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On Switched Controller Design for Robust Control of Uncertain Polynomial Nonlinear Systems Using Sum of Squares
9th IFAC Symposium on Robust Control Design 9th IFAC Symposium on Robust Control Design Florianopolis, Brazil, September 3-5, 2018 9th IFAC Symposium on Robust Control Designonline at www.sciencedirect.com Available Florianopolis, Brazil, September 3-5, 2018 9th IFAC Symposium on Robust Control Florianopolis, Brazil, September 3-5, 2018Design Florianopolis, Brazil, September 3-5, 2018
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IFAC PapersOnLine 51-25 (2018) 269–274
On Switched Controller Design for Robust On On Switched Switched Controller Controller Design Design for for Robust Robust On Switched Controller Design for Robust Control of Uncertain Polynomial Nonlinear Control of Uncertain Polynomial Nonlinear Control of Uncertain Polynomial Nonlinear Control of Uncertain Polynomial Nonlinear Systems Using Sum of Squares Systems Using Sum of Squares Systems Using Sum of Squares Systems Using Sum of Squares∗∗∗ ∗
Igor T. M. Ramos ∗ Uiliam N. L. T. Alves ∗∗∗ Igor T. M. Ramos Uiliam N. T. Alves ∗∗∗ ∗∗∗ ∗ Igor R. T. de M.Oliveira Ramos ∗∗∗∗,∗∗ Uiliam N. L. L. T. Alves Diogo Marcelo C. M. Teixeira ∗,∗∗ ∗∗∗ ∗ Diogo R. de Oliveira Marcelo C. M. Teixeira ∗,∗∗ ∗ Igor T. M. Ramos Uiliam N. L. T. Alves ∗,∗∗ ∗ ∗ ∗ Diogo R. de Oliveira Marcelo C. M. Teixeira ∗ Erica ∗ ∗ Rodrigo Cardim R. M. D. Machado ∗,∗∗ Rodrigo Cardim Erica R. M. D. Machado ∗ ∗ Diogo R. de Oliveira Marcelo C. M. Teixeira ∗ ∗ ∗ Rodrigo Cardim R. M. D. Machado ∗ Edvaldo Assunção ∗ ∗ Erica Edvaldo Assunção ∗ Rodrigo Cardim Erica R. M. D. Machado ∗ Edvaldo Assunção Edvaldo Assunção ∗ ∗ ∗ São Paulo State University (UNESP), School of Engineering, Ilha ∗ São Paulo State University (UNESP), School of Engineering, Ilha ∗ SãoSolteira Paulo -State University (UNESP), School of Engineering, Ilha SP, Brazil, of Electrical Engineering, ∗ -State SP, University Brazil, Department Department of School Electrical Engineering, Ilha SãoSolteira Paulo (UNESP), of Engineering, Solteira SP, Brazil, Department of Electrical Engineering, José Carlos Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo José Carlos- Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo Solteira SP, Brazil, Department of Electrical Engineering, José Carlos [email protected], Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo (e-mail: [email protected], (e-mail: [email protected], [email protected], José(e-mail: Carlos Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]). [email protected]). [email protected], [email protected], ∗∗ ∗∗ Federal Institute [email protected]). of of Education, Education, Science Science and and Technology Technology of of Mato Mato ∗∗ Federal Institute [email protected]). ∗∗ Federal Institute of Education, Science and Technology of Mato Grosso do Sul (IFMS), Angelo Melão St, 790, 79641-162, ∗∗ Grosso do Sul (IFMS), Angelo Melão St, 790, 79641-162, Federal Institute ofBrazil Education, and Technology of Mato Grosso do -Sul (IFMS), AngeloScience Melão St, 790, 79641-162, Três Lagoas MS, (e-mail: [email protected]) Lagoas --Sul MS, Brazil (e-mail: [email protected]) Grosso do (IFMS), Angelo Melão St, 790, 79641-162, ∗∗∗ Três Três Lagoas MS, Brazil (e-mail: [email protected]) Federal Institute Institute of of Education, Education, Science Science and and Technology Technology of of Paraná Paraná ∗∗∗ Federal ∗∗∗ Três Lagoas -Dr. MS,of Brazil (e-mail: [email protected]) ∗∗∗ Institute Education, Science and Technology of Paraná (IFPR), Tito Ave, Jardim Panorama, 86400-000, ∗∗∗ Federal (IFPR), Dr. Tito Ave, Jardim Panorama, 86400-000, Federal Institute of Education, Science and Technology of Paraná (IFPR), Dr. Tito Ave, (e-mail Jardim Panorama, 86400-000, Jacarezinho -- PR, Brazil [email protected]) Jacarezinho (IFPR), Dr. TitoBrazil Ave, (e-mail Jardim [email protected]) Panorama, 86400-000, Jacarezinho - PR, PR, Brazil (e-mail [email protected]) Jacarezinho - PR, Brazil (e-mail [email protected])
Abstract: Abstract: Abstract: This manuscript manuscript presents presents aa switched switched control control design design for for robust robust control control of of aa class class of of uncertain uncertain This Abstract: This manuscript a switched control design control of awhose class dynamics of uncertain nonlinear systems.presents In particular, particular, it is is considered considered a class classfor of robust nonlinear systems whose dynamics are nonlinear systems. In it a of nonlinear systems are This manuscript presents a switched control control awhose class of uncertain nonlinear systems. In particular, it is considered a class of robust nonlinear dynamics are described by polynomial polynomial functions, that dependdesign on thefor state vector ofsystems the of system, with uncertain described by functions, that depend on the state vector of the system, with uncertain nonlinear systems. In particular, it can is considered a class ofterms nonlinear whosewith dynamics are described byThe polynomial functions, that depend on the in state vector ofsystems the uncertain parameters. The design procedure be represented in of sum sum of system, squares. The proposed parameters. design procedure can be represented terms of of squares. The proposed described by polynomial functions, that depend on the state vector of the system, with uncertain parameters. The design procedure can be represented in terms of sum of squares. The proposed switched control control procedure procedure uses uses aa switching switching law law to to select select at at each each instant instant of of time time aa state-feedback state-feedback switched parameters. The procedure design procedure can in at terms sum ofofsquares. Theprocedure. proposed switched control uses ato law toobtained select eachofthe instant timedesign a state-feedback polynomial gain, which belongs belongs toswitching a set setbeof ofrepresented gains obtained from the proposed design polynomial gain, which a gains from proposed procedure. switched control procedure uses a switching law to select at each instant of time a state-feedback polynomial which belongs a set of gains obtained frompolynomial the proposed design The aim aim of of gain, the switching switching law is istoto to choose state-feedback polynomial gain that procedure. minimizes The the law choose aa state-feedback gain that minimizes polynomial gain, which belongs toto a Lyapunov set of gains obtainedAfrom the proposed design procedure. The aim of the switching law is choose a state-feedback polynomial gain that minimizes the time derivative of a polynomial function. numerical example, using chaotic the time derivative of a polynomial function. A numerical example, usingminimizes aa chaotic The aimsystem of thewith switching law is to Lyapunov choose a illustrates state-feedback polynomial gain that the time derivative of a polynomial Lyapunov function. the A numerical example, using approach. a chaotic Lorenz parametric uncertainties, validity of the proposed Lorenz with of parametric uncertainties, validity ofexample, the proposed the timesystem derivative a polynomial Lyapunovillustrates function. the A numerical using approach. a chaotic Lorenz system with parametric uncertainties, illustrates the validity of the proposed approach. Lorenz with parametric uncertainties, illustrates the validity of theLtd. proposed approach. © 2018, system IFAC (International Federation of Automatic Control) Hosting by Elsevier All rights reserved. Keywords: Sum Sum of of squares squares (SOS), (SOS), Robust Robust control, control, Uncertain Uncertain nonlinear nonlinear systems, systems, Switched Switched Keywords: Keywords: Sum of squares (SOS), Robust control, Uncertain nonlinear systems, Switched systems, Polynomial Polynomial systems. systems, systems. Keywords: Sum of squares (SOS), Robust control, Uncertain nonlinear systems, Switched systems, Polynomial systems. systems, Polynomial systems. nonlinear systems whose dynamics are described by by polypoly1. INTRODUCTION INTRODUCTION described 1. nonlinear systems systems whose whose dynamics dynamics are are described nonlinear byofpoly1. INTRODUCTION nomial functions, that depend on the state vector the nomial functions, functions, that depend on the the state vector ofpolythe nonlinear systems whosedepend dynamics are described bycontrol 1. INTRODUCTION nomial that on state vector of the system, namely polynomial nonlinear systems. The The control design for nonlinear systems is an important system, namely polynomial nonlinear systems. The control The control design for nonlinear systems is an important nomial functions, that depend on the state vector of the namely polynomialusing nonlinear systems. The control design be the of theory, The design forcontrol nonlinear systems is et anal., important topiccontrol of the the nonlinear nonlinear theory (Prajna 2004b). system, design can can be approached approached using the sum sum of squares squares theory, topic of control theory (Prajna et al., 2004b). system, namely polynomial nonlinear systems. Theorigin control The control design for nonlinear systems is an important design can be approached using the sum of squares theory, and it is based on conditions to assure that the is topic of the nonlinear control theory (Prajna et al., 2004b). There are several approaches to deal with this issue, conand it is based on conditions to assure that the origin is Thereofare several approaches to deal(Prajna with this issue, con- design be approached usingasymptotically sum that of squares theory, topic the nonlinear control theory et al., 2004b). and it can is based on conditions tothe assure the origin is an equilibrium point globally stable, even There are several approaches to deal with this issue, considering different classes of dynamic systems and many an equilibrium point globally asymptotically stable, even sidering different classes of dynamic systems and many and it is based on conditions to assure that the origin is There aredifferent several approaches to2001; deal with this con- an equilibrium point globally(Prajna asymptotically stable,Chesi, even with polytopic et sidering classes of al., dynamic systems and many performance indexes (Lee et et al., Prajna et issue, al., 2004b; 2004b; with polytopic uncertainties uncertainties (Prajna et al., al., 2004b; 2004b; Chesi, performance indexes (Lee 2001; Prajna et al., an equilibrium point Huang globallyet asymptotically stable, even sidering different classes of dynamic systems and many with polytopic uncertainties (Prajna et al., 2004b; Chesi, 2004; Xu et al., 2009; al., 2013). Besides that, the performance indexes (Lee et al., 2001; Prajna et al., 2004b; Alves et et al., al., 2016; 2016; de de Oliveira Oliveira et et al., al., 2018). 