Design of shifting state-feedback controllers for LPV systems subject to time-varying saturations via parameter-dependent Lyapunov functions

This paper considers the problem of designing a shifting state-feedback controller via quadratic parameter-dependent Lyapunov functions (QPDLFs) for systems subject to symmetric time-varying saturations. By means of the linear parameter varying (LPV) framework and the use of the shifting paradigm and the ellipsoidal invariant theory, it is shown that the solution to this problem can be expressed with linear matrix inequalities (LMIs) which can efficiently be solved via available solvers. Specifically, three hyper-ellipsoidal regions are defined in the state-space domain for ensuring that the control action remains in the linearity region of the actuators where saturation does not occur. Furthermore, the closed-loop convergence speed is regulated online according to the instantaneous saturation limit values through the shifting paradigm concept. The main characteristics of the proposed approach are validated by means of two illustrative examples. © 2021 The Authors. Published by Elsevier Ltd on behalf of ISA. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
All real-world systems' performances are affected negatively by the inherent physical limitations of actuators. Among these limitations, actuator saturations arise in control systems as progressively decreasing actuation signals due to, for example, temporary shortages in the availability of the required power. If saturations are not taken into account properly, they may cause actuator performance deterioration and even lead to the instability of the closed-loop system. For this reason, in the last decades this phenomenon has attracted a strong attention by several researchers, which is supported by the large amount of books about this topic, see [1][2][3]. Actuator saturations can be taken into account through the anti-windup compensator approach, in which a pre-designed controller can handle the saturation constraints after a compensator is added [4,5]. On the other hand, alternative solutions use the direct control design approach where the input saturation constraints are considered directly in the controller design stages [6,7].
In recent years, the linear parameter varying (LPV) framework has been applied widely to address the presence of nonlinearities in various fields such as automotive, robotics, or aerospace, see [8] and references therein. These systems were introduced by Shamma in 1988 as a particular case of linear time varying (LTV) systems where the time-varying elements are dependent upon quantifiable parameters which vary over time [9,10]. Nonlinear systems can be represented within this framework through an appropriate definition of the time-varying parameters, which let us embed the nonlinearities through parameters that are dependent on endogenous signals. In this situation, the system is referred to as quasi-LPV, in order to emphasize the fact that the varying parameters are not dependent on exogenous signals. Three different ways to model these systems have emerged as the most successful ones in the literature, i.e. linear fractional transformation (LFT), LPV input-output models and the polytopic approach. Furthermore, the LPV framework has been applied successfully in many plants where, e.g., the parametrization of the nonlinearities allows the controller to take into account the generator shaft speed and wind speed variations improving the conversion efficiency in variable-speed wind energy conversion systems [11] or the suspension's deflections in a semi-active suspension control of a vehicle [12].
Note that the existing works assume commonly that the actuator saturation limits are constant in time. For instance, a health-aware control based on the remaining useful life estimation of a battery has been designed for an autonomous racing vehicle assuming that the input/output limits are constant [13]. The same assumption holds for [14], where a model predictive control (MPC) algorithm has been developed for polytopic LPV systems subject to input saturations. On the other hand, some works have proposed to use a saturation indicator parameter to schedule the input constraints whose limits are still constant in time [15,16]. The lack of literature addressing time-varying saturations motivates us to consider the possible degradation of the actuators over time, which would affect the online performance of the designed controller. It is worth highlighting that, practically speaking, time-varying saturation limits could occur in control systems due to the natural wear and tear of the machinery or to the temporary loss of energy availability. This fact would provide a progressively decreasing availability of the actuation signal in the case of multi-rotor vehicles, such as the loss of thrust force in each rotor due to the discharge of the battery [17] or the variations of the aerodynamic coefficients as a consequence of environment changes [18].
With the goal of designing a state-feedback controller with the ability to regulate the closed-loop convergence speed online according to the instantaneous saturation limits, a linear matrix inequality (LMI)-based methodology has been proposed in [19]. Similarly, a shifting H ∞ LPV state-feedback controller has been developed in [20], where the designed controller could adapt its disturbance rejection performance online taking into account the control action availability. In both works, the overall design has been performed using a quadratic Lyapunov function (QLF) and the ellipsoidal invariant theory [21]. Furthermore, these works have exploited the shifting paradigm concept which was introduced in [22] for polytopic LPV systems, thus defining a scheduling parameter vector that allows changing online the closed-loop performance index under consideration. Nevertheless, it is well known that the use of QLFs introduces some conservativeness that can lead to sub-optimal performance or even make the design problem infeasible. This issue can be alleviated by considering parameter-dependent Lyapunov functions (PDLFs) [23][24][25][26], piecewise functions [27] or polyhedral functions [28], although at the cost of increasing the computational burden.
The main contribution of this work lies in proposing an LMIbased methodology for designing a shifting state-feedback controller via a quadratic parameter-dependent Lyapunov function (QPDLF), thus overcoming the conservativeness of the design methodology presented in [19]. The design conditions are obtained through the application of the invariant ellipsoidal theory and the shifting paradigm concept. In particular, three hyperellipsoidal regions are defined in the state-space domain for ensuring that the control action remains in the linearity region of the actuators. Furthermore, the shifting paradigm is used to schedule the convergence speed of the closed-loop system response according to the instantaneous saturation limit values. The LMI conditions are obtained by applying Schur complements [21] together with the Pólya's relaxation inherited from [29]. The final solution is a set of LMIs that can be solved using available solvers, and which are less conservative than the set provided in [19], as demonstrated by the examples, showcasing both a comparison using a numerical example and a proof of concept with a nonlinear quadrotor model.
The paper is organized as follows. The problem formulation is provided in Section 2. In Section 3, the LMI-based methodology for the controller design is given. In Section 4, the design implementation procedure is summarized. In Section 5, simulation results are presented using a numerical system and the nonlinear quadrotor model. Finally, the main conclusions and perspectives on future research are outlined in Section 6.

