + ATi X − Wi Bi − BiT WiT < 0 under the following constraint: + [ XAi − Wi Bi Ri S ] = XΣ2i Σ+ 1i − Wi (I − Σ1i Σ1i ) (20) The matrices Zi , Ki and E are respectively given by Zi = X −1 Wi , Ki = X −1 Ri and E = X −1 S. Proof 4. From lemmas 1, 2 and their respective proves; replacing Ni with its value (Ai − Zi Bi ) and by the change of variables Wi = XZi , Ri = XKi , S = XE and left multiplying (15) by the matrix X, the proof of theorem 5 is straightforwardly obtained. To summarize the proposed procedure, the following design algorithm can be carried out for the design of a polytopic functional observer. Algorithm 1 Polytopic functional observer design for systems with unknown inputs Step 1) Verify if condition (13) is satisfied. Step 2) From (17), define the matrices Ai and Bi Step 3) Solve the LMIs (19) under the constraints (20) and deduce the gains X, Wi , Ri and S. Step 4) From the LMIs solution, deduce the observer gains given by (21). The polytopic functional observer design is thus completed. The observer gains given by: Ni = Ai − X −1 Wi Bi E = X −1 S Ji = Ki + Ni E = X −1 Ri + Ni E Hi = P Bi = (L − EC)Bi

(21a) (21b) (21c) (21d)

2.2 Multiple Polytopic Lyapunov Matrix In order to introduce more relaxation in the design procedure, a multiple polytopic Lyapunov matrix is considered: M  −1 V (ea (t)) = eT (t) e(t) (22) µk (ξ(t))Xk k=1

where M 

and

µk (ξ(t)))Xk

k=1 M 

−1

=

M 

k=1

µk (ξ(t)))Xk

−T

≥0

(23)

ˆ µk (ξ(t))) is required to be at least C1 . This re-

k=1

quirement is satisfied for T-S polytopic models constructed via a sector nonlinearity approach (SnT) if the original nonlinear system is at least C1 (Tanaka and Wang [2001]), (Tanaka et al. [2007]). Please note that (23) holds if XkT = Xk ≥ 0 (Wang et al. [2000]). Lemma 3. Under conditions (13), there exists an asymptotic stable polytopic functional observer of the form (2) for the system (1), if there exists a symmetric positive matrices Xi = XiT > 0 solution of the inequalities: M  − βk Xk + Xi NjT + Nj Xi < 0 (24) k=1

for i, j = 1, . . . , M

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Proof 5. The Lyapunov function derivative is given by: M  −1 d V˙ (e(t)) = eT (t) dt µk (ξ(t)))Xk e(t) +e˙ T (t) +eT (t)

M 

k=1

µk (ξ(t)))Xk

k=1 M 

µk (ξ(t)))Xk

k=1

−1 −1

e(t) ˙

µ˙ k (ξ(t)))Xk

k=1

M 

k=1

µk (ξ(t)))Xk

k=1

From (26), (25) becomes:

V˙ (e(t)) = −eT (t) M 

µ˙ k (ξ(t)))Xk

k=1 T

+e˙ (t) +eT (t)

M 

k=1 M 

µk (ξ(t)))Xk

k=1

M 

M 

−1

µk (ξ(t)))Xk

k=1

µk (ξ(t)))Xk

−1

e(t) ˙

µk (ξ(t)))Xk

−1

k=1

e(t) (27)

e(t)

−1

µk (ξ(t)))Xk

(26)

−1

−1

Knowing (23) and from (8), we obtain: M   −T µk (ξ(t)))Xk V˙ (e(t)) = eT (t) − M 

k=1

µ˙ k (ξ(t)))Xk

k=1 M 

+ +

M 

µj (ξ(t))NjT

j=1 M 

µk (ξ(t)))Xk

k=1 M 

µk (ξ(t)))Xk

k=1 M −T 

(28)

k=1

µk (ξ(t)))Xk , we obtain:

k=1 M   µ˙ k (ξ(t)))Xk V˙ (e(t)) = eT (t) −

+ +

M  M 

i=1 j=1 M  M  i=1 j=1

k=1

µi (ξ(t))µj (ξ(t))Xi NjT

 µi (ξ(t))µj (ξ(t))Nj Xi e(t)

