2. State Key Laboratory of Synthetical Automation for Process Industries(Northeastern University), College of Information Science and Engineering, Northeastern University, Shenyang 110819, China;
3. School of Engineering, Dogus University, and also with the School of ElectricalElectronics Engineering and Information Technologies, St. Cyril and St. Methodius University, Skopje, Macedonia, Istanbul TR34722, R. Turkey
Aeroengines are fairly complex multivariable nonlinear systems with a large variation in the system dynamics. Since an accurate analytical engine model is almost impossible to be obtained, the system analysis and synthesis have to be conducted using an approximate analytical model [1]. Many techniques for aeroengine control have been presented in literatures (see, [2][5] and the references therein). Nonlinear approaches are often hard to apply to a multidimensional control system. Approximate linearization techniques of nonlinear systems only ensure the performance around specific operating points, while exact inputoutput feedback linearization methods lack robustness. Zhao et al. proposed an approximate nonlinear engine model and a feedback linearization control strategy for local linear inputoutput performance [3]. A widely applied nonlinear control technique is the gain scheduling [6], [7], however it has no stability or performance guarantee for offdesign points [8]. Linear parametervarying (LPV) control methods can provide a systematic gainscheduling method with guaranteed stability performance [8]. Analysis and synthesis of LPV systems for aeroengines have been studied widely [4], [9]. A model matching problem in the
An alternative method is switching control where a family of controllers are designed at different operating points and the system performs controller switching based on the switching logic. The applications of this strategy are stimulated by the recent development of switched systems. A switched system consists of a finite number of subsystems and a switching law which usually depends on time, states, or both that determines switching between these subsystems [10]. Stability analysis and synthesis methods for switched systems have been widely studied in many literatures [11][16]. References [11], [13] studied the stability for linear and nonlinear switched systems in lower triangular form under an arbitrary switching law. As shown in [14], multiple Lyapunov functions technique was used to deal with a hybrid nonlinear control problem of switched systems. Besides that, the average dwelltime approach was also employed to investigate the stability and stabilization of switched systems [15]. The global robust stabilization problem for a class of uncertain switched nonlinear systems in lower triangular form was considered by [16] under any switching signal with dwell time specifications.
An LPV system is characterized as a smooth linear system with timevarying parameters. Modern aeroengines usually work in a large parameter variation range. Similar to the control of aircraft, the control of aeroengines is different in different parameter subregions [17], nevertheless these are more complicated than the control of aircrafts. The aircraft systems have been constructed as switched LPV models in some papers, such as [18], [19], therefore a single LPV model may not give sufficient approximation to nonlinear engine dynamics over the entire operating range. A reasonable and natural idea is to adopt several LPV models and corresponding controllers, each suitable for a specific parameter subregion, and switching among them [17], such LPV systems then become switched LPV systems. The switching LPV control approach can obtain a better approximation of the nonlinear dynamics and better performance than a single LPV control method [20]. Switched LPV systems have received considerable attention in the recent literatures [19], [21]. A complete overview of the stability results for LPV and switched LPV systems was given in [21]. Based on multiple parameterdependent Lyapunov functions, a switching LPV control technique was presented for an F16 aircraft via controller state reset using hysteresis and average dwell time switching law in [18]. Switching control for LPV systems in aeroengines is still an open and interesting issue.
On the other hand, due to the emergence of switching control in robotic systems and many other manufacturing processes, tracking control research on switched systems has received increasing attention. Based on the statedependent switching method, sufficient conditions for the solvability of the state tracking control problem were given in [22].
Motivated by the above discussions, this technical note studies the problem of switching
The technical note is organized as follows. Section Ⅱ gives the problem formulation and preliminaries. Section Ⅲ gives a tracking control design technique. In Section Ⅳ we apply the designed method to a switched LPV aeroengine model. Finally, the conclusion is given in Section Ⅴ.
The notations used in this paper are fairly standard. For a matrix
In this section, the problem will be formulated, and some preliminaries about the switched LPV systems will be given.