Notation
For a real symmetric matrix M ∈ R m×m , the notation M ≻ 0 (M ⪰ 0) stands for a positive (semi-)definite matrix and indicates that all the eigenvalues of M are positive (non-negative). In the same way, M ≺ 0 (M ⪯ 0) denotes a negative (semi-)definite matrix, which implies that all the eigenvalues of M are negative (non-positive). The symbols I n , diag(. . .) and ⊗ denote the ndimensional identity matrix, a diagonal matrix and the Kronecker product of matrices [30], respectively. Moreover, where A T is the transpose of A. Finally, given the Pólya's relaxation degree d ∈ N with d ≥ 2 and the number of vertices N, the symbols P (d,N) and P + (d,N) denote the following sets: whereas P(⃗ q) ⊂ P (d,N) corresponds to the set of permutations, with possible repeated elements, of multi-index ⃗ q.

Problem statement
Let us look at the following continuous-time saturated LPV system: where x(t) ∈ R nx is the state vector and u(t) ∈ R nu denotes the control input vector. A(ϑ (t)) ∈ R nx×nx and B(ϑ (t)) ∈ R nx×nu denote the state and the input matrices, respectively, which depend on a scheduling parameter vector ϑ(t) ∈ Θ ⊂ R n ϑ , with set Θ known, closed and bounded. Moreover, ϑ(t) is assumed throughout this work to be continuously differentiable for t ≥ 0 in order to guarantee the existence ofθ (t), and it is assumed thaṫ ϑ(t) ∈ Θ d ⊂ R n ϑ , with set Θ d known, closed and bounded.
The input u(t) in (3) is affected by symmetric input saturations with time-varying limits: where h = 1, . . . , n u and σ h (t) ∈ R nu + is the instantaneous saturation limit value. σ h (t) takes values within the interval [σ h , σ h ], where σ h is the lowest possible saturation limit and σ h is the highest possible saturation limit for each input u h (t). Hence, the linear region of the actuators L(t) ⊂ R nu , defined as the hypercube region of the input space in which u(t) does not saturate, is time-varying and given by: where × denotes the Cartesian product and {−σ h (t), σ h (t)} indicates the corresponding set of each instantaneous saturation limit value ∀h = 1, . . . , n u .
Before stating the problem formally (Section 2.5), let us introduce the polytopic representation, the shifting paradigm, the performance criterion and the established region constraints.