(30)

Remark 2. In some practical applications, it may not be easy to select βk to satisfy the above assumption (µ˙ k (ξ(t)) ≥ βk ), ∀t and k = 1, . . . , M . For these cases where it is difficult to select a proper βk , one can resort to a small value for βk ; of course extremely small values can lead to some conservative results. However, the obtained conditions always guarantee less conservatism than those deduced from a single common Lyapunov function. Then, the following theorem is proposed: Theorem 2. Under conditions (13), there exists an asymptotic stable polytopic functional observer of the form (2) for the system (1), if there exists a symmetric positive matrices Xi = XiT > 0, solution of the following inequalities:  M   T  − βk Xk + Xi Aj + Aj Xi Zj Bj αXi     k=1  < 0 (31)  (Zj Bj )T −αI 0 

0 −αI αXi for i, j = 1, . . . , M and under the constraints: + (32) [ Ai − Zi Bi Ki E ] = Σ2i Σ+ 1i − Zi (I − Σ1i Σ1i ) where α is an arbitrary positive scalar to be fixed. Proof 6. Let us consider the following lemma (Bezzaoucha et al. [2013]): Lemma 4. Consider two matrices X and Y with appropriate dimensions, Σ and G symmetric positive definite matrices. The following property is verified −X T ΣX −Y T Σ−1 Y ≤ X T Y +Y T X ≤ X T GX +Y T G−1 Y (33)

−

M 

βk Xk + Xi ATj + Aj Xi + Xi GXi + Zj Bj G−1 BjT ZjT < 0

k=1

(34) Applying Schur’s complement and the congruence principle, by setting G = αI, (31) is straightforwardly obtained, which ends the proof.



Applying the congruence principle by left multiplying M  T (28) by µk (ξ(t)))Xk and right multiplying it by M 

βk Xk + Xi NjT + Nj Xi < 0

From lemmas 3 and 4, condition (30) becomes:

µj (ξ(t))Nj e(t)

j=1

k=1

−1

M 

k=1

(25)

e(t)

It is known that: M M  −1 −1  d µ (ξ(t)))X = − µk (ξ(t)))Xk k k dt k=1 M 

−

(29)

We assume that µ˙ k (ξ(t))) ≥ βk , ∀t and k = 1, . . . , M and from Xk ≥ 0, knowing the convex sum property of the weighting functions, then the stability condition for (29) is satisfied if:

To summarize the proposed procedure, the following design algorithm can be carried out for the design of a polytopic functional observer. Algorithm 2 Polytopic functional observer design for systems with unknown inputs Step 1) Verify if condition (13) is satisfied. Step 2) From (17), define the matrices Ai and Bi Step 3) Solve the LMIs (31) under the constraints (32) and deduce the gains Zi , E and Ki . Step 4) From the LMIs solution, deduce the observer gains given by (35). The polytopic functional observer design is thus completed. The observer gains given by: N i = Ai − Z i B i Ji = Ki + Ni E Hi = P Bi = (L − EC)Bi

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(35a) (35b) (35c)

Proceedings of the 20th IFAC World Congress Souad Bezzaoucha et al. / IFAC PapersOnLine 50-1 (2017) 9753–9759 Toulouse, France, July 9-14, 2017

 x˙   1       x˙ 2      x˙ 3    x˙ 4

3. ILLUSTRATIVE EXAMPLE In order to illustrate the efficiency of the proposed approach design, a nonlinear model of a quadrotor aerial robots landing on a moving platform is considered. Based on the work presented in (Voos and Bou-Ammar [2010]) for the nonlinear tracking and landing controller of quadrotor aerial robots, we consider in the following the part where the system should land on a moving platform. The platform is moving on the surface of the underlying terrain at an altitude of zs (t) with regard to the inertial frame. The overall tracking and landing procedure can be decomposed into two independent control tasks: a tracking procedure in a pure x − y-plane and an altitude control problem in pure z-direction. In the pure x − y-plane, only the planar mappings of the center of mass of the quadrotor and the platform and their respective motions are considered. The 2D-tracking controller has the task to reduce the planar distance between the quadrotor and the platform in this two-dimensional plane to zero and to maintain the zero distance even if disturbances occur (Voos and Bou-Ammar [2010]). For that purpose, we consider a platform that is moving with the two velocity components vP x and vP y in x- and y-direction, respectively. The quadrotor is moving with the two velocity components vQx and vQy , where the dynamics between the desired velocities vQxd , vQyd and the actual velocities is given by a first order system. The engagement geometry is depicted in Fig. 1, where σ is the line-of-sight angle and R is the distance or range between the quadrotor and the moving platform. It can be derived from classical missile guidance problems (Voos and Bou-Ammar [2010]).