A. Problem FormulationConsider the following switched LPV system:
$ \begin{align} \dot x(t) = A_{\sigma} \left(\rho \right)x(t) + B_{\sigma}\left(\rho\right)u(t)+\omega(t), \quad x(0)=0 \end{align} $  (1) 
where
Suppose that the parameter
The reference state
$ \begin{align} \dot{x}_r(t)=A_{r}x_{r}(t)+r(t), \quad x_r(0)=0 \end{align} $  (2) 
where
Remark 1: The reference signal can also be given by a parameterdependent model. For this case we can replace (2) by
Combining the system (1) with the system (2), one can get the augmented system
$ \begin{align} \left[ {\begin{array}{*{20}c} {\dot x(t)} \\ {\dot x_r (t)} \\ \end{array}} \right] = \left[ {\begin{array}{*{20}c} {A_\sigma (\rho)x(t) + B_\sigma (\rho)u(t)} \\ {A_r x_r (t)} \\ \end{array}} \right] + \left[ {\begin{array}{*{20}c} {\omega (t)} \\ {r(t)} \\ \end{array}} \right]. \end{align} $  (3) 
Give the following performance index:
$ \begin{align} \int_0^{t_f } {e_r^{T} (t)} e_r (t){d}t < \gamma \int_0^{t_f } {\bar{\omega}^{T} (t)} \bar{\omega}(t){d}t \end{align} $  (4) 
where
The
1) the system (3) is asymptotically stable, when the
2) under the zeroinitial condition, the inequality (4) holds for all
Our objective is to design both controller
The following assumptions are adopted which are useful in our later development.
Assumption 1 [17]: The matrix function triple
Assumption 2 [12]: The switching signal
Lemma 1 [26]: Let
$ \begin{align} DFE+E^{T}F^{T}D^{T} \leq {\gamma}^{1}DD^{T}+{\gamma}E^{T}E. \end{align} $  (5) 
In this section, we will design the state feedback parameterdependent controller and a switching law
The state feedback controller is given as
$ \begin{align} u(t)=K_i(\rho)e_r(t), \quad {i\in {\mathbb{Z}_N}}. \end{align} $  (6) 
From augmented system (3) with controller (6) we have the following closedloop system:
$ \begin{align} \dot{\bar{x}}(t)=\bar A_{i}(\rho){\bar x}(t)+\bar{\omega}(t) \end{align} $  (7) 
where
$ \begin{align*} &\bar x(t) = [{x^{T}(t)}, {x_r^{T}(t)}]^{T}\\ &\bar A_{i}(\rho) = \left[{\begin{array}{*{20}c} {A_i (\rho) + B_i (\rho)K_i (\rho)} & {B_i (\rho)K_i (\rho)} \\ 0 & {A_r } \\ \end{array}} \right]. \end{align*} $ 
Now we deal with the issue of parameterdependent switching law
Firstly, we choose multiple parameterdependent Lyapunov function candidate for the system (1) with the system (2) as
$ \begin{align} V_{i}(\bar {x}, \rho)=\bar x^{T}X_{i}(\rho)\bar{x} \end{align} $  (8) 
where
Hysteresis Switching Law: As previously mentioned, the set
$ \begin{align} &\text{When}~ t=0, \ \ \ \, \sigma(0)=i, \quad \text{if} \ \rho(0)\in \mathscr P_{i}\notag\\ &\text{When}~ t>0, \ \begin{cases} \sigma(t)=i, &\text{if} \ \sigma(t^{})=i\, ~ \text{and}~ \rho(t)\in \mathscr P_{i}\\ \sigma(t)=j, & \text{if} \ \sigma(t^{})=i\, ~ \text{and} ~ \rho(t)\in \mathscr P_{j}\mathscr P_{i}. \end{cases} \end{align} $  (9) 
For the switched closedloop system (7), if on the switching surface
Remark 2: The parameter subsets are partitioned by switching surfaces and the switching law is parameterdependent in [17]. In [27], [28] the switching law is not parameterdependent but rather is modedependent for dealing with the switching LPV control problem. In this paper, we studied the tracking problem for aeroengines, considering the character of practical problem, we choose the parameterdependent switching law.