Polytopic representation
In the rest of the work, a polytopic representation of the system (3) will be used, according to the following definition. [31]). The LPV system (3) is said to be polytopic if it can be represented by state-space matrices whose dependence on the scheduling parameter vector ϑ(t) satisfies:

Definition 1 (Apkarian and Gahinet
Note that, taking into account the knowledge of the bounds of Θ and Θ d ,μ i (ϑ (t)) ∀i = 1, . . . , N can be bounded as follows: with known bounds v i , v i ∈ R.

Shifting paradigm
The shifting paradigm was presented in [22] as a possible approach to take advantage of the properties of polytopes and LMIs, enabling the design of a gain-scheduling controller that adapts the performance of the closed-loop system online depending on some chosen criteria, e.g., guaranteed decay rate, pole clustering or H ∞ /H 2 bounds. This paradigm has been used recently in [19] and [20] for describing the time-varying saturation in polytopic form by introducing a new scheduling parameter vector ϕ(t) ∈ Φ ⊂ R nϕ . Roughly speaking, ϕ(t) was used to increase/decrease online the convergence speed [19] or the disturbance effectiveness [20] of the closed-loop system according to the instantaneous value of σ h (t). In this way, a faster response or a better disturbance rejection can be guaranteed when a large control action is available (σ h (t) → σ h ) whereas the controller will provide a more conservative performance when the instantaneous value of σ h (t) is closer to σ h . Then, the time-varying limit changes in (4) can be described by the scheduling parameter ϕ(t), which is chosen to be a weighted function of σ h (t), for example taking values within the interval [0, 1].
Similarly to the case of ϑ(t), the matrices which depend on ϕ(t) and satisfy T (ϕ(t)) ∈ Co { T j } can be described as follows: where M is the number of vertices of the polytope Φ and, T (ϕ(t)) and T j correspond to the chosen polytope and vertex matrices, respectively. Additionally, the time-derivative of this new scheduling parameter vector is assumed to satisfyφ(t) ∈ Φ d ⊂ R nϕ , with set Φ d known, closed and bounded, so that the following holds ∀j = 1, . . . , M: with known bounds p j , p j ∈ R.

Performance criterion
Let us consider the following LPV state-feedback control law for (3) and the quadratic parameter-dependent Lyapunov function (QPDLF) which depends on the scheduling parameter vectors ϑ(t) and ϕ(t): where K (ϑ (t), ϕ(t)) ∈ R nu×nx is the parameter-dependent controller gain and P(ϑ (t), ϕ(t)) ∈ R nx×nx denotes a positive definite parameter-dependent Lyapunov matrix.
Then, based on the shifting paradigm idea and the above defined QPDLF (12), let us define the performance criterion considered throughout this work. 1 Definition 2 (Guaranteed Shifting Decay Rate). The LPV system (3) with state-feedback control law (11) is said to satisfy the guaranteed shifting decay rate λ(ϑ, ϕ) with the parameter-dependent matrix P(ϑ , ϕ), if: By means of the above defined performance criterion the shifting state-feedback controller (11) ensures the closed-loop exponential stability of the LPV system (3) Moreover, the closed-loop response of (3) adapts its behaviour online, in the sense that the convergence speed will change online according to the instantaneous value taken by λ(ϑ, ϕ). However, the fulfilment of the Lyapunov condition (13) is conditioned to the use of a nonlinear expression of u(t) complicating the obtention of computationally applicable design conditions. This issue can be alleviated by ensuring that u(t) ∈ L(t) ∀t ≥ 0, thus allowing to neglect the saturation nonlinearity during the design stage.

