LMI-based design of state-feedback controllers for pole clustering of LPV systems in a union of -regions

LMI-based design of state-feedback controllers for pole clustering of LPV systems in a union of DR-regions Ruicong Yanga, Damiano Rotondoband Vicen¸c Puiga,c aInstitut de Rob`otica i Inform`atica Industrial, CSIC-UPC, Llorens i Artigas 4-6, 08028, Barcelona, Spain; bDepartment of Electrical Engineering and Computer Science (IDE), University of Stavanger, Kristine Bonnevies vei 22, 4021, Stavanger, Norway; cDepartment of Automatic Control (ESAII), Technical University of Catalonia (UPC), Rambla de Sant Nebridi 10, 08222 - Terrassa, Spain ARTICLE HISTORY Compiled October 12, 2020 ABSTRACT This paper introduces an approach for the design of a state-feedback controller for LPV systems that achieves pole clustering in a union of DR-regions. The design conditions, obtained using a partial pole placement theorem, are eventually expressed in terms of linear matrix inequalities, which can be solved efficiently using available solvers. In addition, it is shown that the approach can be modified in a shifting sense, which means that the controller gain is computed such that different values of the varying parameters imply different regions of the complex plane where the closed-loop poles are situated, thus enabling online modification of the closed-loop performance. The effectiveness of the proposed method is demonstrated by means of simulations. KEYWORDS Linear parameter varying (LPV) systems, pole placement, union of regions, linear matrix inequalities (LMIs), state-feedback control. 1. Introduction Linear parameter varying (LPV) systems have received a lot of attention from the control community in the last decades. They were first introduced in Shamma (1988), in order to distinguish such systems from linear time invariant (LTI) and linear time varying (LTV) (Shamma, 2012). The LPV paradigm has proved to be suitable for controlling nonlinear systems by embedding the nonlinearities in the varying parameters, and it has become a standard formalism in systems and control, for analysis, synthesis of controllers and even system identification. In this case, since the varying parameters depend on some endogenous signals, such as states and/or inputs, the system is commonly referred to as quasi-LPV (Rugh & Shamma, 2000). LPV systems are closely related to the Takagi-Sugeno (TS) approach (Takagi & Sugeno, 1985), and some recent works have discussed the existing similarities between the two approaches (L´opez-Estrada, Rotondo, & Valencia-Palomo, 2019; Rotondo, Puig, & Nejjari, 2016; Rotondo, Puig, Nejjari, & Witczak, 2015), stating that the main remarkable differ- ence lies in the use of fuzzy logic by the latter, whereas the former relies on traditional mathematics. More recently, there has been a growing interest in extending these techniques to nonlinear parameter varying systems (NLPV), see e.g. Larimore (2013), Blesa, Jim´enez, Rotondo, Nejjari, and Puig (2014), Rotondo and Johansen (2018), R. Yang, Rotondo, and Puig (2019), since in many practical applications there are time-varying nonlinearities that can be dealt with using ad hoc approaches. In recent years, there has been an important progress in the development of analysis and design techniques for LPV system, and this concept has been further investigated by several researchers, who brought different innovations (Hoffmann & Werner, 2014; Rotondo, 2017). LPV techniques have found application in many fields, such as bicycle (Brizuela Mendoza, Sorcia V´azquez, Guzm´an Valdivia, Osorio S´anchez, & Mart´ınez Garc´ıa, 2018), robotics (San Miguel, Puig, & Aleny`a, 2019), aerospace (D. Yang, Zong, & Karimi, 2019), ground vehicles (Zhang, Zhang, & Wang, 2016), wind turbines (P´erez-Estrada, Osorio-Gordillo, Alma, Darouach, & Olivares-Peregrino, 2018) and power system (El-Guindy, Schaab, Sch¨urmann, Stursberg, & Althoff, 2017). Remarkable applications can be mentioned as machine learning (Rizvi, Velni, Abbasi, T´oth, & Meskin, 2018), and model predictive control (MPC) (Ding, Dong, & Hu, 2019), and the research is currently undergoing theoretical development (Morato, NormeyRico, & Sename, 2020). Among the considered specifications for the design, pole clustering in linear matrix inequality (LMI) regions, also known as D-stability, has received a lot of interest. Initially characterized by Chilali and Gahinet (1996) using a quadratic Lyapunov function with constant matrix, this idea was further developed by Peaucelle, Arzelier, Bachelier, and Bernussou (2000), who considered uncertain systems by means of a parameter dependent Lyapunov function, and is still investigated nowadays, see e.g. the recent improvements in Nguyen, M´arquez, Guerra, and Dequidt (2017) and Chesi (2017). However, LMI regions have some limitations, such that they are not able of describing non-convex regions or the union of different regions. For this reason, Peaucelle et al. (2000) proposed a new characterization of LMI regions referred to as DR-regions and considered uncertain systems by means of a parameter-dependent Lyapunov function. In Peaucelle et al. (2000), DR-regions were shown to be able to describe non-convex regions but only represented symmetrically, which motivated Bosche, Bachelier, and Mehdi (2005) to extend the concept further to consider non-symmetrical regions. On the other hand, Bachelier and Pradin (1999) developed an approach that allows specifying not only a simple convex region, but also a non-convex region, defined as a union of convex subregions. Then, Maamri, Bachelier, and Mehdi (2006) proposed a technique in order to achieve partial pole placement via aggregation in such regions. This method can influence strongly the performance, in terms of settling time and damping ratio. In Tornil-Sin, Theilliol, Ponsart, and Puig (2010), this method was applied to fault-tolerant control. Although the concept of the pole is not formally defined for LPV systems, Ghersin and Pena (2002) showed that by including pole clustering specification in LPV design, the performance of the LPV control systems could be improved. Moreover, R. Yang et al. (2019) showed the existence of a relationship between pole placement and the Lyapunov function. In fact, pole placement for gain-scheduled systems has progressed strongly in the last decades, with several results concerning the design of observers (Nejjari, Puig, de Oca, & Sadeghzadeh, 2009), state-feedback controllers (Bouazizi, Kochbati, & Ksouri, 2001; R. Yang et al., 2019), H∞controllers (Rotondo, Nejjari, & Puig, 2014; Yu, Chen, & Woo, 2002), and application in many fields, such as aerospace vehicles (Ghersin & Pena, 2002), UAV (L´opez-Estrada, Ponsart, Theilliol, Zhang, & 2 Astorga-Zaragoza, 2016), missile (Shen, Yu, Luo, & Mei, 2017), power system (Jabali & Kazemi, 2017a), fuel cells (Rotondo, Fernandez-Canti, Tornil-Sin, Blesa, & Puig, 2016) and robotics (Jabali & Kazemi, 2017b). However, based on the literature review, it seems that the aforementioned partial pole placement technique has not been applied yet to gain-scheduled systems, such as LPV and TS. In fact, such an extension is not trivial, as the design of controller gain through aggregation introduces nonlinearities that destroy the polytopic decomposition usually exploited in the LPV controller design. Motivated by this fact, the main goal of this paper is to consider the problem of designing an LPV state-feedback controller for LPV systems that can guarantee some desired closed-loop poles clustering in a region defined as the union of disjoint and non-symmetric subregions. It is shown that it is possible to exploit the aggregation technique initially proposed for LTI systems in