Remark 3: For the switching surface
For the closedloop switched LPV system (7), we state the synthesis condition of switching LPV control in the following theorem.
Theorem 1: Consider the augmented closedloop system (7) with the parameter set
$ \begin{align} \left[{\begin{array}{*{20}c} {\text {He} \{X_i (\rho) \bar A_{i} (\rho)\}+\dot X_i (\rho, \dot{\rho})+ Q } & {X_i (\rho)} \\ {X_i (\rho)} & {\gamma I} \\ \end{array}} \right] < 0 \end{align} $  (10) 
and
$ \begin{align} X_i (\rho) X_j (\rho)\ge 0, \quad \rho \in {{\pmb S}_{ij}} \end{align} $  (11) 
where
Proof: For the multiple parameterdependent Lyapunov functions candidate (8), computing time derivative along the state variables trajectory of the system (7), we have
$ \begin{align} \dot{V}_{i}(\bar x, \rho)=&\ \bar x^{T}(t)\left[X_i (\rho)\bar A_{i}(\rho) +\bar A^{T}_{i}(\rho)X_i(\rho)+\dot{X}_i(\rho, \dot{\rho})\right]\bar x(t)\nonumber \\ &\, +2\bar{\omega}^{T}(t)X_i(\rho)\bar x(t) \end{align} $  (12) 
when
The matrix inequality (10) implies
$ \begin{align*} X_i (\rho)\bar A_{i}(\rho)+\bar A^{T}_{i}(\rho)X_i(\rho)+\dot{X}_i(\rho, \dot{\rho}) < Q. \end{align*} $ 
Because
$ \begin{align*} X_i (\rho)\bar A_{i}(\rho)+\bar A^{T}_{i}(\rho)X_i(\rho)+\dot{X}_i(\rho, \dot{\rho}) < 0{\color{blue}{}} \end{align*} $ 
which tell us that
Now we show the performance index (4) under the zero initial condition with
From (12), applying Lemma 1, we have
$ \begin{align*} 2 \bar \omega^{T}(t)X_i(\rho)\bar x(t) \leq \gamma^{1} \bar x^{T}(t)X_i(\rho)X_i(\rho)\bar x(t)+\gamma \bar{\omega}^{T}(t)\bar{\omega}(t). \end{align*} $ 
Then,
$ \begin{align*} \dot{V}_{i}(\bar x, \rho)\leq &\ \bar x^{T}(t)[\text{He}\{X_i (\rho)\bar A_{i} (\rho)\} + \dot X_i (\rho, \dot{\rho}) \\&\, + \gamma ^{1} X_i (\rho)X_i (\rho)] \bar x(t) + \gamma \bar{\omega}^{T}(t)\bar{\omega}(t). \end{align*} $ 
From the inequality (10) and with Schur complement, we have
$ \begin{align} \text{He}\{X_i (\rho)\bar A_{i} (\rho)\}+\dot X_i(\rho, \dot{\rho})+\gamma^{1}X_i(\rho)X_i(\rho) <  Q \end{align} $  (13) 
that is
$ \begin{align} \dot{V}_{i}(\bar x, \rho) < \bar x^{T} (t)Q\bar x(t) + \gamma {\bar \omega}^{T} (t)\bar \omega(t) \end{align} $  (14) 
since
$ \begin{align*} \bar x^{T} (t)Q\bar x(t) = \left[{\begin{array}{*{20}c} {x(t)} \\ {x_r (t)} \\ \end{array}} \right]^{T} \left[{\begin{array}{*{20}c} I & {I} \\ {I} & I \\ \end{array}} \right]\left[{\begin{array}{*{20}c} {x(t)} \\ {x_r (t)} \\ \end{array}} \right] =e_r^{T} (t)e_r (t) \end{align*} $ 