Region constraints
The approach used to guarantee that u(t) ∈ L(t) is based on the definition of appropriate hyper-ellipsoidal regions. Consider the parameter-varying set U (ϕ) ⊂ R nu defined as the maximal hyper-ellipsoidal region contained in L(t), as follows: where S(ϕ) ∈ R nu×nu is a known matrix function that defines the parameter-varying orientation and size of the region U (ϕ). Due to the LPV state-feedback control law (11), the region U (ϕ) can be mapped to a corresponding state-space region U x (ϑ , ϕ), as follows: Let us define V(ϑ , ϕ) as the parameter-dependent region of the state-space that is delimited by the unit level curves of the QPDLF (12) and E as the region of the state-space which contains the initial conditions of interest for the system: where X 0 ∈ R nx×nx with X 0 ≻ 0 defines the size and orientation of E. The hyper-ellipsoidal regions described in (15)-(17) must satisfy inclusions: in order to guarantee that u(t) does not saturate ∀t > 0 during the transient response of x(t) no matter the initial state x(0) ∈ E.
If (18) holds, the LPV system with saturations (3) can be reduced to the following non-saturated LPV system for design purposes:

Problem definition
Finally, on the basis of the performance criterion ( Definition 2) and the region constraints established in (18), the design problem considered in this paper can be formulated as follows: Problem 1. Given the polytopic LPV system (3) and (6) subject to the time-varying saturations (4), a desired guaranteed shifting decay rate λ(ϑ, ϕ) and the regions (15)-(17), find P(ϑ , ϕ) and K (ϑ , ϕ) such that for any x(0) ∈ E the response of the closed-loop LPV system satisfies (13).

Remark 1.
In this paper, in order to keep the mathematical complexity somehow limited, the presence of external disturbances is not considered. The interested reader is referred to [20], where a shifting H ∞ LPV state-feedback controller has been designed to ensure that a polytopic LPV system subject to symmetric time-varying input saturation is quadratically bounded with respect to unknown disturbances. The extension to the parameter-dependent case of quadratic boundedness or similar robust techniques goes beyond the scope of this paper and will be addressed in future work.

Shifting LPV state-feedback synthesis
Let us start by introducing the following theorem which gives a parameter-dependent LMI for designing a parameter-dependent controller gain that ensures some desired guaranteed shifting decay rate. (19), the control law (11) and the QPDLF in (12), and assume that there exist a positive symmetric matrix function Q (ϑ , ϕ) ∈ R nx×nx and a parameter-dependent matrix Γ (ϑ , ϕ) ∈ R nu×nx such that the following parameter-dependent LMI is satisfied:
Proof of Theorem 1. By introducing the control law (11) into the system's equation (19), the following LPV closed-loop system representation is obtained: where A cl (ϑ , ϕ) denotes the closed-loop parameter-dependent state matrix.
Then, by replacing (21) in (26), the following is obtained: which is a bilinear matrix inequality (BMI) due to the product between the decision variables K (ϑ , ϕ) and Q (ϑ , ϕ). In order to transform it into a parameter-dependent LMI, the change (20). ■ Note that the parameter-dependent LMI (20) represents an infinite number of constraints, which can be converted into a finite number of LMIs by considering the polytopic assumption. To this end, a suitable polytopic representation for the terms For λ(ϑ, ϕ), Γ (ϑ , ϕ) and Q (ϑ , ϕ) such representations can be obtained straightforwardly by assuming that: where λ ij ∈ R + , Γ ij ∈ R nu×nx and Q ij ∈ R nx×nx correspond to a desired decay rate value and two decision variables, respectively, for the pair (i, j). Then, by differentiating the expression (30), we find out thaṫ Q (ϑ , ϕ) can be expressed as: whereμ i (ϑ ) andη j (ϕ) vary within the defined bounds (8) and (10). According to [23][24][25][26], note that µ i (ϑ ) and η j (ϕ) fulfil (7) and (9), respectively, implying that: Moreover, it is possible to compute a finite number of O vectors f {k} and R vectors g {l} that satisfy: where O and R are the number of vertices of the polytopes obtained as the intersections of the hyper-rectangles defined by the constraints (8) and (10) and the hyperplanes defined in (32) and (33), respectively. f . . .
Thus, the polytopic representation ofQ (ϑ , ϕ) is: for some coefficients α k (ϑ ,θ) and β l (ϕ,φ) satisfying the constraints appearing in (36)-(37). Given the above discussion, we can now introduce the following theorem which provides a finite number of LMIs that can be used to assess the parameter-dependent LMI (20).