Maamri et al. (2006) to achieve partial pole placement in LPV systems, so that disjoint regions can be assigned for the closed-loop distribution of the poles. The proposed design conditions are formulated through an LMI approach. In order to deal with the nonlinearities introduced by the eigendecomposition required to achieve partial pole placement, new varying parameters are introduced so that a polytopic representation can be recovered. In addition to the classical pole clustering problem, in this paper, we consider also an extension referred to as shifting pole placement (Rotondo, Nejjari, & Puig, 2013, 2015). This approach allows designing the controller gain in such a way that different values of the varying parameters imply different regions where the closed-loop poles are situated. By means of the shifting paradigm, the online modification of the performance can be achieved, as demonstrated for example by Ruiz, Rotondo, and Morcego (2019) and Ruiz, Rotondo, and Morcego (2020), who have applied this concept to saturated system showing that it is possible to schedule the closed-loop performance according to changes in the saturation function. The main contributions of this paper can be summarized as follows: •The partial pole placement originally developed in Maamri et al. (2006) for LTI systems is extended to work with LPV systems. •A procedure for the design of a state-feedback controller which achieves pole clustering in a union of DR-regions is proposed for LPV systems. •It is shown that in spite of the nonlinearities introduced by the aggregation technique, it is possible to introduce new sets of varying parameters in terms of which polytopic representations suitable for reducing the number of design LMIs from infinite to finite can be obtained. The rest of the paper is organized as follows. In Section 2, partial pole placement background information is introduced. Section 3 explains in detail the pole clustering in a union of regions for LPV systems. Section 4 discusses how the approach can be extended according to the shifting paradigm. Section 5 shows the application of the developed technique to a numerical example. Finally, Section 6 summarizes the conclusions and suggests possible future work. Notation:Rn×mand Cn×mdenote the set of real and complex matrices with nrows and mcolumns; AT,A∗and A+denote the transpose, the conjugate and the pseudoinverse of A, respectively; AHis the Hermitian matrix defined as AH=A+A∗; the Euclidean norm is indicated by ||A||;⊗represents the Kronecker product; in matrix inequalities, negative (semi-)definiteness is indicated by ≺0 (0), whereas 0 (0) denotes positive (semi-)definiteness. 3 2. Background The idea of LMI regions was first introduced by Chilali and Gahinet (1996) in order to provide a Lyapunov-based characterization of pole clustering in stable subregions of the complex plane. Their formal definition is given as follows: Definition 1:(LMI region) A subset Dof the complex plane is called an LMI region if there exist a matrix α= [αkl]∈Sm×mand a matrix β= [βkl]∈Rm×msuch that: D={s∈C:fD≺0}(1) with the characteristic function given by: fD(s) = α+sβ +s∗βT= [αkl +βkls+βlks∗]1≤k,l≤m(2) In other words, LMI regions are subsets of the complex plane that are represented by an LMI in sand s∗. In addition, a new characterization of regions was proposed in Peaucelle et al. (2000) called DR-regions. Definition 2:(DR-regions) Let Rbe a 2d×2dHermitian matrix defined as: R=R00 R10 R∗ 10 R11∈C2d×2d(3) Then, the subset of the complex plane defined according to DR={s∈C:R00 + (R10s)H+R11s∗s≺0}(4) is called a DR-region of degree d. The class of DR-regions is a class of open convex subsets of the complex plane that includes (among others) half-planes, disks, conic sectors, vertical and horizontal strips and ellipses, symmetrical or not with respect to the real axis. For