2018). The The reprerepre- with 2004; polytopic Xu et al., uncertainties 2009; Huang et al., 2013). Besides that, the Alves (Prajna et al., 2004b; Chesi, performance indexes (Lee et al., 2001; Prajna et al., 2004b; 2004; Xu et al., 2009; Huang et al., 2013). Besides that, the design procedure can be developed considering polynomial Alves et al., 2016; de Oliveira et al., 2018). The representation of of the the nonlinear nonlinear system system using using Takagi-Sugeno Takagi-Sugeno 2004; design procedure can be developed considering polynomial sentation Xu etfunctions al., 2009; Huang et al., 2013). Besides that, the Alves et al., 2016;nonlinear de et al., 2018). The repredesign procedure canand be developed considering polynomial Lyapunov matrices polynomial terms, sentation of models the system using Takagi-Sugeno (T-S) fuzzy fuzzy models is aaOliveira well-established methodology to design Lyapunov functions and matrices with with polynomial terms, (T-S) is well-established methodology to procedure can be developed considering polynomial sentation of the nonlinear system using Takagi-Sugeno Lyapunov functions and matrices with polynomial terms, which the theory an methodology (T-S) fuzzy a well-established methodology to Lyapunov deal with with themodels controlisproblem problem class of of nonlinear nonlinear systems which makes makes the SOS SOS theory an attractive attractive methodology deal the control aa class systems functions andtheory matrices with polynomial terms, (T-S) fuzzy a well-established methodology to which makes the SOS an attractive methodology for the design of controllers (Prajna et al., 2004b; Tanaka deal with themodels controlis1985). problem a class of nonlinear systems (Takagi and Sugeno, The T-S fuzzy model makes it for the design of controllers (Prajna et al., 2004b; Tanaka (Takagi and Sugeno, 1985). The T-S fuzzy model makes it which makes the SOS theory an attractive methodology deal with the control problem a class of nonlinear systems the 2009). design of controllers (Prajna al., 2004b; Tanaka et al., Finally, Tanaka et al. (2016) presented a (Takagi Sugeno, the 1985). The T-S model makes it for possible and to represent represent the dynamics of fuzzy the nonlinear nonlinear system et al., al., 2009). Finally, Tanaka et al. al.et (2016) presented possible to dynamics of the system the design of controllers (Prajna et al., 2004b; Tanaka (Takagi and Sugeno, the 1985). The T-S fuzzy model makes it for et 2009). Finally, Tanaka et (2016) presented aa new SOS design for robust control of polynomial fuzzy possible to represent dynamics of the nonlinear system by a convex combination of local linear models. This reprenewal.,SOS design for robust control of(2016) polynomial fuzzy by a convex combination of local linear models. Thissystem repre- et 2009). Finally, Tanaka etconsidering al. of presented possible tocan represent dynamics of athe nonlinear SOS design for robust control polynomial fuzzya systems with uncertainties and, an uncertain by a convex combination of local linear models. repre- new sentation be done donethe exactly within region ofThis operation systems with uncertainties and, considering an uncertain sentation can be exactly within a region of operation new SOS design for robust control of polynomial fuzzy by a convex combination of local linear repre- systems withfuzzy uncertainties and, considering an uncertain polynomial system to and sentation can be exactly within a models. region ofThis operation (Taniguchi et al.,done 2001). Usually, using this methodology, methodology, polynomial fuzzy system subject subject to disturbance, disturbance, and Yu Yu (Taniguchi et al., 2001). Usually, using this systems withfuzzy uncertainties and, considering an uncertain sentation can be done exactly within a region of operation polynomial system subject to disturbance, and Yu et al. (2016) proposed conditions for a robust H control ∞ (Taniguchi et al., 2001). Usually, using this methodology, the design design procedure procedure is is given given by by Linear Linear Matrix Matrix Inequalities Inequalities et al. (2016) proposed conditions for a robust H control ∞ the polynomial fuzzy system subject to disturbance, and Yu (Taniguchi et al., 2001). Usually, usingMatrix this methodology, et al. (2016) proposed conditions for a robust H∞ control design in of the design procedure is given by Linear Inequalities et (LMIs). design in terms terms of SOS. SOS.conditions for a robust H∞ (LMIs). al. (2016) proposed ∞ control the design procedure is given by Linear Matrix Inequalities design in terms of SOS. (LMIs). The less conservative design procedure is an aim for the design in terms of SOS. In Parrilo (2000) it is presented a methodology for the (LMIs). design is for the In Parrilo (2000) it is presented a methodology for the The The less less conservative conservative design procedure procedure is an anInaim aim for the development of recent control methodologies. this sense, In Parrilo (2000) it is presented a methodology for the sum of squares (SOS) decomposition for multivariable development of recent control methodologies. In this sense, sum of squares (SOS) decomposition for multivariable The less conservative design procedure is an aim for the In Parrilo (2000) it is presented a methodology for the development of recent control methodologies. In this sense, the switched control is a technique that is widely applied to sum of squares (SOS) decomposition for multivariable polynomials. In this this procedure, it is is considered considered class of of the switched control is control a technique that is widely applied to polynomials. In procedure, it aa class development of recent methodologies. In this sense, sum of squares (SOS) decomposition for multivariable polynomials. In this procedure, it is considered a class of the switched control is a technique that is widely applied to switched control is a technique that is widely applied to polynomials. In IFAC this procedure, is considered a class of the 2405-8963 © 2018, (InternationalitFederation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © under 2018 IFAC 380 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 380 Copyright © 2018 IFAC 380 10.1016/j.ifacol.2018.11.117 Copyright © 2018 IFAC 380
IFAC ROCOND 2018 270 Florianopolis, Brazil, September 3-5, 2018Igor T.M. Ramos et al. / IFAC PapersOnLine 51-25 (2018) 269–274
address this issue. Souza et al. (2013) proposed a switched control design for linear time-invariant systems with polytopic uncertainties where the switched control law uses the state vector to select a state-feedback gain, at each instant of time. In Souza et al. (2014) a switched control design for a class of uncertain nonlinear systems represented by T-S fuzzy models, with unknown membership functions was proposed. In addition, the procedure presented in Alves et al. (2016) proposes a smoothing switched control law, to deal with the local stability problem and control signal saturation. An example presented in Alves et al. (2016) showed that the switched controller can provide an improvement in the performance of the controlled system when compared to the closed-loop system with a timeinvariant controller. Recently, in de Oliveira et al. (2018) it was proposed a local H∞ switched controller design for uncertain nonlinear T-S fuzzy systems subject to external disturbances. A comparative example and a practical implementation, using an active suspension system, illustrate the advantages of the switched control design. This work proposes a switched control design for robust control of uncertain polynomial nonlinear systems. The design procedure is represented in terms of SOS. The proposed switched control methodology uses a switching law to select at each instant of time a state-feedback polynomial gain, which belongs to a set of gains obtained from the design technique. The aim of the switching law is to choose a state-feedback polynomial gain that minimizes the time derivative of a polynomial Lyapunov function. A numerical example, using a chaotic Lorenz system with parametric uncertainties, illustrates the validity of the proposed approach. The remaining of the manuscript is organized as follows: In Section 2 some terminologies are defined concerning the SOS theory. Section 3 introduces the polynomial nonlinear systems with polytopic uncertainties. The proposed switched control design for robust control of uncertain polynomial nonlinear systems is presented in Section 4. In Section 5, a numerical example illustrates the validity of the proposed approach. Finally, Section 6 draws the conclusions and presents the future perspectives. In this manuscript, the following notation will be adopted: Kr = {1, 2, . . . , r}, r ∈ N, x(t) = x and V (x(t)) = V . Finally, consider for an uncertain constant vector α = [α1 α2 . . . αr ]T , r ∑ αi (Ai (x), Bi (x), Ki (x)), (A(x, α), B(x, α), K(x, α)) = i=1
αi ≥ 0,
∀i ∈ Kr ,
r ∑
αi = 1,
(1)
i=1
where r = 2s and s is the number of distinct uncertain parameters in A(x) and B(x). 2. SUM OF SQUARES In this section it is defined some terminologies concerning the SOS theory. A multivariable polynomial p(x(t)), x(t) ∈ Rn , with degree 2d, is a finite combination of monomials, with real coefficients, in which the higher degree of the monomials in this combination is 2d. In addition, a monomial of x(t) is a product among xdi i (t), i = 1, 2, · · · , n, 381
where di is a nonnegative integer. ∑nIn this case, the degree of xd11 (t)xd22 (t) . . . xdnn (t) is d = i=1 di (Lam et al., 2013; Parrilo, 2000).
A polynomial p(x) with degree 2d is an SOS exist ∑if there 2 q (x(t)). polynomials qj (x(t)) such that p(x(t)) = j=1 j Therefore, being p(x(t)) an SOS, then p(x(t)) ≥ 0. An important result, as pointed out in Parrilo (2000), is that the polynomial p(x(t)) is an SOS if, and only if, there exist a semidefinite matrix Q such that x ˆT (x(t))Qˆ x(x(t)), where x ˆ(x(t)) is a column vector whose elements are all monomials in x(t) with degree no greater than d. In this case, the problem of finding this matrix Q can be formulated as a semidefinite problem (Tanaka et al., 2009). In this manuscript, the authors have been used the SeDuMi solver (Sturm, 1999) and the SOSTOOLS toolbox (Prajna et al., 2004a) with the software MatLab in order to solve the SOS conditions. 3. NONLINEAR SYSTEMS WITH POLYTOPIC UNCERTAINTIES
Consider a polynomial uncertain nonlinear system, whose dynamics is described only by polynomials functions of the state variables of the system and its uncertainties have a polytopic representation, such that this nonlinear system can be described by x˙ = A(α, x)ˆ x(x) + B(α, x)u(t), (2) where x ∈ Rn is the state vector, x ˆ(x) ∈ RN ×1 is a vector whose entries are monomials of the state vector x, u ∈ Rm is the input vector, A(α, x) ∈ Rn×N and B(α, x) ∈ Rn×m are described in (1) are known, where i ∈ Kr . Consider the auxiliary control law, defined as follows r ∑ αi Ki (x)ˆ x(x) = −K(α, x)ˆ x(x), (3) u(t) = uα (t) = i=1 m×N
where Ki (x) ∈ K , i ∈ Kr . From (2) and (3), one has x˙ = A(α, x)ˆ x(x) − B(α, x)K(α, x)ˆ x(x) r r ∑∑ = αi αj (Ai (x) − Bi (x)Kj (x))ˆ x(x). (4) i=1 j=1
Remark 1. Note that the auxiliary control law (3) can not be implemented for controlling the plant, since it depends on the uncertain parameters αi , i ∈ Kr . However, it plays an important rule in the understanding of the switched control law, which does not depend on the αi , as it will be see in the Section 4 (Souza et al., 2013). Before proceeding further with the stability analysis of the closed-loop system, some definitions will be set. Let T (x) ∈ Rn×N be the Jacobian matrix of x ˆ(x), whose entries (i, j) of the polynomial matrix T (x) are given by ∂x ˆi T ij (x) = (x), (5) ∂xj for all i ∈ KN , j ∈ Kn . To lighten the notation, let Ak (x) denotes k-th row of the matrix A(x), and let κ = {k1 , k2 , . . . , km } be the set of row indexes of B(x) whose corresponding rows are identically equal to zero. x ˜ denotes a column vector in which each entry is a monomial in x(t) T such that x ˜ = [xk1 xk2 · · · xkm ] , ki ∈ κ.