Remark 2.
Note that the possible codependence between the coefficients µ i (ϑ ) and α k (ϑ ,θ) has been neglected. The same applies to the codependence between η j (ϕ) and β l (ϕ,φ). Albeit introducing some conservativeness, this assumption has enabled the application of Pólya's theorem [29], so that the computational complexity during the design stage has been reduced.

Remark 3.
According to [32], the fact of considering arbitrarily large values for the bounds (8) and (10)  Taking into account the theoretical results obtained so far, the following theorem provides a set of LMIs for solving Problem 1, i.e., a set of LMIs that can be used for obtaining a controller that ensures that the closed-loop response of the LPV system (3) is adapted online according to the instantaneous saturation limit values of σ (t) through the scheduling parameter vector ϕ(t).
Furthermore, this theorem guarantees that the control action u(t) will remain inside the linearity region of the actuators L(t) due to the inclusion chain described in (18). Note that the inclusion E ⊆ V(ϑ , ϕ) for x(0) ∈ E ensures that x(t) ∈ V(ϑ , ϕ) ∀t as long as the system works in the linear region described by (4).
Also, taking into account V(ϑ , ϕ) ⊆ U x (ϑ , ϕ) any state trajectory x(t) contained in V(ϑ , ϕ) will also be located in U x (ϑ , ϕ), so no saturation happens and, therefore, the convergence of x(t) → 0 when t → ∞ is guaranteed for any x(0) ∈ V(ϑ , ϕ), and hence for any x(0) ∈ E. (19), the control law (11), the QPDLF in (12), the hyper-ellipsoidal regions (15)-(17) with the given matrices X 0 and S(ϕ), a guaranteed decay rate λ(ϑ, ϕ) and finite number of vectors f {k} and g {l} for which (34)-(37) hold. Consider also that the Pólya's relaxation degree d ∈ N and s ∈ N have been selected. Furthermore, suppose that the inverse of the parameter-dependent matrix function S(ϕ) can be expressed as:
Proof of Theorem 3. The fact that (39) ensures that the closedloop system (19) satisfies the guaranteed decay rate λ(ϑ, ϕ) as long as it works in the region of linearity of the actuators, L(t), has been shown in the proof of Theorem 2. Hence, let us demonstrate that if the LMIs (42)-(43) hold, then u(t) remains inside the region U (ϕ) for all t, as long as x(0) ∈ E.
To this end, let us consider the inclusions in (18). The inclusion E ⊆ V(ϑ , ϕ) may be formulated as: which is equivalent to: and that, by applying Schur complements, becomes: Thereupon, as the parameter-dependent matrix Q (ϑ , ϕ) is described by means of (30), the set of LMIs (42) is obtained.

Design implementation procedure
Problem 1 is solved using the methodology given by Theorem 3, which corresponds to an LMI-based feasibility problem subject to the performance criterion (13), the region constraints (15)-(17) and the inclusion chain (18). The design and implementation procedure can be summarized as follows: Off-line computation:

Remark 4.
Previous knowledge of the plant is required to define regions E and U (ϕ) described in (14) and (17) through the matrices X 0 and S(ϕ), so that they have a physical meaning. These regions define the initial conditions of interest and the control action space, respectively. As a consequence of this fact, the solution of Theorem 3 is conditioned by trade-offs related to these regions, for example, a choice of the expected initial conditions closer to the origin facilitates the feasibility of the LMI-based problem, although it is desirable that the controller operates over a region of possible initial conditions as big as possible. However, from a practical point of view, this is constrained by the available range of the control action. For instance, when a wide range of control action is available, a larger region of initial conditions could be considered while maintaining feasibility of the solution.