example, the vertical left half-plane defined by Re(s)< λ is characterized by a matrix Requal to: R=−2λ1 1 0(5) while the interior of a disk with center c=c1+c2iand radius ris characterized by: R=c2 1+c2 2−r2−c1+c2i −c1−c2i1(6) In contrast to LMI regions, DR-regions are able to represent non-convex region. Without any assumption on the matrix R11,DR-regions are not convex, but with R11 0, DR-regions become a slight modification of the characterization provided by LMI regions (Rotondo, 2017). In Bachelier and Pradin (1999), non-convex regions were considered as unions of convex subregions. Before introducing the partial pole placement for LPV systems, let us recall the existing approach for LTI system (Maamri et al., 2006). 4 Let us consider the following system: ˙x(t) = Ax(t) + Bu(t) (7) Then, the following lemmas can be applied: Lemma 1 (Peaucelle et al., 2000): The matrix Ais said to be DR-stable, or in other words all its eigenvalues lie inside the region DRif and only if there exists a matrix P0such that: R00 ⊗P+ (R10 ⊗(PA))H+R11 ⊗(A∗PA)≺0 (8) In Maamri et al. (2006), Lemma 1 has been extended to partial pole placement in DR, which means that only peigenvalues are wished to be affected by the feedback. To do so, some matrices are defined as follows: Λ = V−1AV =Λ10 0 Λ2C=Ip0V−1 C+=VIp 0ˆ A=CAC+ˆ B=CB where Vis the modal matrix of A, that is, the matrix whose columns are the eigenvectors of A. Λ1∈Cp×pis associated to the set of peigenvalues desired to be affected by the feedback, whereas Λ2denotes the remaining eigenvalues. Then, the following lemma can be applied: Lemma 2 (Maamri et al., 2006): There exists a state-feedback gain ˆ Kthat assigns ppoles in DRif and only if there exist an matrix ˆ X0and a matrix ˆ Ssuch that the LMI: R00 ⊗ˆ X+ (R10 ⊗(ˆ Aˆ X+ˆ Bˆ S))HZ∗⊗(ˆ Aˆ X+ˆ Bˆ S)∗ Z⊗(ˆ Aˆ X+ˆ Bˆ S)−Id⊗ˆ X≺0 (9) holds, where Zis deduced from the Cholesky factorization R11 =Z∗Z. In this case, a suitable state-feedback gain is: ˆ K=ˆ Sˆ X−1(10) 3. Pole clustering in a union of regions for LPV system In this section, the aggregation technique is extended to LPV systems to compute a state-feedback controller gain which performs pole clustering in a union of regions for LPV systems. Let us recall that an LPV system is defined as a finite-dimensional time-varying system whose state equation, although linear, is described by matrices which are function of some varying parameters θ(t)∈Θ⊂Rnθ(with Θ known closed set), that are 5 assumed to be unknown a priori, but that can be measured or estimated in real-time: ˙x(t) = A(θ(t))x(t) + B(θ(t))u(t) (11) where A(θ(t)) ∈Rnx×nx,B(θ(t)) ∈Rnx×nuare the state and input matrices, x(t)∈ Rnxdenotes the system state, u(t)∈Rnuis the control input. The system (11) is said to be polytopic if it can be represented by state-space matrices A(θ(t)) and B(θ(t)) which range over a convex set: ˙x= w X n=1 µn(θ(t))Anx(t) + Bnu(t)(12) where µnare the non-negative coefficients of the polytopic decomposition such that: w X n=1 µn(θ(t)) = 1 µn(θ(t)) ≥0∀n= 1, ..., w ∀θ∈Θ Let us use a gain-scheduled state-feedback control law given by: u(t) = K(θ(t))x(t) (13) where K(θ(t)) is the controller gain to be designed. 3.1. Aggregation technique The main idea of the algorithm employed to achieve pole clustering in a union of regions is to use a structured feedback gain K(θ(t)) that modifies just a subset of the system poles. Consider the Jordan canonical form for the matrix A(θ(t)): Λ(θ(t)) = V(θ(t))−1A(θ(t))V(θ(t)) (14) where V(θ(t)) is the modal matrix of A(θ(t)), which means that the columns of V(θ(t)) are the eigenvectors of A(θ(t)). Meanwhile Λ(θ(t)) can be rearranged in a form such that: Λ(θ(t)) = Λ1(θ(t)) 0 0 Λ2(θ(t))(15) where Λ1(θ(t)) ∈Cp×pis associated to the set of peigenvalues that are wished to be affected by the feedback. Let us define the following matrices: C(θ(t)) = Ip0V(θ(t))−1(16) with