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Considering the auxiliary control law (3), the next theorem follows from Yu and Huang (2013), which considers a decay rate equal to or greater than β ≥ 0. Theorem 1. Consider the closed-loop system (4). Assume that there exist a symmetric polynomial matrix X(˜ x) ∈ RN ×N , polynomial matrices Mi (x) ∈ Rm×N and a scalar β ≥ 0 such that the conditions x) − ϵ1 (x))υ is SOS, (6) υ T (X(˜ ( −υ T T (x)Ai (x)X(˜ x) − T (x)Bi (x)Mj (x) + X(˜ x)ATi (x)T T (x) − MjT (x)BiT (x)T T (x) + T (x)Aj (x)X(˜ x) − T (x)Bj (x)Mi (x)
+ X(˜ x)ATj (x)T T (x) − MiT (x)BjT (x)T T (x) ∑ ∂X ( ) − (˜ x) Aki (x)ˆ x(x) ∂xk k∈κ ∑ ∂X ( ) (˜ x) Akj (x)ˆ x(x) − ∂xk k∈κ ) x) υ is SOS, ∀ i ≤ j, + ϵ2ij (x)I + 4βX(˜
( x) + X(˜ x)ATi (x)T T (x) −υ T T (x)Ai (x)X(˜
− T (x)B(x)Mi (x) − MiT (x)B T (x)T T (x) ∑ ∂X (˜ x)(Aki (x)ˆ x(x)) − ∂xk k∈κ ) (11) + 2βX(˜ x) + ϵ2i (x)I υ is SOS,
hold for all i ∈ Kr , where ϵ1 (x) > 0 for x ̸= 0 and ϵ2i (x) ≥ 0 for all x; υ ∈ RN is a vector that is independent of x and T (x) ∈ RN ×n is a polynomial matrix given in (5). Then the origin x = 0 is a stable equilibrium point for the closed-loop system (9), where the polynomial gains are given by Ki (x) = Mi (x)X −1 (˜ x), i ∈ Kr .
In addition, if (11) holds with ϵ2i (x) > 0 ∀ x ̸= 0 is an asymptotically stable equilibrium point, with a decay rate equal to or greater than β ≥ 0 for the closed-loop system (9). If X(˜ x) is a constant matrix, then stability properties of the equilibrium point hold globally.
(7)
hold for all i ∈ Kr and i ≤ j ∈ Kr , where ϵ1 (x) > 0 and ϵ2ij (x) ≥ 0, for all x ̸= 0, are nonnegative polynomials; υ ∈ RN is a vector that is independent of x and T (x) ∈ RN ×n is a polynomial matrix given in (5). Then the origin x = 0 is a stable equilibrium point for the closedloop system (4), where the polynomial gains are given by Ki (x) = Mi (x)X −1 (˜ x), i ∈ Kr . In addition, if (7) holds with ϵ2ij (x) > 0 ∀ x ̸= 0, then the origin x = 0 is an asymptotically stable equilibrium point, with a decay rate equal to or greater than β ≥ 0, for the closed-loop system (4). If X(˜ x) is a constant matrix, then the stability properties of the equilibrium point hold globally.
Proof. The proof can be found in Yu and Huang (2013), where it is shown if the conditions in the Theorem 1 hold with ϵ2ij (x) > 0. Then, considering a polynomial Lyapunov function candidate Vuα (x) = x ˆT (x)X −1 (˜ x)ˆ x(x), along the state trajectory of the closed-loop system (4) one has V˙ uα + 2βVuα < 0 ∀ x ̸= 0.
As a special case, assume that B(α, x) = B(x) is a known polynomial matrix, in other words, B(x) is a function of the state variables of the system and is not a function of the system uncertainties. Then, the system (2) becomes x˙ = A(α, x)ˆ x(x) + B(x)u. (8)
In this case, the closed-loop system (8) and (3) is x˙ = A(α, x)ˆ x(x) − B(x)K(α, x)ˆ x(x) r ∑ αi (Ai (x) − B(x)Ki (x))ˆ x(x) =
271
Proof. The proof follows from the proof of Theorem 1, considering B(α, x) = B(x). 4. SWITCHED CONTROL OF UNCERTAIN POLYNOMIAL NONLINEAR SYSTEMS 4.1 Case 1: Uncertain polynomial nonlinear system with known matrix B(α, x) = B(x) The switched control law selects a state-feedback polynomial gain Kσ (x), which belongs to the set of gains {Ki (x) ∈ Rm×N , i ∈ Kr }. The switched control law is defined by x(x), u = uσ = −Kσ (x)ˆ { T } ∗ σ = arg min −ˆ x (x)X −1 (˜ x)T (x)B(x)Ki (x)ˆ x(x) , i∈KN
(12) { T } where arg min ∗ −ˆ x (x)X −1 (˜ x)T (x)B(x)Ki (x)ˆ x(x) dei∈KN
notes the smallest index σ ∈ Kr , such that
x)T (x)B(x)Kσ (x)ˆ x(x) −x ˆT (x)X −1 (˜ { T } ∗ −1 = min −ˆ x (x)X (˜ x)T (x)B(x)Ki (x)ˆ x(x) . i∈KN
(13)
At this point, it will be considered the closed-loop system (8) and (12), whose the system dynamics is given by x˙ = A(α, x)ˆ x(x) + B(x)uσ r ( ) ∑ = αi Ai (x) − B(x)Kσ (x) . (14) i=1
(9)
i=1
and, from Theorem 1, it follows the next Corollary. Corollary 1. Consider the closed-loop system (9). Assume that there exist a symmetric polynomial matrix X(˜ x) ∈ RN ×N , polynomial matrices Mi (x) ∈ Rm×N and a scalar β ≥ 0 such that the conditions x) − ϵ1 (x))υ is SOS, (10) υ T (X(˜ 382
Theorem 2. Suppose that the conditions of Corollary 1 with a constant matrix X(˜ x) and ϵ2ij (x) > 0, x ̸= 0, hold for the system (8) with the control law (3). Then, the switched control law (12), where the polynomial gains are given by Ki (x) = Mi (x)X −1 (˜ x), i ∈ Kr , makes the equilibrium point x = 0 globally asymptotically stable for the closed-loop system (14). Proof. Consider a polynomial Lyapunov function candix)ˆ x(x). Then, define V˙ uα (x) and date V (x) = x ˆT (x)X −1 (˜ V˙ uσ (x) as the time derivatives of V (x) for the system (8) with the control laws (3) and (12), respectively.
IFAC ROCOND 2018 272 Florianopolis, Brazil, September 3-5, 2018Igor T.M. Ramos et al. / IFAC PapersOnLine 51-25 (2018) 269–274
Consequently, from (14), xT (x)X −1 (˜ x)x ˆ˙ + x ˆT (x)X˙ −1 (˜ x)ˆ x(x) V˙ uσ (x) = 2ˆ
T
+x ˆ (x)
= 2ˆ xT (x)X −1 (˜ x)T (x) [A(α, x) − B(x)Kσ (x)] x ˆ(x) ( ) ∑ ∂X −1 +x ˆT (x) (˜ x)x˙ k x ˆ(x). (15) ∂xk k∈κ
Since Bik (x) = 0 for all k ∈ κ and x ∈ Rn , from (14), it follows that r ∑ ( ) x˙ k = αi (x) Aki (x)ˆ x(x) , (16) i=1
/ κ, and for all k ∈ κ. On the other hand, considering xi , i ∈ it is aware that the symmetric polynomial matrix X(˜ x) does not contain such state variable, thus, ∂X −1 (˜ x) = 0. (17) ∂xi Therefore, from (15), (16) and (17), one has xT (x)X −1 (˜ x)T (x) [A(α, x) − B(x)Kσ (x)] x ˆ(x) V˙ uσ (x) = 2ˆ ) ( n ∑ ∂X −1 ) ( +x ˆT (x) x(x) x ˆ(x) (˜ x) Ak (α, x)ˆ ∂xk = 2ˆ xT (x)X
= V˙ uα (x).
(
n ∑ ∂X −1
k=1
∂xk
(
k
(˜ x) A (α, x)ˆ x(x)
)
)
x ˆ(x)
Therefore, V˙ uσ (x) ≤ V˙ uα (x). As the conditions of Corollary 1 ensure that V˙ uα (x) < 0 for all x ̸= 0, the proof is concluded. Remark 2. Theorem 2 shows that if the conditions of Corollary 1 hold with ϵ2ij (x) > 0 ∀ x ̸= 0, then V˙ uα (x) < 0 for all x ̸= 0 and, therefore, V˙ uσ (x) < 0 for all x ̸= 0. In this case, the equilibrium point x = 0 is globally asymptotically stable for the closed-loop system (8) and (12). Thus, Corollary 1 can be used to design the polynomials gains K1 (x), K2 (x), · · · , Kr (x) and the matrix X −1 (˜ x) from the switched control law (12). Moreover, the switched control law (12) does not use the uncertain parameters αi , i ∈ Kr to compose the control signal. Hence, considering these interesting results, one can see that was very useful to specify the switched control law to minimize the time derivative of a polynomial Lyapunov function. 4.2 Case 2: Uncertain polynomial nonlinear system with uncertain matrix B(α, x)
k=1 −1
(˜ x)T (x)A(α, x)ˆ x(x)
− 2ˆ xT (x)X −1 (˜ x)T (x)B(x)Kσ (x)ˆ x(x) ( n ) ∑ ∂X −1 ( k ) T +x ˆ (x) (˜ x) A (α, x)ˆ x(x) x ˆ(x). ∂xk k=1 (18) As a result, note that from (12) and it is aware that the minimum in a set of real numbers is less than or equal to any convex combination of the elements of this set, { T } x (x)X −1 (˜ x)T (x)B(x)Ki (x)ˆ x(x) min −ˆ i∈Kr ( r ) ∑ T −1 x)T (x)B(x) αi Ki (x) x ˆ(x). ≤ −ˆ x (x)X (˜ i=1
Then, from (18) and the control laws in (3) and (12), observe that xT (x)X −1 (˜ x)T (x)A(α, x)ˆ x(x) V˙ uσ (x) = 2ˆ { T } −1 + 2 min −ˆ x (x)X (˜ x)T (x)B(x)Ki (x)ˆ x(x) i∈Kr ( n ) ∑ ∂X −1 ( k ) T (˜ x) A (α, x)ˆ x(x) x ˆ(x) +x ˆ (x) ∂xk
In this subsection, it will be considered the uncertain polynomial nonlinear system (2), where αi , i ∈ Kr , are uncertain parameters, as defined in (1), namely, ˇ x)x ˇ ˆˇ(ˇ x ˇ˙ = A(α, x) + B(α, x)u(t), ˇ x) = A(α,
r ∑ i=1
αi Aˇi (x),
ˇ B(α, x) =
r ∑
ˇi (x). αi B
(19)
i=1
ˇ In order to deal with the uncertain matrix B(α, x), it will be used the same analysis presented in Souza et al. (2013). Consider the time derivative of the control input u ∈ Rm , denoted by ν ∈ Rm . Then, define xn+l and νl such that ∫t x˙ n+l = u˙ l = νl , in other words, xn+l = ul = 0 νl (τ )dτ , l = 1, 2, · · · , m. Hence, one has the following dynamics ˙ ˇ x)x ˇ ˆˇ + B(α, x ˇ = A(α, x)u, x˙ n+1 = ν1 , (20) .. . x˙ n+m = νm , which can be rewritten as (Barmish, 1983),
x˙ = A(α, x)ˆ x + B(x)ν, (21) k=1 T T T T ≤ 2ˆ xT (x)X −1 (˜ x)T (x)A(α, x)ˆ x(x) ˆˇ u ] = [x ˆˇ xn+1 · · · xn+m ] , and where x ˆ = [x ) ( r ∑ x ˇ ] [ [ ] αi Ki (x) x ˆ(x) + 2ˆ xT (x)X −1 (˜ x)T (x)B(x) − ˇ x) B(α, ˇ 0n×m xn+1 A(α, x) , B = , A(α, x) = . x = i=1 ... ( n ) Im×m 0m×N 0m×m ∑ ∂X −1 ( ) xn+m (˜ x) Ak (α, x)ˆ x(x) x ˆ(x) +x ˆT (x) ∂xk k=1 As a result, note that the system (21) is similar to the = 2ˆ xT (x)X −1 (˜ x)T (x) [A(α, x) − B(x)K(α, x)] x ˆ(x) system (8), and after this transformation, the control ( n ) ∑ ∂X −1 design for the Case 2 can be done using the procedure ( k ) T +x ˆ (x) (˜ x) A (α, x)ˆ x(x) x ˆ(x) presented in the Case 1, considering (21), where x ˆ ∈ ∂xk k=1 R(N +m)×1 and the switched control law is given by ν = = 2ˆ xT (x)X −1 (˜ −Kσ (x)ˆ x)T (x) [A(α, x) + B(x)uα ] x ˆ(x) x(x), Kσ ∈ Rm×(N +m) . 383
IFAC ROCOND 2018 Florianopolis, Brazil, September 3-5, 2018Igor T.M. Ramos et al. / IFAC PapersOnLine 51-25 (2018) 269–274
273
5. A NUMERIC EXAMPLE Consider the chaotic Lorenz system described by (Lee et al., 2001): x˙ 1 = −λ1 x1 + λ1 x2 + u, x˙ 2 = λ2 x1 − x2 − x1 x3 , (22) x˙ 3 = x1 x2 − λ3 x3 , with the nominal values of (λ1 , λ2 , λ3 ) = (10, 28, 8/3).