Illustrative examples
In this section, the proposed design approach is demonstrated through two examples comparing the advantages of using a QPDLF versus the LMI methodology proposed in [19], where a constant quadratic Lyapunov function (QLF) was used. Section 5.1 presents a numerical example where the influence of the saturation limit variation range and the bounds ofη j (ϕ) is shown. Then, Section 5.2 shows the closed-loop response of the attitude control of a quadrotor with a performance that varies online according to the value of the instantaneous saturation limits given by the maximum angular rotor speed.
The simulations are obtained in MATLAB environment, by implementing Theorem 3 via the YALMIP toolbox [34] using the solver SeDuMi [35]. Additionally, the Multi-Parametric Toolbox 3.0 (MPT3) [36] was used in both examples in order to obtain a finite number of vectors (34)-(35) by solving a vertex enumeration problem through the function Polyhedron, whose input arguments are A, b, A e and b e . These arguments can be obtained by expressing the constraints (8) and (32) in the form of Ax ≤ b and A e x ≤ b e as follows: and similarly for the constraints (10) and (33).
For both examples, the polytopic weights µ i (ϑ ) were obtained by means of the simplest polytopic approximation, known as bounding box, which relies on bounding each scheduling parameter of ϑ by an interval [31]. Similarly, η j (ϕ) were obtained applying the same approach except for Section 5.2, where a less conservative approach was used, as described therein.

Example 1: Numerical example
Consider an open-loop unstable LPV system modelled as in (3), ϑ(t) ∈ [0, 1] and the following state-space matrices: where the parameter-dependent matrix A(ϑ (t)) and the matrix B can be written in polytopic form by means of the Definition 1, assuming that B has the same value for all the N = 2 vertices: ] . (53)

Methodology comparison
Let us establish the following set conditions with the purpose of evaluating the advantage of using the QPDLF (12) instead of a QLF. To this end, σ 1 and the maximum desired decay rate value λ are generated as follows: Furthermore, the desired vertex values of (28) are fixed to where λ i2 will be evaluated for each combination of the selected values of σ 1 and λ. Table 1 shows the selected specifications for the different established scenarios. Scenario A corresponds to use the QLF defined in [19], whereas Scenarios B-D correspond to use the QPDLF defined as in (12) and the polytopic representations (30) and (38).
Finally, Theorem 3 is evaluated for each element of σ 1 and λ under the different scenarios described in Table 1. Fig. 1 shows, for each fixed value of σ 1 , the maximum value obtained for λ so that the set of LMIs (39) and (42)-(43) described in Theorem 3 is feasible. The benefits of using a QPDLF instead of using a QLF can be observed, as a larger feasible guaranteed decay rate is obtained when a larger range of variation of σ 1 (t) is considered. However, note that for small variations of σ 1 (t), the maximum feasible values of λ are practically the same for both cases. Furthermore, when large bounds are considered forη j (ϕ), the QLF approach and the proposed QPDLF approach provide similar performance for big values of σ 1 .
Similarly to Section 5.1, let us assume that the axes of u(t) are aligned with the axes of the hyper-ellipsoidal region U (ϕ(t)) (14). Then, let us define the squared expression of Eq. (63) as follows Thereupon, let us introduce the following scheduling parameters ϕ 1 (t) and ϕ 2 (t) which are linked to Ω(t) 2 and Ω(t) 4 , respectively, thus obtaining expressions of σ 1 (t) 2 , σ 2 (t) 2 and σ 3 (t) 2 as a function of ϕ(t) = [ϕ 1 (t), ϕ 2 (t)] T : Once Eq. (65) is obtained, the axis magnitudes of U (ϕ(t)) are established through the matrix function S(ϕ(t)) as follows: For the purpose of solving Problem 1, let us define the polytope Φ (where the scheduling parameters ϕ 1 (t) and ϕ 2 (t) lie) generating a set of possible values for both parameters taking into account the bounds of Ω(t). Note that these sets reach large values due to ϕ 1 (t) and ϕ 2 (t) being linked to square and fourth power of Ω(t), respectively. Hence, it is necessary to normalize both parameters in the range [0, 1] in order to avoid numerical issues when computing η j (ϕ(t)). Then, taking into account the bounds of the polytope Φ, it is possible to compute η j (ϕ(t)) ∀j = 1, 2, 3 as the solution of the  corresponds to V (x(t), ϑ(t), ϕ(t)); denotes λ(ϑ(t), ϕ(t))).