pseudo-inverse given by: C(θ(t))+=V(θ(t)) Ip 0(17) 6 and: ˆ A(θ(t)) = C(θ(t))A(θ(t))C(θ(t))+(18) ˆ B(θ(t)) = Ip0V(θ(t))−1B(θ(t)) (19) ¯ B(θ(t)) = 0In−pV(θ(t))−1B(θ(t)) (20) and finally, consider a feedback gain defined as: K(θ(t)) = ˆ K(θ(t))C(θ(t)) (21) The closed-loop matrix then satisfies: A(θ(t)) + B(θ(t)) ˆ K(θ(t))C(θ(t)) = ... ... =V(θ(t)) Λ1(θ(t)) + ˆ B(θ(t)) ˆ K(θ(t)) 0 ¯ B(θ(t)) ˆ K(θ(t)) Λ2(θ(t))V(θ(t))−1(22) which means that the eigenvalues associated to Λ1(θ(t)) are modified whereas the eigenvalues of Λ2(θ(t)) are invariant with respect to the feedback K(θ(t)) = ˆ K(θ(t))C(θ(t)), i.e. such feedback gain will only modify the peigenvalues of interest. 3.2. Partial DR-stability In order to extend the pole clustering in a union of regions to LPV system, let us consider the DR-stability of the LPV system (11), in the sense of all the frozen poles of (11) lying in DR. By applying Lemma 1, DR-stability holds if there exists a matrix P0 such that ∀θ∈Θ: R00 ⊗P+ (R10 ⊗(PA(θ(t))))H+R11 ⊗(A(θ(t))∗PA(θ(t))) ≺0 (23) The above means that for partial pole placement in DRbeing achieved by the state-feedback gain ˆ K(θ(t)), the gain has to be chosen in such a way that it satisfies: R00 ⊗ˆ P+ (R10 ⊗(ˆ P(ˆ A(θ(t)) + ˆ B(θ(t)) ˆ K(θ(t)))))H+... ... +R11 ⊗(( ˆ A(θ(t)) + ˆ B(θ(t)) ˆ K(θ(t)))∗ˆ P(ˆ A(θ(t)) + ˆ B(θ(t)) ˆ K(θ(t)))) ≺0 (24) for some matrix ˆ P0. Deduced from the above, the next theorem states LMI conditions for obtaining a controller gain that achieve partial pole placement. Theorem 1: There exists a state-feedback gain ˆ K(θ) that assigns peigenvalues of the closed-loop matrix A(θ) + B(θ)ˆ K(θ)C(θ) in the region DRdefined by (4) if there 7 exist matrices ˆ X0 and ˆ S(θ) of appropriate dimensions such that ∀θ∈Θ: R00 ⊗ˆ X+ (R10 ⊗(ˆ A(θ)ˆ X+ˆ B(θ)ˆ S(θ)))HZ∗⊗(ˆ A(θ)ˆ X+ˆ B(θ)ˆ S(θ))∗ Z⊗(ˆ A(θ)ˆ X+ˆ B(θ)ˆ S(θ)) −Id⊗ˆ X≺0 (25) where Zis deduced from the Cholesky factorization R11 =Z∗Z. Then, the statefeedback gain is given by: ˆ K(θ(t)) = ˆ S(θ(t)) ˆ X−1(26) Proof: It can be derived by following the steps in the proof of Theorem 2.1 in Maamri et al. (2006).  Hence, the partial pole placement procedure boils down in obtaining a solution for the LMI (25), and then using (26) to recover the appropriate ˆ K(θ(t)). It is necessary to mention that the design condition (25) requires satisfying an infinite number of conditions, which leads to a computational issue. In order to reduce the number of conditions from infinite to finite, the most common way to solve this problem is to use the polytopic assumption. However, the non-linearity introduced by the multiplications in (18) implies that even if a polytopic representation is available for A(θ(t)), it does not necessarily hold that the same polytopic coefficients describe how ˆ A(θ(t)) varies with respect to θ(t). However, it is possible to introduce new varying parameters, that are some nonlinear function of the original varying parameters, hereafter denoted by ˆ θ(t), and then obtain a polytopic representation for the matrix ˆ Awith coefficients that depend on ˆ θ. This can be done using available methods in the literature, such as the bounding box (Sun & Postlethwaite, 1998) the singular value decomposition boxing (Baranyi, 2009) or identification approaches (Fujimori & Ljung, 2005). Hence the matrices ˆ A(θ(t)) and ˆ B(θ(t)) are expressed as polytopic combination of matrices ˆ Aiand ˆ Bias follows: ˆ A(θ(t)) = ˆ A(ˆ θ(t)) = r X i=1 αi(ˆ θ(t)) ˆ Ai(27) ˆ B(θ(t)) = ˆ B(ˆ θ(t)) = r X i=1 αi(ˆ θ(t)) ˆ Bi(28) where αiare the non-negative coefficients of the polytopic decomposition such that: r X i=1 αi(ˆ θ(t)) = 1 αi(ˆ θ(t)) ≥0∀i= 1, ..., r ∀ˆ θ∈ˆ Θ⊂Rnˆ θ(29) and the matrix function ˆ S(θ(t)) is constrained to satisfy: ˆ S(θ(t)) = ˆ S(ˆ θ(t)) = r X i=1 αi(ˆ θ(t)) ˆ Si(30) 8 Then, the approach proposed by Sala and Ari˜no Sala and Ari˜no (2007) to check the definiteness