it follows from (14) that
$ \begin{align} \dot{V}_{i}(\bar x, \rho) < e_r^{T} (t)e_r(t) + \gamma \bar \omega^{T}(t)\bar \omega(t). \end{align} $  (15) 
Integrating both sides of (15) from zero to
$ \begin{align*} \int_0^{t_f } {\sum\limits_{i \in \mathbb{Z}_N } {\dot V(\bar x, \rho){d}t} } &= \sum\limits_{j = 0}^{t_f }{\sum\limits_{i_{j} \in \mathbb{Z}_N } {\int_{t_{i_j}}^{t_{i_{j} + 1}}{\dot V(\bar x, \rho){d}t}}}\\ &= V(\bar x(t_f), \rho)  V(\bar x(0), \rho) \\ & <  \int_0^{t_f } {e_r^{T} (t)e_r(t){d}t + } \gamma \int_0^{t_f } {\bar{\omega} ^{T} (t)\bar{\omega}(t){d}t}. \end{align*} $ 
According to the zero initial condition and
$ \begin{align*} \int_0^{t_f } {e_r^{T} (t)} e_r (t){d}t < \gamma \int_0^{t_f } {\bar{\omega}^{T} (t)}\bar{\omega}(t){d}t. \end{align*} $ 
Therefore, under the switching law (9), the
Remark 4: The appearance of matrix
Since the matrix inequalities condition (10) of Theorem 1 are nonconvex in
Theorem 2: Consider the system (7) with the parameter set
$ \begin{align} \left[{\begin{array}{*{20}c} {\Psi _{i11} } & {\Psi _{i12}} & {Y_{i}(\rho)} & {I} & {0} \\ * & \Psi _{i22} & {Y_{i}(\rho)} & {0} & {I} \\ * & * & {{I}} & {0} & {0} \\ * & * & * & {\gamma {I}} & {0} \\ * & * & * & * & {\gamma {I}} \\ \end{array}} \right] < 0 \end{align} $  (16) 
and for
$ \begin{align} {Y_{i}(\rho)}{Y_{j}(\rho)}\leq 0 \end{align} $  (17) 
where
$ \begin{align*} &\Psi _{i11} = \text {He}\{A_i (\rho)Y_i(\rho) + B_i (\rho)W_i (\rho)\}  \sum\limits_{k = 1}^s \left\{\underline{v}_k, \bar{v}_k \right\}{{\frac{\partial Y_i(\rho)} {\partial \rho_k}} }\\ &\Psi _{i12} = { B_i (\rho)W_i (\rho)}\\ & \Psi _{i22} = \text {He} \{A_r Y_i(\rho)\}  \sum\limits_{k = 1}^s \left\{\underline{v}_k, \bar{v}_k \right\}{ {\frac{\partial Y_i(\rho)} {\partial \rho_k}} }. \end{align*} $ 
Then, under the switching law (9), the controller (6) with controller gain given by
Proof: Choosing
$ \begin{align} \left[{\begin{array}{*{20}c} {\Upsilon _{i, 11} } & {\Upsilon _{i, 12} } & I & 0 \\ {\Upsilon _{i, 21} } & {\Upsilon _{i, 22} } & 0 & I \\ I & 0 & {\gamma I} & 0 \\ 0 & I & 0 & {\gamma I} \\ \end{array}} \right] < 0 \end{align} $  (18) 
where