In this case, it is possible to use Corollary 1 directly, because the matrix B(x) is known. Then, solving the conditions in Corollary 1 with ϵ1 = 10−9 and ϵ2i = 10−9 × (x21 + x22 + x23 ), i ∈ Kr , using a decay rate β = 0.5 and considering a constant matrix X(˜ x), it is possible to design the switched control (12) for the system (22) till γ = 0.8, that is, when the uncertainties in the nominal values of the parameters are 80% of its nominal values. The obtained matrix X and polynomial gains Ki (x) ∈ Rm×N , i ∈ Kr , for γ = 0.8 are shown in (25). Note that the entries of each of these gains are polynomials of monomials with degree 0 and/or 1 in x. 0.3465×103 −0.4149×103 8.524×10−9 X =−0.4149×103 3.5050×103 1.7×10−11 . (25a) 8.524×10−9 1.7×10−11 3.5050×103
In order to perform a time simulation, it was adopted the parameters λ1 = 15, λ2 = 36.4 and λ3 = 4.8, and T the initial condition x(0) = [5 5 10] . The simulation was performed with the open-loop system from 0 till 5 seconds, and after that the controller (12) with the gains (25) was used, that is, the system was simulated in closed-loop after t = 5 seconds. The results of the simulations are presented as follows. Fig. 1 shows the state trajectory with its initial condition indicated by a red circle and its final condition indicated by a blue square. The state variables of the system, the control signal u(t) and the switching rule σ(t) (12) are presented in Fig. 2. 6. CONCLUSION In this paper, it was established a new design procedure for a switched controller applied to uncertain polynomial 384
x3 (t) x1 (t)
40 20 0 -20 -40
x2 (t)
λl3
-20
0
20
40
-40
-20
0
40
20
x2 (t)
Fig. 1. The state trajectory of the system (22), considering the uncertain parameters (λ1 , λ2 , λ3 ) = (15, 36.4, 4.8). The initial condition and final condition are indicated by a red circle (◦) and a blue square ( ), respectively.
40 20 0 -20 -40
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
t(s)
0
1
2
3
4
5
t(s) x3 (t)
λk2
20
x1 (t)
60 45 30 15 0
0
1
2
3
4
5
t(s) u(t)
The values of and are obtained as following λ11 = λ1 × (1 + γ), λ21 = λ1 × (1 − γ), (24) λ12 = λ2 × (1 + γ), λ22 = λ2 × (1 − γ), λ13 = λ3 × (1 + γ), λ23 = λ3 × (1 − γ), where γ denotes the level of uncertainty with respect to the nominal values of the parameters. λj1 ,
40
0 -40
1500 1000 500 0
0
1
2
3
4
5
t(s)
σ(t)
In this example, it will be considered that the parameters (λ1 , λ2 , λ3 ) are uncertain and they can assume values within a range corresponding to a percentage of their nominal values. Then, the system (22) can be represented by a convex combination, as described in (8), considering T x ˆ(x) = [x1 x2 x3 ] and the following matrices: j j [ ] −λ1 λ1 0 1 Ai (x) = λk2 −1 −x1 , B(x) = 0 , (23) l 0 0 x1 λ 3 where i ∈ K8 and j, k, l ∈ K2 .
60
8 7 6 5 4 3 2 1 0
1
2
3
4
5
t(s)
Fig. 2. The state variables of the system (22) and the switched control law (12) (control signal u(t) and the switching rule σ(t)), for x(0) = [5 5 10]T , considering the uncertain parameters (λ1 , λ2 , λ3 ) = (15, 36.4, 4.8). For 0 ≤ t < 5 seconds: open-loop system (u(t) = 0). For t ≥ 5 seconds: closed-loop system using the switched control law (12). nonlinear systems by means of the sum of squares theory. The SOS theory can be seen as an extension of the Linear Matrix Inequalities one, which is commonly used for stability and stabilization analysis of linear systems and of nonlinear systems described by T-S fuzzy models. When T-S fuzzy models are used, the stability analysis is usually limited to an operating region. In the proposed methodology, the nonlinear systems analysis, as well as the controller design, are not limited to one point nor an operation region. In future works, the authors intend to extend the obtained results to uncertain fuzzy polynomial models, which allows a more general representation for nonlinear systems compared to the well-known T-S fuzzy models.
IFAC ROCOND 2018 274 Florianopolis, Brazil, September 3-5, 2018Igor T.M. Ramos et al. / IFAC PapersOnLine 51-25 (2018) 269–274
[
� �
[
[
]
[
−14
−15
� � � �
−17
−15
−15
]
−15
1.1701 × 10 x1 − 4.9591 × 10 x2 + 1.7752 × 10 x3 + 43.1096 −8.2853 × 10 x1 − 1.5708 × 10 x2 − 2.1382 × 10 x3 + 43.0844 = −2.4236 × 10−12 x1 + 2.4745 × 10−15 x2 + 5.5896 × 10−16 x3 + 27.2428 −2.4236 × 10−12 x1 + 1.3026 × 10−16 x2 + 2.1221 × 10−16 x3 + 27.0225 , −15 −15 −10 −17 −16 −10 x2 − 1.1408 × 10 x3 − 6.7643 × 10 x2 + 3.7163 × 10 x3 − 2.4176 × 10 −0.11839x1 − 1.9652 × 10 −0.11839x1 − 6.2248 × 10 (25b) −15 −15 −15 −14 −15 −16 x1 − 4.8347 × 10 x2 + 7.9921 × 10 x3 + 21.7011 3.9678 × 10 x1 − 4.8718 × 10 x2 − 3.4795 × 10 x3 + 33.1541 −7.6148 × 10 T T K3 K4 = −2.428 × 10−12 x1 − 1.2038 × 10−15 x2 + 6.6931 × 10−16 x3 + 23.6977 −2.4317 × 10−12 x1 + 4.7757 × 10−16 x2 + 7.1725 × 10−16 x3 + 21.5056 , T
T
K1 K2
T K5
� �
]
[
−0.11838x1 + 9.3762 × 10
−16
x2 + 2.751 × 10
−16
x3
� � � −09� − 3.0014 × 10 −0.11838x
1
− 1.3269 × 10
−15
x2 + 1.4064 × 10
−15
−10
x3 − 3.201 × 10
]
(25c) � ] [ � T ] 4.8383 × 10−15 x1 + 1.0781 × 10−15 x2 − 8.9096 × 10−15 x3 + 40.0365 � 4.6355 × 10−14 x1 − 7.0796 × 10−15 x2 + 1.7979 × 10−15 x3 + 30.2244 �K6 = −2.428 × 10−12 x1 − 7.9127 × 10−17 x2 − 3.3215 × 10−15 x3 + 13.7926�−2.3965 × 10−12 x1 − 2.8128 × 10−15 x2 + 9.0987 × 10−16 x3 + 11.3168 , −15 −16 −09� −17 −16 −09 x + 9.9279 × 10 x + 7.8998 × 10 x + 3.8851 × 10 x + 1.1741 × 10 −0.11839x − 2.772 × 10 −0.11839x − 4.2568 × 10 1
2
3
1
2
3
2
3
(25d) � ] −14 −16 −17 x2 + 2.8708 × 10 x3 + 50.8373 � 2.3228 × 10 x1 − 2.3125 × 10 x2 + 9.0743 × 10 x3 + 50.7379 [ T� T] 1 + 1.5497 × 10 −12 −17 −16 −12 −15 −17 � � K7 K8 = −2.427 × 10 x1 + 9.4641 × 10 x2 + 1.1826 × 10 x3 + 8.7501 −2.4128 × 10 x1 − 1.3853 × 10 x2 + 2.6199 × 10 x3 + 8.6768 . −17 −16 −09� −17 −17 −10 −0.11838x − 6.3243 × 10 −0.11838x + 4.813 × 10 x − 3.1737 × 10 x − 4.9225 × 10 x + 8.7142 × 10 x + 1.3599 × 10
[ 4.9878 × 10−15 x 1
−16
2
−16
3
1
ACKNOWLEDGEMENTS
The authors would like to thank the Brazilian agencies CNPq, CAPES and FAPESP. REFERENCES Alves, U.N.L.T., Teixeira, M.C.M., de Oliveira, D.R., Cardim, R., Assunção, E., and Souza, W.A. (2016). Smoothing switched control laws for uncertain nonlinear systems subject to actuator saturation. International Journal of Adaptive Control and Signal Processing, 30(810), 1408–1433. Barmish, B. (1983). Stabilization of uncertain systems via linear control. IEEE Transactions on Automatic Control, 28(8), 848–850. Chesi, G. (2004). Estimating the domain of attraction for uncertain polynomial systems. Automatica, 40(11), 1981–1986. de Oliveira, D.R., Teixeira, M.C.M., Alves, U.N.L.T., de Souza, W.A., Assunção, E., and Cardim, R. (2018). On local H∞ switched controller design for uncertain T-S fuzzy systems subject to actuator saturation with unknown membership functions. Fuzzy Sets and Systems, 344(1), 1–26. Huang, W.C., Hong-Fei, S., and Jian-Ping, Z. (2013). Robust control synthesis of polynomial nonlinear systems using sum of squares technique. Acta Automatica Sinica, 39(6), 799–805. Lam, H., Narimani, M., Li, H., and Liu, H. (2013). Stability analysis of polynomial-fuzzy-model-based control systems using switching polynomial lyapunov function. IEEE Transactions on Fuzzy Systems, 21(5), 800–813. Lee, H.J., Park, J.B., and Chen, G. (2001). Robust fuzzy control of nonlinear systems with parametric uncertainties. IEEE Transactions on Fuzzy Systems, 9(2), 369– 379. Parrilo, P.A. (2000). Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. Ph.D. thesis, California Institute of Technology, Pasadena. Prajna, S., Papachristodoulou, P.S., and Parrilo, P.A. (2004a). SOSTOOLS: Sum of Squares Optimization Toolbox for MATLAB, version 2.00. Pasadena. Prajna, S., Papachristodoulou, A., and Wu, F. (2004b). Nonlinear control synthesis by sum of squares optimization: A Lyapunov-based approach. In Asian Control Conference, 157–165. Souza, W.A., Teixeira, M.C.M., Cardim, R., and Assunçao, E. (2014). On switched regulator design of 385
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uncertain nonlinear systems using Takagi-Sugeno fuzzy models. IEEE Transactions on Fuzzy Systems, 22(6), 1720–1727. Souza, W.A., Teixeira, M.C.M., Santim, M.P.A., Cardim, R., and Assunção, E. (2013). On switched control design of linear time-invariant systems with polytopic uncertainties. Mathematical Problems in Engineering, 2013(1), 1–10. Sturm, J.F. (1999). Using sedumi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimization methods and software, 11(1-4), 625–653. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, 15(1), 116–132. Tanaka, K., Tanaka, M., Chen, Y.J., and Wang, H.O. (2016). A new sum-of-squares design framework for robust control of polynomial fuzzy systems with uncertainties. Fuzzy Systems, IEEE Transactions on, 24(1), 94–110. Tanaka, K., Yoshida, H., Ohtake, H., and Wang, H.O. (2009). A sum-of-squares approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems. IEEE Transactions on Fuzzy Systems, 17(4), 911–922. Taniguchi, T., Tanaka, K., Ohtake, H., and Wang, H.O. (2001). Model construction, rule reduction, and robust compensation for generalized form of Takagi-Sugeno fuzzy systems. IEEE Transactions on Fuzzy Systems, 9(4), 525–538. Xu, J., Xie, L., and Wang, Y. (2009). Simultaneous stabilization and robust control of polynomial nonlinear systems using SOS techniques. IEEE Transactions on Automatic Control, 54(8), 1892–1897. Yu, G.R. and Huang, H.T. (2013). A sum-of-squares approach to synchronization of chaotic systems with polynomial fuzzy systems. International Journal of Fuzzy Systems, 15(2), 159. Yu, G.R., Huang, Y.C., and Cheng, C.Y. (2016). Robust H∞ controller design for polynomial fuzzy control systems by sum-of-squares approach. Control Theory Applications, IET, 10(14), 1684–1695.