of double polytopic sums (as the ones arising from the terms ˆ B(θ)ˆ S(θ) in (25)) can be applied by choosing a scalar s∈Nand using the symbols Psand P+ sto denote the following sets: Ps=n~p = [~p1, . . . , ~ps]T∈Ns|1≤~pk≤s∀k= 1, . . . , so(31) P+ s={~p ∈Ps|~pk≤~pk+1, k = 1, . . . , s −1}(32) whereas P(~p)⊂Psdenotes the set of permutations, with possible repeated elements, of the multi-index ~p, thus obtaining the following corollary. Corollary 1: For any s∈N, with s≥2, there exist a matrix ˆ X0 and matrices ˆ S1,ˆ S2,..., ˆ Srsuch that: X ~m∈P(~p)R00 ⊗ˆ X+ (R10 ⊗(ˆ A~m1ˆ X+ˆ B~m2ˆ S~m1))HZ∗⊗(ˆ A~m1ˆ X+ˆ B~m2ˆ S~m1)∗ Z⊗(ˆ A~m1ˆ X+ˆ B~m2ˆ S~m1)−Id⊗ˆ X≺0 (33) holds ∀~p ∈P+ s, where Zis obtained from the Cholesky factorization R11 =Z∗Z, then the state-feedback gain given by (26), with ˆ Scomputed using (30), assigns p eigenvalues of the closed-loop matrix A(θ) + B(θ)ˆ K(θ)C(θ) in the region DRdefined by (4). Proof: Taking into account the definition of ˆ A(θ(t)), ˆ B(θ(t)) and ˆ S(θ(t)) in (27)- (30), the parameter-dependent LMI (25) is equivalent to: r X i=1 r X j=1 αi(ˆ θ)αj(ˆ θ)R00 ⊗ˆ X+ (R10 ⊗(ˆ Aiˆ X+ˆ Bjˆ Si))HZ∗⊗(ˆ Aiˆ X+ˆ Bjˆ Si)∗ Z⊗(ˆ Aiˆ X+ˆ Bjˆ Si)−Id⊗ˆ X≺0 (34) which corresponds to the problem of verifying the negativity of a double polytopic sum. By applying Polya’s theorem on definite quadratic forms (Sala & Ari˜no, 2007), (33) is obtained.  As discussed by Sala and Ari˜no (2007), the sufficient conditions obtained through the application of Polya’s theorem become progressively less conservative when sincreases, and actually exact, i.e. necessary and sufficient, for a finite value of s. 3.3. Pole clustering in a union of regions for LPV systems Assume now that the region of interest Dis obtained as follows: D= q [ k=1 DRk(35) where each subregion DRkis a DR-region defined in (4). Then, the pole clustering in the region Dcan be performed by successive partial pole clustering in the subregions DRk, for k= 1, ..., q. This can be achieved according to the following algorithm. Algorithm 9 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -200 -100 0 100 200 300 400 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 100 200 300 400 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Time(seconds) 0 100 200 300 400 Figure 4. The closed-loop response by pole clustering in D1(blue) and D2(red). 16 in the least-squares sense, using as input data (αk−1,i(ˆ θk−1(t)); rk(θ(t))2) where αk−1,i(ˆ θ(t)) are the known coefficients of the polytopic decomposition obtained beforehand and rk(θ(t))2is the radius of the circular region Rk(θ(t)). The coefficient vector βk−1= [ βk−1,0βk−1,1· · · βk−1,r ]Tis then computed as: βk−1=I αk−1+rk(θ(t))2(54) Fig. 5 shows the frozen-parameter poles position after shifting pole clustering in D3, where the circles in red correspond to Rk(θ(t)) when θ= 0 while those in blue to Rk(θ(t)) when θ= 1. Moreover, Fig.6 shows that the desired regions and the actual positions of the closed-loop poles vary according to the value of θ. -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 Re(s) -50 -40 -30 -20 -10 0 10 20 30 40 50 Im(s) (a) Poles position in D3at iteration k= 1 -180 -160 -140 -120 -100 -80 -60 -40 Re(s) -50 -40 -30 -20 -10 0 10 20 30 40 50 Im(s) (b) Poles position in D3at iteration k= 2 Figure 5. Position of the closed-loop frozen poles using the shifting pole clustering approach. Fig.7 shows that the behavior of the closed-loop state variables depends upon the trajectory of θ. These responses have been obtained starting from the initial state x(0) = −5.6 5.4 6.1Tin four different cases, three of which corresponding to constant values of the scheduling parameter θ= 0.2, θ= 0.6 and θ= 1 (red, green and blue line, respectively), and the remaining case corresponding to a time-varying scheduling parameter trajectory as follows: θ(t) = 0.6−0.4 cos(5πt) (purple line). It can