Fig. 1. Engagement geometry of quadrotor and mobile platform The relative kinematics can be described by the two differential equations:   R˙ = vP x cos(σ) + vP y sin(σ) − vQx cos(σ) − vQy sin(σ) 

σ˙ =

1 R (vP y

cos(σ) − vP x sin(σ) −vQy cos(σ) + vQx sin(σ))

(36) T T For a state vector x = [ R σ vQx vQy ] with the input variable u1 = vQxd and u2 = vQyd and the two measurable disturbance variable d1 = vP x and d2 = vP y , the following state equations are obtained:

= −x3 cos(x2 ) − x4 sin(x2 ) +d1 cos(x2 ) + d2 sin(x2 ) = x11 (x3 sin(x2 ) − x4 cos(x2 ) −d1 sin(x2 ) + d2 cos(x2 )) = − T11 x3 + T11 u1 = − T12 x4 + T12 u2

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(37)

We recall that the considered system is a part of an overall control tracking and landing problem where T1 and T2 result of the assumption that the inner attitude control loops are sufficiently fast and could be approximated by a static system (see (Voos and Bou-Ammar [2010]) for more details). The overall vehicle control system consists of independent velocity control loops which can be approximated by linear first-order system such that 1 Vx (s) , i = 1, 2 (38) ≈ Vxd(s) T is + 1 Since the control part has been taken care in the reference (Voos and Bou-Ammar [2010]) and due to limitation space constraint, the control algorithm is simply given as (for proof and details, see (Voos and Bou-Ammar [2010])):  u1 = d1 + T1 x1 cos(x2 ) − T1 xx21 sin(x2 ) (39) u2 = d2 + T2 x1 cos(x2 ) + T2 xx21 cos(x2 )

In order to apply this 2D-tracking controller, the range x1 = R, the line-of-sight angle x2 = σ as well as the velocity components d1 = vP x , d2 = vP y of the platform must be measured. Both R and σ can be easily calculated if the positions of the quadrotor and the platform in the inertial frame are measured. In addition it is assumed that the platform also measures its velocity components. Both position and velocity components of the platform are transmitted via communication to the quadrotor, resulting in a cooperative approach. Regarding the measurements, a DGPS is applied for the determination of the positions, respectively, during the approach phase. However, more accurate measurements are necessary during the landing phase. There are some possible solutions for this problem such as a vision based or ultrasonic based sensor system. The velocity components of the platform could be measured with a suitable inertial measurement unit on-board. Since the main focus of this work is on the development of the observer system, we do not go into further details. Our objective is to synthesize a functional observer through the proposed approach in order to estimate the states x1 (t), x3 (t) and x4 (t). Let us first choose the following premise variables in order to write the nonlinear system (37) in a polytopic T-S form: 1 ξ1 (t) = cos(x2 ), ξ2 (t) = sin(x2 ), ξ3 (t) = (40) x1 (t) Remark 3. For numerical reasons, the third premise variable ξ3 (t) is defined for a Range x1 (t) = 0, if x1 (t) = 0, 1 then ξ3 (t) is set to be equal to 0.01 . The quasi-LPV model is then deduced x(t) ˙ = A(ξ)x(t) + Bu(t) + F (ξ)d(t) with   0 0 −ξ1 (t) −ξ1 (t)  0 0 ξ3 (t)ξ2 (t) −ξ3 (t)ξ1 (t)   A(ξ) =   0 0 −1/T1  0 0 0 0 −1/T2

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(41)

(42)