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On Switched Controller Design for Robust On On Switched Switched Controller Controller Design Design for for Robust Robust On Switched Controller Design for Robust Control of Uncertain Polynomial Nonlinear Control of Uncertain Polynomial Nonlinear Control of Uncertain Polynomial Nonlinear Control of Uncertain Polynomial Nonlinear Systems Using Sum of Squares Systems Using Sum of Squares Systems Using Sum of Squares Systems Using Sum of Squares∗∗∗ ∗
Igor T. M. Ramos ∗ Uiliam N. L. T. Alves ∗∗∗ Igor T. M. Ramos Uiliam N. T. Alves ∗∗∗ ∗∗∗ ∗ Igor R. T. de M.Oliveira Ramos ∗∗∗∗,∗∗ Uiliam N. L. L. T. Alves Diogo Marcelo C. M. Teixeira ∗,∗∗ ∗∗∗ ∗ Diogo R. de Oliveira Marcelo C. M. Teixeira ∗,∗∗ ∗ Igor T. M. Ramos Uiliam N. L. T. Alves ∗,∗∗ ∗ ∗ ∗ Diogo R. de Oliveira Marcelo C. M. Teixeira ∗ Erica ∗ ∗ Rodrigo Cardim R. M. D. Machado ∗,∗∗ Rodrigo Cardim Erica R. M. D. Machado ∗ ∗ Diogo R. de Oliveira Marcelo C. M. Teixeira ∗ ∗ ∗ Rodrigo Cardim R. M. D. Machado ∗ Edvaldo Assunção ∗ ∗ Erica Edvaldo Assunção ∗ Rodrigo Cardim Erica R. M. D. Machado ∗ Edvaldo Assunção Edvaldo Assunção ∗ ∗ ∗ São Paulo State University (UNESP), School of Engineering, Ilha ∗ São Paulo State University (UNESP), School of Engineering, Ilha ∗ SãoSolteira Paulo -State University (UNESP), School of Engineering, Ilha SP, Brazil, of Electrical Engineering, ∗ -State SP, University Brazil, Department Department of School Electrical Engineering, Ilha SãoSolteira Paulo (UNESP), of Engineering, Solteira SP, Brazil, Department of Electrical Engineering, José Carlos Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo José Carlos- Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo Solteira SP, Brazil, Department of Electrical Engineering, José Carlos [email protected], Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo (e-mail: [email protected], (e-mail: [email protected], [email protected], José(e-mail: Carlos Rossi Ave, 1370, 15385-000, Ilha Solteira, São Paulo [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected]). [email protected]). [email protected], [email protected], ∗∗ ∗∗ Federal Institute [email protected]). of of Education, Education, Science Science and and Technology Technology of of Mato Mato ∗∗ Federal Institute [email protected]). ∗∗ Federal Institute of Education, Science and Technology of Mato Grosso do Sul (IFMS), Angelo Melão St, 790, 79641-162, ∗∗ Grosso do Sul (IFMS), Angelo Melão St, 790, 79641-162, Federal Institute ofBrazil Education, and Technology of Mato Grosso do -Sul (IFMS), AngeloScience Melão St, 790, 79641-162, Três Lagoas MS, (e-mail: [email protected]) Lagoas --Sul MS, Brazil (e-mail: [email protected]) Grosso do (IFMS), Angelo Melão St, 790, 79641-162, ∗∗∗ Três Três Lagoas MS, Brazil (e-mail: [email protected]) Federal Institute Institute of of Education, Education, Science Science and and Technology Technology of of Paraná Paraná ∗∗∗ Federal ∗∗∗ Três Lagoas -Dr. MS,of Brazil (e-mail: [email protected]) ∗∗∗ Institute Education, Science and Technology of Paraná (IFPR), Tito Ave, Jardim Panorama, 86400-000, ∗∗∗ Federal (IFPR), Dr. Tito Ave, Jardim Panorama, 86400-000, Federal Institute of Education, Science and Technology of Paraná (IFPR), Dr. Tito Ave, (e-mail Jardim Panorama, 86400-000, Jacarezinho -- PR, Brazil [email protected]) Jacarezinho (IFPR), Dr. TitoBrazil Ave, (e-mail Jardim [email protected]) Panorama, 86400-000, Jacarezinho - PR, PR, Brazil (e-mail [email protected]) Jacarezinho - PR, Brazil (e-mail [email protected])
Abstract: Abstract: Abstract: This manuscript manuscript presents presents aa switched switched control control design design for for robust robust control control of of aa class class of of uncertain uncertain This Abstract: This manuscript a switched control design control of awhose class dynamics of uncertain nonlinear systems.presents In particular, particular, it is is considered considered a class classfor of robust nonlinear systems whose dynamics are nonlinear systems. In it a of nonlinear systems are This manuscript presents a switched control control awhose class of uncertain nonlinear systems. In particular, it is considered a class of robust nonlinear dynamics are described by polynomial polynomial functions, that dependdesign on thefor state vector ofsystems the of system, with uncertain described by functions, that depend on the state vector of the system, with uncertain nonlinear systems. In particular, it can is considered a class ofterms nonlinear whosewith dynamics are described byThe polynomial functions, that depend on the in state vector ofsystems the uncertain parameters. The design procedure be represented in of sum sum of system, squares. The proposed parameters. design procedure can be represented terms of of squares. The proposed described by polynomial functions, that depend on the state vector of the system, with uncertain parameters. The design procedure can be represented in terms of sum of squares. The proposed switched control control procedure procedure uses uses aa switching switching law law to to select select at at each each instant instant of of time time aa state-feedback state-feedback switched parameters. The procedure design procedure can in at terms sum ofofsquares. Theprocedure. proposed switched control uses ato law toobtained select eachofthe instant timedesign a state-feedback polynomial gain, which belongs belongs toswitching a set setbeof ofrepresented gains obtained from the proposed design polynomial gain, which a gains from proposed procedure. switched control procedure uses a switching law to select at each instant of time a state-feedback polynomial which belongs a set of gains obtained frompolynomial the proposed design The aim aim of of gain, the switching switching law is istoto to choose state-feedback polynomial gain that procedure. minimizes The the law choose aa state-feedback gain that minimizes polynomial gain, which belongs toto a Lyapunov set of gains obtainedAfrom the proposed design procedure. The aim of the switching law is choose a state-feedback polynomial gain that minimizes the time derivative of a polynomial function. numerical example, using chaotic the time derivative of a polynomial function. A numerical example, usingminimizes aa chaotic The aimsystem of thewith switching law is to Lyapunov choose a illustrates state-feedback polynomial gain that the time derivative of a polynomial Lyapunov function. the A numerical example, using approach. a chaotic Lorenz parametric uncertainties, validity of the proposed Lorenz with of parametric uncertainties, validity ofexample, the proposed the timesystem derivative a polynomial Lyapunovillustrates function. the A numerical using approach. a chaotic Lorenz system with parametric uncertainties, illustrates the validity of the proposed approach. Lorenz with parametric uncertainties, illustrates the validity of theLtd. proposed approach. © 2018, system IFAC (International Federation of Automatic Control) Hosting by Elsevier All rights reserved. Keywords: Sum Sum of of squares squares (SOS), (SOS), Robust Robust control, control, Uncertain Uncertain nonlinear nonlinear systems, systems, Switched Switched Keywords: Keywords: Sum of squares (SOS), Robust control, Uncertain nonlinear systems, Switched systems, Polynomial Polynomial systems. systems, systems. Keywords: Sum of squares (SOS), Robust control, Uncertain nonlinear systems, Switched systems, Polynomial systems. systems, Polynomial systems. nonlinear systems whose dynamics are described by by polypoly1. INTRODUCTION INTRODUCTION described 1. nonlinear systems systems whose whose dynamics dynamics are are described nonlinear byofpoly1. INTRODUCTION nomial functions, that depend on the state vector the nomial functions, functions, that depend on the the state vector ofpolythe nonlinear systems whosedepend dynamics are described bycontrol 1. INTRODUCTION nomial that on state vector of the system, namely polynomial nonlinear systems. The The control design for nonlinear systems is an important system, namely polynomial nonlinear systems. The control The control design for nonlinear systems is an important nomial functions, that depend on the state vector of the namely polynomialusing nonlinear systems. The control design be the of theory, The design forcontrol nonlinear systems is et anal., important topiccontrol of the the nonlinear nonlinear theory (Prajna 2004b). system, design can can be approached approached using the sum sum of squares squares theory, topic of control theory (Prajna et al., 2004b). system, namely polynomial nonlinear systems. Theorigin control The control design for nonlinear systems is an important design can be approached using the sum of squares theory, and it is based on conditions to assure that the is topic of the nonlinear control theory (Prajna et al., 2004b). There are several approaches to deal with this issue, conand it is based on conditions to assure that the origin is Thereofare several approaches to deal(Prajna with this issue, con- design be approached usingasymptotically sum that of squares theory, topic the nonlinear control theory et al., 2004b). and it can is based on conditions tothe assure the origin is an equilibrium point globally stable, even There are several approaches to deal with this issue, considering different classes of dynamic systems and many an equilibrium point globally asymptotically stable, even sidering different classes of dynamic systems and many and it is based on conditions to assure that the origin is There aredifferent several approaches to2001; deal with this con- an equilibrium point globally(Prajna asymptotically stable,Chesi, even with polytopic et sidering classes of al., dynamic systems and many performance indexes (Lee et et al., Prajna et issue, al., 2004b; 2004b; with polytopic uncertainties uncertainties (Prajna et al., al., 2004b; 2004b; Chesi, performance indexes (Lee 2001; Prajna et al., an equilibrium point Huang globallyet asymptotically stable, even sidering different classes of dynamic systems and many with polytopic uncertainties (Prajna et al., 2004b; Chesi, 2004; Xu et al., 2009; al., 2013). Besides that, the performance indexes (Lee et al., 2001; Prajna et al., 2004b; Alves et et al., al., 2016; 2016; de de Oliveira Oliveira et et al., al., 2018). 