be seen from the figure that the closed-loop system behaves as expected, in the sense that for the value of θwhich corresponds to poles located farther from the imaginary axis, a faster dynamics of the closed-loop system is obtained. The purple curve shows that at the beginning of the simulation, the system behaves in an underdamped way due to the value of the varying parameter being approximately equal to θ= 0.2 (exactly equal at t= 0); then, as the value of θ(t) increases with time, the closed-loop frozen pole get closer to the real axis, which increases the damping and causes the oscillations to fade away. 6. Conclusions This paper has provided a procedure for pole clustering in a union of DR-regions for LPV systems. From this technique, a method to compute a state-feedback gain that achieves the desired closed-loop pole location has been deduced. Furthermore, 17 -180 -160 -140 -120 -100 -80 -60 -40 Re(s) -60 -40 -20 0 20 40 60 Im(s) Figure 6. Position of the closed-loop poles for different values of θusing the shifting approach. 18 0 0.05 0.1 0.15 0.2 0.25 0.3 -10 -5 0 5 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0 5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 Time(seconds) 0 5 10 15 Figure 7. System response for different values/trajectories of θ(t) using the shifting approach. 19 this technique has been extended to the shifting case, thus allowing pole clustering in parameter-dependent varying regions which enables online modification of the transient performance. The proposed technique has been validated using simulations on a numerical system, demonstrating the main characteristics and the effectiveness of the developed control strategy. Future work will be devoted to developing equivalent techniques for LPV systems with residual nonlinearities, such as parameter-varying Lipschitz or quadratic terms. 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Numerical values for pole clustering in D1 The matrices ˆ Ak−1,i and ˆ Bk−1,i in (39)-(40) during iteration of Algorithm 1 k= 1 are given as follows: ˆ A0,1=2−10 10.36 2 ˆ A0,2=2−10 10.36 1 ˆ A0,3=2−10 10 2  ˆ A0,4=2−10 10 1 ˆ A0,5=2−10.36 10.36 2 ˆ A0,6=2−10.36 10.36 1  ˆ A0,7=2−10.36 10 2 ˆ A0,8=2−10.36 10 1 ˆ A0,9=1−10 10.36 2  ˆ A0,10 =1−10 10.36 1 ˆ A0,11 =1−10 10 2 ˆ A0,12 =1−10 10 1  ˆ A0,13 =1−10.36 10.36 2 ˆ A0,14 =1−10.36 10.36 1 ˆ A0,15 =1−10.36 10 2  ˆ A0,16 =1−10.36 10 1  ˆ B0,1=−0.43 −0.10ˆ B0,2=−0.45 −0.11ˆ B0,3=−0.43 −0.11ˆ B0,4=−0.45 −0.10 then, the obtained controller gains are ˆ Kk,i are: ˆ K1,1=91.68 192.69ˆ K1,2=91.97 187.00ˆ K1,3=88.38 187.32 ˆ K1,4=89.11 185.33ˆ K1,5=91.44 191.78ˆ K1,6=91.78 186.36 ˆ K1,7=88.18 185.52ˆ K1,8=89.03 184.72ˆ K1,9=90.88 193.00 ˆ K1,10 =90.87 186.08ˆ K1,11 =88.27 191.49ˆ K1,12 =88.50 186.24 ˆ K1,13 =91.03 192.73ˆ K1,14 =90.89 185.82ˆ K1,15 =87.92 190.19 ˆ K1,16 =88.58 186.09 During the iteration k= 2, the matrices ˆ Ak−1,i,ˆ Bk−1,i and ˆ Kk,i are obtained as follows: ˆ A1,1= 11 ˆ A1,2= 1 ˆ B1,1= 0.18 ˆ B1,2= 0.16 ˆ K2,1=−405.98 ˆ K2,2=−343.60 Appendix B. Numerical information of pole placement in D2 When the region of interest is the one denoted as D2in Table 1, at iteration k= 1 of Algorithm 1, the matrices ˆ Ak−1,i and ˆ Bk−1,i have the same values as the ones previously shown for the case of D1. Then, the obtained controller gains ˆ Kk,i are as 23 follows: ˆ K1,1=102.96 549.39ˆ K1,2=102.32 539.21ˆ K1,3=99.73 539.80 ˆ K1,4=99.44 533.60ˆ K1,5=102.48 550.38ˆ K1,6=102.98 538.35 ˆ K1,7=100.40 541.86ˆ K1,8=98.61 531.05ˆ K1,9=103.57 550.09 ˆ K1,10 =104.04 536.49ˆ K1,11 =100.49 547.46ˆ K1,12 =99.79 537.14 ˆ K1,13 =104.01 548.72ˆ K1,14 =104.29 535.83ˆ K1,15 =99.43 547.46 ˆ K1,16 =99.24 536.97 At iteration k= 2, the matrices ˆ Ak−1,i and ˆ Bk−1,i are obtained as follows: ˆ A1,1= 11 ˆ A1,2= 1 ˆ B1,1= 0.08 ˆ B1,2= 0.06 and the controller gains ˆ Kk,i are: ˆ K2,1=−1611.80 ˆ K2,2=−1465.50 24