 0 0  0 0   B=  1/T1 0  0 1/T2 

The matrices C and L are defined as follow:     1000 1000 C= , L = 0 0 1 0 0100 0001

Velocity VQx (m/s)

and

(43)

(44)

Velocity VQy (m/s)

 ξ2 (t) ξ2 (t)  −ξ. (t)ξ2 (t) −ξ3 (t)ξ2 (t)   F (ξ) =    0 0 0 0 

Range (m)

Proceedings of the 20th IFAC World Congress 9758 Souad Bezzaoucha et al. / IFAC PapersOnLine 50-1 (2017) 9753–9759 Toulouse, France, July 9-14, 2017

(45)

1

z (t) system 1

z1(t) polytopic Lyapunov matrix

0.5

z (t) unique Lyapunov matrix 1

0

0

5

10

µi (ξ(t))(Ai x(t) + Bu(t) + Fi d(t))

(46)

i=1

z2(t) system z (t) polytopic Lyapunov matrix

0.2 0

2

z2(t) unique Lyapunov matrix

0

5

10

25

z3(t) system

0.2 0.1 0

z (t) polytopic Lyapunov matrix 3

z3(t) unique Lyapunov matrix

0

5

10

15

20

ξj,2 = min {ξj (x, u, d)} (47) x,u,d

 ξj (x, u, d) − ξj,2    Fj,1 (ξj (x, u, d)) = ξj,1 − ξj,2 ξj,1 − ξj (x, u, d)    Fj,2 (ξj (x, u, d)) = ξj,1 − ξj,2

25

0

5

10

(50)

Applying the proposed synthesis methods, the functional polytopic observer is designed for both single and multiple Lyapunov function. For space limitation reason, the numerical values of the observer gains are not given. The dynamic model has been implemented in Matlab for the simulative evaluation of the controlled system. The identification and control dynamic have been considered in (Voos and Bou-Ammar [2010]), we only present the results of the vehicle estimation dynamics. The system state z(t) = Lx(t) and their estimate for each proposed approach are depicted in figure 2.

0.04 0.02 0

20

25

unique Lyapunov matrix polytopic Lyapunov matrix

0

5

10

15

20

25

Time (s) e 3(t)

where the indexes σij (i = 1, . . . , 2q and j = 1, . . . , q) are equal to 1 or 2 and indicates which partition of the j th premise variable (Fj,1 or Fj,2 ) is involved in the ith submodel. The constant matrices Xi are obtained by replacing the variables ξj in the matrices A, B, C, and F with the scalars defined in (47):

15

Time (s)

i

X ∈ {A, B, C, F }

unique Lyapunov matrix polytopic Lyapunov matrix

0 -0.2 -0.4

(48)

For q = 3 premise variables, M = 2q = 8 sub-models are obtained. The weighting functions µi (t) are defined by: q (49) µi (t) = j=1 Fj,σ j (ξj (x, u, d))

Xi = X(ξ1,σi1 , . . . , ξq,σiq ),

20

Fig. 2. System state z(t) = Lx(t) and their estimate

e 1(t)

x,u,d

15

Time (s)

The estimation errors ei (t) = zi (t) − zˆi (t), i = 1, 2, 3 are depicted in figure 3.

with: ξj,1 = max {ξj (x, u, d)},

25

Time (s)

e 2(t)

x(t) ˙ =

20

0.4

From (41) and applying the SNT transformation, the following polytopic T-S model is obtained: 8 

15

Time (s)

unique Lyapunov matrix polytopic Lyapunov matrix

0.04 0.02 0

0

5

10

15

20

25

Time (s)

Fig. 3. Estimation error e(t) = z(t) − zˆ(t) As it can be seen, the state estimate zˆ(t) converges asymptotically to its real value z(t), which confirms the efficiency of the proposed approaches. It is also clear that the approach based on the polytopic multiple Lyapunov function is more efficient than the method based on the classical single Lyapunov matrix, which shows the relaxation effect introduced by the second approach. 4. CONCLUSION AND FUTURE WORKS In the present paper, sufficient conditions for the existence of a polytopic functional observer for nonlinear systems with unknown inputs are given. A new constructive and systematic algorithm for the observer design is proposed. Through an LMI based solution and under structural and