2018). The The reprerepre- with 2004; polytopic Xu et al., uncertainties 2009; Huang et al., 2013). Besides that, the Alves (Prajna et al., 2004b; Chesi, performance indexes (Lee et al., 2001; Prajna et al., 2004b; 2004; Xu et al., 2009; Huang et al., 2013). Besides that, the design procedure can be developed considering polynomial Alves et al., 2016; de Oliveira et al., 2018). The representation of of the the nonlinear nonlinear system system using using Takagi-Sugeno Takagi-Sugeno 2004; design procedure can be developed considering polynomial sentation Xu etfunctions al., 2009; Huang et al., 2013). Besides that, the Alves et al., 2016;nonlinear de et al., 2018). The repredesign procedure canand be developed considering polynomial Lyapunov matrices polynomial terms, sentation of models the system using Takagi-Sugeno (T-S) fuzzy fuzzy models is aaOliveira well-established methodology to design Lyapunov functions and matrices with with polynomial terms, (T-S) is well-established methodology to procedure can be developed considering polynomial sentation of the nonlinear system using Takagi-Sugeno Lyapunov functions and matrices with polynomial terms, which the theory an methodology (T-S) fuzzy a well-established methodology to Lyapunov deal with with themodels controlisproblem problem class of of nonlinear nonlinear systems which makes makes the SOS SOS theory an attractive attractive methodology deal the control aa class systems functions andtheory matrices with polynomial terms, (T-S) fuzzy a well-established methodology to which makes the SOS an attractive methodology for the design of controllers (Prajna et al., 2004b; Tanaka deal with themodels controlis1985). problem a class of nonlinear systems (Takagi and Sugeno, The T-S fuzzy model makes it for the design of controllers (Prajna et al., 2004b; Tanaka (Takagi and Sugeno, 1985). The T-S fuzzy model makes it which makes the SOS theory an attractive methodology deal with the control problem a class of nonlinear systems the 2009). design of controllers (Prajna al., 2004b; Tanaka et al., Finally, Tanaka et al. (2016) presented a (Takagi Sugeno, the 1985). The T-S model makes it for possible and to represent represent the dynamics of fuzzy the nonlinear nonlinear system et al., al., 2009). Finally, Tanaka et al. al.et (2016) presented possible to dynamics of the system the design of controllers (Prajna et al., 2004b; Tanaka (Takagi and Sugeno, the 1985). The T-S fuzzy model makes it for et 2009). Finally, Tanaka et (2016) presented aa new SOS design for robust control of polynomial fuzzy possible to represent dynamics of the nonlinear system by a convex combination of local linear models. This reprenewal.,SOS design for robust control of(2016) polynomial fuzzy by a convex combination of local linear models. Thissystem repre- et 2009). Finally, Tanaka etconsidering al. of presented possible tocan represent dynamics of athe nonlinear SOS design for robust control polynomial fuzzya systems with uncertainties and, an uncertain by a convex combination of local linear models. repre- new sentation be done donethe exactly within region ofThis operation systems with uncertainties and, considering an uncertain sentation can be exactly within a region of operation new SOS design for robust control of polynomial fuzzy by a convex combination of local linear repre- systems withfuzzy uncertainties and, considering an uncertain polynomial system to and sentation can be exactly within a models. region ofThis operation (Taniguchi et al.,done 2001). Usually, using this methodology, methodology, polynomial fuzzy system subject subject to disturbance, disturbance, and Yu Yu (Taniguchi et al., 2001). Usually, using this systems withfuzzy uncertainties and, considering an uncertain sentation can be done exactly within a region of operation polynomial system subject to disturbance, and Yu et al. (2016) proposed conditions for a robust H control ∞ (Taniguchi et al., 2001). Usually, using this methodology, the design design procedure procedure is is given given by by Linear Linear Matrix Matrix Inequalities Inequalities et al. (2016) proposed conditions for a robust H control ∞ the polynomial fuzzy system subject to disturbance, and Yu (Taniguchi et al., 2001). Usually, usingMatrix this methodology, et al. (2016) proposed conditions for a robust H∞ control design in of the design procedure is given by Linear Inequalities et (LMIs). design in terms terms of SOS. SOS.conditions for a robust H∞ (LMIs). al. (2016) proposed ∞ control the design procedure is given by Linear Matrix Inequalities design in terms of SOS. (LMIs). The less conservative design procedure is an aim for the design in terms of SOS. In Parrilo (2000) it is presented a methodology for the (LMIs). design is for the In Parrilo (2000) it is presented a methodology for the The The less less conservative conservative design procedure procedure is an anInaim aim for the development of recent control methodologies. this sense, In Parrilo (2000) it is presented a methodology for the sum of squares (SOS) decomposition for multivariable development of recent control methodologies. In this sense, sum of squares (SOS) decomposition for multivariable The less conservative design procedure is an aim for the In Parrilo (2000) it is presented a methodology for the development of recent control methodologies. In this sense, the switched control is a technique that is widely applied to sum of squares (SOS) decomposition for multivariable polynomials. In this this procedure, it is is considered considered class of of the switched control is control a technique that is widely applied to polynomials. In procedure, it aa class development of recent methodologies. In this sense, sum of squares (SOS) decomposition for multivariable polynomials. In this procedure, it is considered a class of the switched control is a technique that is widely applied to switched control is a technique that is widely applied to polynomials. In IFAC this procedure, is considered a class of the 2405-8963 © 2018, (InternationalitFederation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © under 2018 IFAC 380 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 380 Copyright © 2018 IFAC 380 10.1016/j.ifacol.2018.11.117 Copyright © 2018 IFAC 380
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address this issue. Souza et al. (2013) proposed a switched control design for linear time-invariant systems with polytopic uncertainties where the switched control law uses the state vector to select a state-feedback gain, at each instant of time. In Souza et al. (2014) a switched control design for a class of uncertain nonlinear systems represented by T-S fuzzy models, with unknown membership functions was proposed. In addition, the procedure presented in Alves et al. (2016) proposes a smoothing switched control law, to deal with the local stability problem and control signal saturation. An example presented in Alves et al. (2016) showed that the switched controller can provide an improvement in the performance of the controlled system when compared to the closed-loop system with a timeinvariant controller. Recently, in de Oliveira et al. (2018) it was proposed a local H∞ switched controller design for uncertain nonlinear T-S fuzzy systems subject to external disturbances. A comparative example and a practical implementation, using an active suspension system, illustrate the advantages of the switched control design. This work proposes a switched control design for robust control of uncertain polynomial nonlinear systems. The design procedure is represented in terms of SOS. The proposed switched control methodology uses a switching law to select at each instant of time a state-feedback polynomial gain, which belongs to a set of gains obtained from the design technique. The aim of the switching law is to choose a state-feedback polynomial gain that minimizes the time derivative of a polynomial Lyapunov function. A numerical example, using a chaotic Lorenz system with parametric uncertainties, illustrates the validity of the proposed approach. The remaining of the manuscript is organized as follows: In Section 2 some terminologies are defined concerning the SOS theory. Section 3 introduces the polynomial nonlinear systems with polytopic uncertainties. The proposed switched control design for robust control of uncertain polynomial nonlinear systems is presented in Section 4. In Section 5, a numerical example illustrates the validity of the proposed approach. Finally, Section 6 draws the conclusions and presents the future perspectives. In this manuscript, the following notation will be adopted: Kr = {1, 2, . . . , r}, r ∈ N, x(t) = x and V (x(t)) = V . Finally, consider for an uncertain constant vector α = [α1 α2 . . . αr ]T , r ∑ αi (Ai (x), Bi (x), Ki (x)), (A(x, α), B(x, α), K(x, α)) = i=1
αi ≥ 0,
∀i ∈ Kr ,
r ∑
αi = 1,
(1)
i=1
where r = 2s and s is the number of distinct uncertain parameters in A(x) and B(x). 2. SUM OF SQUARES In this section it is defined some terminologies concerning the SOS theory. A multivariable polynomial p(x(t)), x(t) ∈ Rn , with degree 2d, is a finite combination of monomials, with real coefficients, in which the higher degree of the monomials in this combination is 2d. In addition, a monomial of x(t) is a product among xdi i (t), i = 1, 2, · · · , n, 381
where di is a nonnegative integer. ∑nIn this case, the degree of xd11 (t)xd22 (t) . . . xdnn (t) is d = i=1 di (Lam et al., 2013; Parrilo, 2000).