Methodology comparison
Let us now describe the performance specifications for the controller design considering that the initial state of the quadrotor is around the hover attitude point (φ(t) = 0, θ (t) = 0) with the purpose of evaluating the LMI methodology described in Theorem 3 versus the procedure proposed in [19] under the scenarios indicated in Table 2.
To this end, let us define the shape of the region (17)   where ξ κ denotes the corresponding κ upper bound value of the interval.
Then, with the aim of controlling online the convergence speed of the closed-loop system, let us define the desired decay rate values in (28) as follows: λ i1 = 0, λ i2 = 0, λ i3 = λ ∀i = 1, . . . , 8, thus specifying the fastest closed-loop system response when the largest saturation limit of u(t) is available.
Once the performance specifications are defined, let us proceed to compare Theorem 3 against Theorem 2 in [19] by evaluating λ for each value of the set {n ∈ R + : 0 ≤ n ≤ 100}. Thus, obtaining the maximum feasible decay rate λ shown in Table 3  The benefits of using Theorem 3 instead of using the procedure proposed in [19] can be observed in Table 3, as an improvement of 20-30% with respect to the largest feasible guaranteed decay rate obtained when QLF is used for the above defined conditions. However, this improvement comes at the cost of increasing significantly the computational cost, as shown in Table 4. Note that the increase in computational burden affects only the off-line computation due to the growth of the number of LMIs that must be satisfied. On the other hand, the online computation is not affected, since the total number of vertices does not depend on which of the methodologies is used.

Closed-loop response
Let As shown in Fig. 3, the closed-loop system stability is guaranteed ∀t ≥ 0. Moreover, note that jumps in the values of φ(t) and ψ(t) were introduced every 10 s in order to show the effectiveness of the controller. It can be seen that the slowest system response corresponds to when Ω(t) → Ω at t ≥ 30, as shown in Fig. 4. Conversely, the fastest closed-loop response corresponds to when the maximum angular speed of each propeller is available. This demonstrates that the designed shifting LPV state-feedback controller adapts online the closed-loop system response in the sense of convergence speed according to the available control action. Fig. 4 shows the evolution of the maximum available propeller angular speed, which decreases its value over time reproducing an incipient discharge of the battery, which limits the instantaneous value of u(t) according to (63). This fact is exemplified in Fig. 5, where smaller values of u(t) were obtained over time preventing saturation to occur. Similarly, Fig. 6 shows the behaviour of the angular speed for each propeller under the scenario described above.
Finally, Fig. 7 shows the adaptability of the control performance through the evolution of the QPDLF (12). Note that the largest value of λ(ϑ(t), ϕ(t)) stands for the fastest closed-loop response showed in Fig. 3, whose instantaneous saturation limits corresponded to the largest possible ones. Furthermore, V (x(t), ϑ(t), ϕ(t)) is under the unit value ∀t ≥ 0, which provides theoretical guarantees that none of the control actions saturates during the transient response, as already shown in Fig. 5.

Conclusions
In this work, we have considered the issues of designing a shifting LPV state-feedback controller via QPDLFs for LPV systems subject to time-varying input saturation limits. The design procedure proposed in [19] has been extended through the use of a QPDLF obtaining a suitable set of LMIs, which can be accurately solved with accessible solvers. Furthermore, Pólya's theorem has been used in order to deal with the multiple polytopic summations, reducing the complexity in the design stage.
The results obtained in the illustrative examples have shown the improvements and the effectiveness brought by the proposed approach. The designed controller has shown its ability to regulate the closed-loop system convergence speed according to the instantaneous saturation limit values of the actuators. The results obtained so far appear to be less conservative, although at the cost of increasing the computational burden and the mathematical complexity, which could hinder the implementation of the proposed design approach in higher-order plants due to the large number of required LMIs. Future research will aim at reducing the computational complexity by focusing on the application of hybrid techniques (for example, MPC with LPV, or feedback linearization with LPV) or more advanced LPV frameworks which incorporate switching elements, in order to reduce the number of LMIs to be handled.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.