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rank constraints, a function of the state is estimated. Two solutions were proposed, the first one is based on a classical single Lyapunov function and the second one is based on a multiple Lyapunov function, which allows more relaxation and better efficiency of the proposed algorithm. The chosen application example is a quadrotor aerial robots landing system. From the nonlinear equations of the system, a polytopic T-S model is derived. The proposed observers are then synthesized following the proposed algorithms and the obtained results illustrate its performance. REFERENCES Bezzaoucha, S., Marx, B., Maquin, D., and Ragot, J. (2013). Stabilization of nonlinear systems subject to uncertainties and actuator saturation. In American Control Conference. Washington, DC, USA. Chadli, M., Maquin, D., and Ragot, J. (2002). Static output feedback for Takagi-Sugeno systems : an LMI approach. In 10th Mediterranean Conference on Control and Automation. Lisbon, Portugal. Darouach, M. (2000). Existence and design of functional observers for linear systems. IEEE Transaction on Automatic Control, 45(5), 940–943. Darouach, M., Zasadzinski, M., and Xu, S. (1994). Fullorder observers for linear systems with unknown inputs. IEEE Transactions on Automatic Control, 39(3), 606– 609. Ezzine, M., Darouach, M., Souley Ali, H., and Messaoud, H. (2011). Unknown inputs functional observers design for descriptor systems with constant time delay. In 18th IFAC World Congress. Guerra, T., Kruszewski, A., Vermeiren, L., and Tirmant, H. (2006). Conditions of output stabilization for nonlinear models in the Takagi-Sugeno’s form. Fuzzy Sets and Systems, 157(9), 1248 –1259. Kawamoto, S., Tada, K., Ishigame, A., and Taniguchi, T. (1992). An approach to stability analysis of second order fuzzy systems. In IEEE International Conference on Fuzzy Systems. San Diego, California, USA. Kruszewski, A., Wang, R., and Guerra, T.M. (2008). Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time ts fuzzy models: A new approach. IEEE Transactions on Automatic Control, 53, 606–611. Luenberger, D. (1971). An introduction to observers. IEEE Transactions on Automatic Control, 16(6), 596– 602. Nagpal, K., Helmick, R., and Sims, C. (1987). Reducedorder estimation. International Journal of Control, 45, 1867–1888. N’Doye, I., Darouach, M., Voos, H., and Zasadzinski, M. (2013). Design of unknown input fractional-order observers for fractional-order systems. International Journal of Applied Mathematics and Computer Science, 23(3), 491–500. Rao, C. and Mitra, S. (1971). Generalized Inverse of Matrices and its Applications. Wiley, New York, USA. Sename, O., Gaspar, P., and Bokor, J. (2013). Lecture Notes in Robust Control and Linear Parameter Varying approaches: Application to Vehicle Dynamics. Tanaka, K., Ohtake, H., and Wang, H. (2007). A descriptor system approach to fuzzy control system design via

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fuzzy Lyapunov functions. IEEE Transactions on Fuzzy Systems, 15(3), 333–341. Tanaka, K. and Wang, H. (2001). Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach. John Wiley & Sons, Inc. Tuan, H.D., Apkarian, P., Narikiyo, T., and Yamamoto, Y. (2001). Parameterized Linear Matrix Inequality techniques in fuzzy control system design. IEEE Transactions on Fuzzy Systems, 9(2), 324–332. Voos, H. and Bou-Ammar, H. (2010). Nonlinear tracking and landing controller for quadrotor aerial robots. In IEEE International Conference on Control Applications (CCA). Yokohama. Wang, H.O., Li, J., Niemann, D., and Tanaka, K. (2000). T-S fuzzy model with linear rule consequence and PDC controller: A universal framework for nonlinear control systems. In 9th IEEE International Conference on Fuzzy Systems, 549–554. San Antonio, USA. Watson, J. and Grigoriadis, K. (1998). Optimal unbiased filtering via linear matrix inequalities. System and Control letters, 35, 111–118. Yoneyama, Y. (2006). Output feedback stabilization of fuzzy systems with unobservable premise variables. In International Conference on System of Systems Engineering. Los Angeles, CA, USA.

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