A polynomial p(x) with degree 2d is an SOS exist ∑if there 2 q (x(t)). polynomials qj (x(t)) such that p(x(t)) = j=1 j Therefore, being p(x(t)) an SOS, then p(x(t)) ≥ 0. An important result, as pointed out in Parrilo (2000), is that the polynomial p(x(t)) is an SOS if, and only if, there exist a semidefinite matrix Q such that x ˆT (x(t))Qˆ x(x(t)), where x ˆ(x(t)) is a column vector whose elements are all monomials in x(t) with degree no greater than d. In this case, the problem of finding this matrix Q can be formulated as a semidefinite problem (Tanaka et al., 2009). In this manuscript, the authors have been used the SeDuMi solver (Sturm, 1999) and the SOSTOOLS toolbox (Prajna et al., 2004a) with the software MatLab in order to solve the SOS conditions. 3. NONLINEAR SYSTEMS WITH POLYTOPIC UNCERTAINTIES
Consider a polynomial uncertain nonlinear system, whose dynamics is described only by polynomials functions of the state variables of the system and its uncertainties have a polytopic representation, such that this nonlinear system can be described by x˙ = A(α, x)ˆ x(x) + B(α, x)u(t), (2) where x ∈ Rn is the state vector, x ˆ(x) ∈ RN ×1 is a vector whose entries are monomials of the state vector x, u ∈ Rm is the input vector, A(α, x) ∈ Rn×N and B(α, x) ∈ Rn×m are described in (1) are known, where i ∈ Kr . Consider the auxiliary control law, defined as follows r ∑ αi Ki (x)ˆ x(x) = −K(α, x)ˆ x(x), (3) u(t) = uα (t) = i=1 m×N
where Ki (x) ∈ K , i ∈ Kr . From (2) and (3), one has x˙ = A(α, x)ˆ x(x) − B(α, x)K(α, x)ˆ x(x) r r ∑∑ = αi αj (Ai (x) − Bi (x)Kj (x))ˆ x(x). (4) i=1 j=1
Remark 1. Note that the auxiliary control law (3) can not be implemented for controlling the plant, since it depends on the uncertain parameters αi , i ∈ Kr . However, it plays an important rule in the understanding of the switched control law, which does not depend on the αi , as it will be see in the Section 4 (Souza et al., 2013). Before proceeding further with the stability analysis of the closed-loop system, some definitions will be set. Let T (x) ∈ Rn×N be the Jacobian matrix of x ˆ(x), whose entries (i, j) of the polynomial matrix T (x) are given by ∂x ˆi T ij (x) = (x), (5) ∂xj for all i ∈ KN , j ∈ Kn . To lighten the notation, let Ak (x) denotes k-th row of the matrix A(x), and let κ = {k1 , k2 , . . . , km } be the set of row indexes of B(x) whose corresponding rows are identically equal to zero. x ˜ denotes a column vector in which each entry is a monomial in x(t) T such that x ˜ = [xk1 xk2 · · · xkm ] , ki ∈ κ.
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Considering the auxiliary control law (3), the next theorem follows from Yu and Huang (2013), which considers a decay rate equal to or greater than β ≥ 0. Theorem 1. Consider the closed-loop system (4). Assume that there exist a symmetric polynomial matrix X(˜ x) ∈ RN ×N , polynomial matrices Mi (x) ∈ Rm×N and a scalar β ≥ 0 such that the conditions x) − ϵ1 (x))υ is SOS, (6) υ T (X(˜ ( −υ T T (x)Ai (x)X(˜ x) − T (x)Bi (x)Mj (x) + X(˜ x)ATi (x)T T (x) − MjT (x)BiT (x)T T (x) + T (x)Aj (x)X(˜ x) − T (x)Bj (x)Mi (x)
+ X(˜ x)ATj (x)T T (x) − MiT (x)BjT (x)T T (x) ∑ ∂X ( ) − (˜ x) Aki (x)ˆ x(x) ∂xk k∈κ ∑ ∂X ( ) (˜ x) Akj (x)ˆ x(x) − ∂xk k∈κ ) x) υ is SOS, ∀ i ≤ j, + ϵ2ij (x)I + 4βX(˜
( x) + X(˜ x)ATi (x)T T (x) −υ T T (x)Ai (x)X(˜
− T (x)B(x)Mi (x) − MiT (x)B T (x)T T (x) ∑ ∂X (˜ x)(Aki (x)ˆ x(x)) − ∂xk k∈κ ) (11) + 2βX(˜ x) + ϵ2i (x)I υ is SOS,
hold for all i ∈ Kr , where ϵ1 (x) > 0 for x ̸= 0 and ϵ2i (x) ≥ 0 for all x; υ ∈ RN is a vector that is independent of x and T (x) ∈ RN ×n is a polynomial matrix given in (5). Then the origin x = 0 is a stable equilibrium point for the closed-loop system (9), where the polynomial gains are given by Ki (x) = Mi (x)X −1 (˜ x), i ∈ Kr .
In addition, if (11) holds with ϵ2i (x) > 0 ∀ x ̸= 0 is an asymptotically stable equilibrium point, with a decay rate equal to or greater than β ≥ 0 for the closed-loop system (9). If X(˜ x) is a constant matrix, then stability properties of the equilibrium point hold globally.
(7)
hold for all i ∈ Kr and i ≤ j ∈ Kr , where ϵ1 (x) > 0 and ϵ2ij (x) ≥ 0, for all x ̸= 0, are nonnegative polynomials; υ ∈ RN is a vector that is independent of x and T (x) ∈ RN ×n is a polynomial matrix given in (5). Then the origin x = 0 is a stable equilibrium point for the closedloop system (4), where the polynomial gains are given by Ki (x) = Mi (x)X −1 (˜ x), i ∈ Kr . In addition, if (7) holds with ϵ2ij (x) > 0 ∀ x ̸= 0, then the origin x = 0 is an asymptotically stable equilibrium point, with a decay rate equal to or greater than β ≥ 0, for the closed-loop system (4). If X(˜ x) is a constant matrix, then the stability properties of the equilibrium point hold globally.
Proof. The proof can be found in Yu and Huang (2013), where it is shown if the conditions in the Theorem 1 hold with ϵ2ij (x) > 0. Then, considering a polynomial Lyapunov function candidate Vuα (x) = x ˆT (x)X −1 (˜ x)ˆ x(x), along the state trajectory of the closed-loop system (4) one has V˙ uα + 2βVuα < 0 ∀ x ̸= 0.
As a special case, assume that B(α, x) = B(x) is a known polynomial matrix, in other words, B(x) is a function of the state variables of the system and is not a function of the system uncertainties. Then, the system (2) becomes x˙ = A(α, x)ˆ x(x) + B(x)u. (8)
In this case, the closed-loop system (8) and (3) is x˙ = A(α, x)ˆ x(x) − B(x)K(α, x)ˆ x(x) r ∑ αi (Ai (x) − B(x)Ki (x))ˆ x(x) =
271
Proof. The proof follows from the proof of Theorem 1, considering B(α, x) = B(x). 4. SWITCHED CONTROL OF UNCERTAIN POLYNOMIAL NONLINEAR SYSTEMS 4.1 Case 1: Uncertain polynomial nonlinear system with known matrix B(α, x) = B(x) The switched control law selects a state-feedback polynomial gain Kσ (x), which belongs to the set of gains {Ki (x) ∈ Rm×N , i ∈ Kr }. The switched control law is defined by x(x), u = uσ = −Kσ (x)ˆ { T } ∗ σ = arg min −ˆ x (x)X −1 (˜ x)T (x)B(x)Ki (x)ˆ x(x) , i∈KN
(12) { T } where arg min ∗ −ˆ x (x)X −1 (˜ x)T (x)B(x)Ki (x)ˆ x(x) dei∈KN
notes the smallest index σ ∈ Kr , such that
x)T (x)B(x)Kσ (x)ˆ x(x) −x ˆT (x)X −1 (˜ { T } ∗ −1 = min −ˆ x (x)X (˜ x)T (x)B(x)Ki (x)ˆ x(x) . i∈KN
(13)
At this point, it will be considered the closed-loop system (8) and (12), whose the system dynamics is given by x˙ = A(α, x)ˆ x(x) + B(x)uσ r ( ) ∑ = αi Ai (x) − B(x)Kσ (x) . (14) i=1
(9)
i=1
and, from Theorem 1, it follows the next Corollary. Corollary 1. Consider the closed-loop system (9). Assume that there exist a symmetric polynomial matrix X(˜ x) ∈ RN ×N , polynomial matrices Mi (x) ∈ Rm×N and a scalar β ≥ 0 such that the conditions x) − ϵ1 (x))υ is SOS, (10) υ T (X(˜ 382
Theorem 2. Suppose that the conditions of Corollary 1 with a constant matrix X(˜ x) and ϵ2ij (x) > 0, x ̸= 0, hold for the system (8) with the control law (3). Then, the switched control law (12), where the polynomial gains are given by Ki (x) = Mi (x)X −1 (˜ x), i ∈ Kr , makes the equilibrium point x = 0 globally asymptotically stable for the closed-loop system (14). Proof. Consider a polynomial Lyapunov function candix)ˆ x(x). Then, define V˙ uα (x) and date V (x) = x ˆT (x)X −1 (˜ V˙ uσ (x) as the time derivatives of V (x) for the system (8) with the control laws (3) and (12), respectively.
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Consequently, from (14), xT (x)X −1 (˜ x)x ˆ˙ + x ˆT (x)X˙ −1 (˜ x)ˆ x(x) V˙ uσ (x) = 2ˆ
T
+x ˆ (x)
= 2ˆ xT (x)X −1 (˜ x)T (x) [A(α, x) − B(x)Kσ (x)] x ˆ(x) ( ) ∑ ∂X −1 +x ˆT (x) (˜ x)x˙ k x ˆ(x). (15) ∂xk k∈κ
Since Bik (x) = 0 for all k ∈ κ and x ∈ Rn , from (14), it follows that r ∑ ( ) x˙ k = αi (x) Aki (x)ˆ x(x) , (16) i=1
/ κ, and for all k ∈ κ. On the other hand, considering xi , i ∈ it is aware that the symmetric polynomial matrix X(˜ x) does not contain such state variable, thus, ∂X −1 (˜ x) = 0. (17) ∂xi Therefore, from (15), (16) and (17), one has xT (x)X −1 (˜ x)T (x) [A(α, x) − B(x)Kσ (x)] x ˆ(x) V˙ uσ (x) = 2ˆ ) ( n ∑ ∂X −1 ) ( +x ˆT (x) x(x) x ˆ(x) (˜ x) Ak (α, x)ˆ ∂xk = 2ˆ xT (x)X
= V˙ uα (x).
(
n ∑ ∂X −1
k=1
∂xk
(
k
(˜ x) A (α, x)ˆ x(x)
)
)
x ˆ(x)
Therefore, V˙ uσ (x) ≤ V˙ uα (x). As the conditions of Corollary 1 ensure that V˙ uα (x) < 0 for all x ̸= 0, the proof is concluded. Remark 2. Theorem 2 shows that if the conditions of Corollary 1 hold with ϵ2ij (x) > 0 ∀ x ̸= 0, then V˙ uα (x) < 0 for all x ̸= 0 and, therefore, V˙ uσ (x) < 0 for all x ̸= 0. In this case, the equilibrium point x = 0 is globally asymptotically stable for the closed-loop system (8) and (12). Thus, Corollary 1 can be used to design the polynomials gains K1 (x), K2 (x), · · · , Kr (x) and the matrix X −1 (˜ x) from the switched control law (12). Moreover, the switched control law (12) does not use the uncertain parameters αi , i ∈ Kr to compose the control signal. Hence, considering these interesting results, one can see that was very useful to specify the switched control law to minimize the time derivative of a polynomial Lyapunov function. 4.2 Case 2: Uncertain polynomial nonlinear system with uncertain matrix B(α, x)
k=1 −1
(˜ x)T (x)A(α, x)ˆ x(x)
− 2ˆ xT (x)X −1 (˜ x)T (x)B(x)Kσ (x)ˆ x(x) ( n ) ∑ ∂X −1 ( k ) T +x ˆ (x) (˜ x) A (α, x)ˆ x(x) x ˆ(x). ∂xk k=1 (18) As a result, note that from (12) and it is aware that the minimum in a set of real numbers is less than or equal to any convex combination of the elements of this set, { T } x (x)X −1 (˜ x)T (x)B(x)Ki (x)ˆ x(x) min −ˆ i∈Kr ( r ) ∑ T −1 x)T (x)B(x) αi Ki (x) x ˆ(x). ≤ −ˆ x (x)X (˜ i=1
Then, from (18) and the control laws in (3) and (12), observe that xT (x)X −1 (˜ x)T (x)A(α, x)ˆ x(x) V˙ uσ (x) = 2ˆ { T } −1 + 2 min −ˆ x (x)X (˜ x)T (x)B(x)Ki (x)ˆ x(x) i∈Kr ( n ) ∑ ∂X −1 ( k ) T (˜ x) A (α, x)ˆ x(x) x ˆ(x) +x ˆ (x) ∂xk
In this subsection, it will be considered the uncertain polynomial nonlinear system (2), where αi , i ∈ Kr , are uncertain parameters, as defined in (1), namely, ˇ x)x ˇ ˆˇ(ˇ x ˇ˙ = A(α, x) + B(α, x)u(t), ˇ x) = A(α,
r ∑ i=1
αi Aˇi (x),
ˇ B(α, x) =
r ∑
ˇi (x). αi B
(19)
i=1
ˇ In order to deal with the uncertain matrix B(α, x), it will be used the same analysis presented in Souza et al. (2013). Consider the time derivative of the control input u ∈ Rm , denoted by ν ∈ Rm . Then, define xn+l and νl such that ∫t x˙ n+l = u˙ l = νl , in other words, xn+l = ul = 0 νl (τ )dτ , l = 1, 2, · · · , m. Hence, one has the following dynamics ˙ ˇ x)x ˇ ˆˇ + B(α, x ˇ = A(α, x)u, x˙ n+1 = ν1 , (20) .. . x˙ n+m = νm , which can be rewritten as (Barmish, 1983),
x˙ = A(α, x)ˆ x + B(x)ν, (21) k=1 T T T T ≤ 2ˆ xT (x)X −1 (˜ x)T (x)A(α, x)ˆ x(x) ˆˇ u ] = [x ˆˇ xn+1 · · · xn+m ] , and where x ˆ = [x ) ( r ∑ x ˇ ] [ [ ] αi Ki (x) x ˆ(x) + 2ˆ xT (x)X −1 (˜ x)T (x)B(x) − ˇ x) B(α, ˇ 0n×m xn+1 A(α, x) , B = , A(α, x) = . x = i=1 ... ( n ) Im×m 0m×N 0m×m ∑ ∂X −1 ( ) xn+m (˜ x) Ak (α, x)ˆ x(x) x ˆ(x) +x ˆT (x) ∂xk k=1 As a result, note that the system (21) is similar to the = 2ˆ xT (x)X −1 (˜ x)T (x) [A(α, x) − B(x)K(α, x)] x ˆ(x) system (8), and after this transformation, the control ( n ) ∑ ∂X −1 design for the Case 2 can be done using the procedure ( k ) T +x ˆ (x) (˜ x) A (α, x)ˆ x(x) x ˆ(x) presented in the Case 1, considering (21), where x ˆ ∈ ∂xk k=1 R(N +m)×1 and the switched control law is given by ν = = 2ˆ xT (x)X −1 (˜ −Kσ (x)ˆ x)T (x) [A(α, x) + B(x)uα ] x ˆ(x) x(x), Kσ ∈ Rm×(N +m) . 383
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273
5. A NUMERIC EXAMPLE Consider the chaotic Lorenz system described by (Lee et al., 2001): x˙ 1 = −λ1 x1 + λ1 x2 + u, x˙ 2 = λ2 x1 − x2 − x1 x3 , (22) x˙ 3 = x1 x2 − λ3 x3 , with the nominal values of (λ1 , λ2 , λ3 ) = (10, 28, 8/3).
In this case, it is possible to use Corollary 1 directly, because the matrix B(x) is known. Then, solving the conditions in Corollary 1 with ϵ1 = 10−9 and ϵ2i = 10−9 × (x21 + x22 + x23 ), i ∈ Kr , using a decay rate β = 0.5 and considering a constant matrix X(˜ x), it is possible to design the switched control (12) for the system (22) till γ = 0.8, that is, when the uncertainties in the nominal values of the parameters are 80% of its nominal values. The obtained matrix X and polynomial gains Ki (x) ∈ Rm×N , i ∈ Kr , for γ = 0.8 are shown in (25). Note that the entries of each of these gains are polynomials of monomials with degree 0 and/or 1 in x. 0.3465×103 −0.4149×103 8.524×10−9 X =−0.4149×103 3.5050×103 1.7×10−11 . (25a) 8.524×10−9 1.7×10−11 3.5050×103
In order to perform a time simulation, it was adopted the parameters λ1 = 15, λ2 = 36.4 and λ3 = 4.8, and T the initial condition x(0) = [5 5 10] . The simulation was performed with the open-loop system from 0 till 5 seconds, and after that the controller (12) with the gains (25) was used, that is, the system was simulated in closed-loop after t = 5 seconds. The results of the simulations are presented as follows. Fig. 1 shows the state trajectory with its initial condition indicated by a red circle and its final condition indicated by a blue square. The state variables of the system, the control signal u(t) and the switching rule σ(t) (12) are presented in Fig. 2. 6. CONCLUSION In this paper, it was established a new design procedure for a switched controller applied to uncertain polynomial 384
x3 (t) x1 (t)
40 20 0 -20 -40
x2 (t)
λl3
-20
0
20
40
-40
-20
0
40
20
x2 (t)
Fig. 1. The state trajectory of the system (22), considering the uncertain parameters (λ1 , λ2 , λ3 ) = (15, 36.4, 4.8). The initial condition and final condition are indicated by a red circle (◦) and a blue square ( ), respectively.
40 20 0 -20 -40
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t(s) u(t)
The values of and are obtained as following λ11 = λ1 × (1 + γ), λ21 = λ1 × (1 − γ), (24) λ12 = λ2 × (1 + γ), λ22 = λ2 × (1 − γ), λ13 = λ3 × (1 + γ), λ23 = λ3 × (1 − γ), where γ denotes the level of uncertainty with respect to the nominal values of the parameters. λj1 ,
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In this example, it will be considered that the parameters (λ1 , λ2 , λ3 ) are uncertain and they can assume values within a range corresponding to a percentage of their nominal values. Then, the system (22) can be represented by a convex combination, as described in (8), considering T x ˆ(x) = [x1 x2 x3 ] and the following matrices: j j [ ] −λ1 λ1 0 1 Ai (x) = λk2 −1 −x1 , B(x) = 0 , (23) l 0 0 x1 λ 3 where i ∈ K8 and j, k, l ∈ K2 .
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Fig. 2. The state variables of the system (22) and the switched control law (12) (control signal u(t) and the switching rule σ(t)), for x(0) = [5 5 10]T , considering the uncertain parameters (λ1 , λ2 , λ3 ) = (15, 36.4, 4.8). For 0 ≤ t < 5 seconds: open-loop system (u(t) = 0). For t ≥ 5 seconds: closed-loop system using the switched control law (12). nonlinear systems by means of the sum of squares theory. The SOS theory can be seen as an extension of the Linear Matrix Inequalities one, which is commonly used for stability and stabilization analysis of linear systems and of nonlinear systems described by T-S fuzzy models. When T-S fuzzy models are used, the stability analysis is usually limited to an operating region. In the proposed methodology, the nonlinear systems analysis, as well as the controller design, are not limited to one point nor an operation region. In future works, the authors intend to extend the obtained results to uncertain fuzzy polynomial models, which allows a more general representation for nonlinear systems compared to the well-known T-S fuzzy models.
IFAC ROCOND 2018 274 Florianopolis, Brazil, September 3-5, 2018Igor T.M. Ramos et al. / IFAC PapersOnLine 51-25 (2018) 269–274
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1.1701 × 10 x1 − 4.9591 × 10 x2 + 1.7752 × 10 x3 + 43.1096 −8.2853 × 10 x1 − 1.5708 × 10 x2 − 2.1382 × 10 x3 + 43.0844 = −2.4236 × 10−12 x1 + 2.4745 × 10−15 x2 + 5.5896 × 10−16 x3 + 27.2428 −2.4236 × 10−12 x1 + 1.3026 × 10−16 x2 + 2.1221 × 10−16 x3 + 27.0225 , −15 −15 −10 −17 −16 −10 x2 − 1.1408 × 10 x3 − 6.7643 × 10 x2 + 3.7163 × 10 x3 − 2.4176 × 10 −0.11839x1 − 1.9652 × 10 −0.11839x1 − 6.2248 × 10 (25b) −15 −15 −15 −14 −15 −16 x1 − 4.8347 × 10 x2 + 7.9921 × 10 x3 + 21.7011 3.9678 × 10 x1 − 4.8718 × 10 x2 − 3.4795 × 10 x3 + 33.1541 −7.6148 × 10 T T K3 K4 = −2.428 × 10−12 x1 − 1.2038 × 10−15 x2 + 6.6931 × 10−16 x3 + 23.6977 −2.4317 × 10−12 x1 + 4.7757 × 10−16 x2 + 7.1725 × 10−16 x3 + 21.5056 , T
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(25c) � ] [ � T ] 4.8383 × 10−15 x1 + 1.0781 × 10−15 x2 − 8.9096 × 10−15 x3 + 40.0365 � 4.6355 × 10−14 x1 − 7.0796 × 10−15 x2 + 1.7979 × 10−15 x3 + 30.2244 �K6 = −2.428 × 10−12 x1 − 7.9127 × 10−17 x2 − 3.3215 × 10−15 x3 + 13.7926�−2.3965 × 10−12 x1 − 2.8128 × 10−15 x2 + 9.0987 × 10−16 x3 + 11.3168 , −15 −16 −09� −17 −16 −09 x + 9.9279 × 10 x + 7.8998 × 10 x + 3.8851 × 10 x + 1.1741 × 10 −0.11839x − 2.772 × 10 −0.11839x − 4.2568 × 10 1
2
3
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(25d) � ] −14 −16 −17 x2 + 2.8708 × 10 x3 + 50.8373 � 2.3228 × 10 x1 − 2.3125 × 10 x2 + 9.0743 × 10 x3 + 50.7379 [ T� T] 1 + 1.5497 × 10 −12 −17 −16 −12 −15 −17 � � K7 K8 = −2.427 × 10 x1 + 9.4641 × 10 x2 + 1.1826 × 10 x3 + 8.7501 −2.4128 × 10 x1 − 1.3853 × 10 x2 + 2.6199 × 10 x3 + 8.6768 . −17 −16 −09� −17 −17 −10 −0.11838x − 6.3243 × 10 −0.11838x + 4.813 × 10 x − 3.1737 × 10 x − 4.9225 × 10 x + 8.7142 × 10 x + 1.3599 × 10
[ 4.9878 × 10−15 x 1
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ACKNOWLEDGEMENTS
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