$ \begin{align*} \left[{\begin{array}{*{20}l} {\Upsilon _{i, 11} } & {\Upsilon _{i, 12} } \\ {\Upsilon _{i, 21} } & {\Upsilon _{i, 22} } \\ \end{array}} \right] & = \left[{\begin{array}{*{20}c} {\Xi _{i, 11}} & {\Xi _{i, 12}} \\ * & {\Xi _{i, 22}} \\ \end{array}} \right]\\ &\, + \left[{\begin{array}{*{20}c} {\tilde {X}_{i}^{1} (\rho)\tilde {X}_{i}^{1} (\rho)} & {\tilde {X}_{i}^{ 1} (\rho)\tilde {X}_{i}^{ 1} (\rho)} \\ { \tilde {X}_{i}^{ 1} (\rho)\tilde {X}_{i}^{ 1} (\rho)} & {\tilde {X}_{i}^{ 1} (\rho)\tilde {X}_{i}^{ 1} (\rho)} \\ \end{array}} \right] \end{align*} $ 
$ \begin{align*} \Xi _{i, 11} =&\ \text{He}\{[A_i (\rho)\tilde {X}_{i}^{1} (\rho) + B_i (\rho)K_i (\rho)\tilde {X}_{i}^{1} (\rho)]\} \\ &\, + \tilde {X}_{i}^{ 1} (\rho)\dot {\tilde {X}}_{i} (\rho, \dot{\rho})\tilde {X}_{i}^{ 1} (\rho)\\ \Xi _{i, 12} =&\, { B_i (\rho)K_i (\rho)\tilde {X}_{i}^{ 1} (\rho)} \\ \Xi _{i, 22} =&\ \text{He}\{[A_r \tilde {X}_{i}^{1} (\rho)]\} + \tilde {X}_{i}^{ 1} (\rho)\dot {\tilde {X}}_{i} (\rho, \dot{\rho})\tilde {X}_{i}^{ 1} (\rho). \end{align*} $ 
Because
$ {\frac{d}{{{d}t}}}[\tilde {X}_{i}(\rho)\tilde {X}_{i}^{1}(\rho)]=0. $ 
Then, we have
$ {\frac{d}{{{d}t}}[\tilde {X}_{i}(\rho)]} \tilde {X}_{i}^{1}(\rho)+\tilde {X}_{i}(\rho)\frac{d}{{\rm d}t}\tilde {X}_{i}^{1}(\rho)]=0. $ 
Finally we can obtain
$ \begin{align*} \tilde {X}_{i}^{1}(\rho){\frac{d}{{d}t}}[\tilde {X}_{i}(\rho)] \tilde {X}_{i}^{1}(\rho)={\frac{d}{{d}t}}[\tilde {X}_{i}^{1}(\rho)]. \end{align*} $ 
Defining
$ \begin{align*} \Xi _{i, 11} =&\ \text{He}\{[A_i (\rho)Y_i(\rho) + B_i (\rho)K_i (\rho)Y_i(\rho)]\} \\ &\sum\limits_{k = 1}^s \left\{\underline{v}_k, \bar{v}_k \right\} {{\frac{\partial Y_i(\rho)} {\partial \rho_k}} }\\ \Xi _{i, 12} =&\, { B_i (\rho)K_i (\rho)Y_i(\rho)} \\ \Xi _{i, 22} =&\ \text{He}\{[A_r Y_i(\rho)]\}  \sum\limits_{k = 1}^s \left\{\underline{v}_k, \bar{v}_k \right\}{{\frac{\partial Y_i(\rho)} {\partial \rho_k}} }. \end{align*} $ 
With Schur complement,
$ \begin{align*} \left[{\begin{array}{*{20}c} {\Upsilon _{i, 11} } & {\Upsilon _{i, 12} } \\ {\Upsilon _{i, 21} } & {\Upsilon _{i, 22} } \\ \end{array}} \right] = \left[{\begin{array}{*{20}c} {\Xi _{i, 11} } & {\Xi _{i, 12}} & {Y_i(\rho)} \\ * & \Xi _{i, 22} & {Y_i(\rho)} \\ * & * & {I} \\ \end{array}} \right]. \end{align*} $ 
Defining
$ \begin{align} \left[{\begin{array}{*{20}c} {X_i (\rho)} & 0 \\ 0 & {X_i (\rho)} \\ \end{array}} \right]  \left[{\begin{array}{*{20}c} {X_j (\rho)} & 0 \\ 0 & {X_j (\rho)} \\ \end{array}} \right] \ge 0 \end{align} $  (19) 
multiplying matrix diag
$ \begin{align} \left[{\begin{array}{*{20}c} {\tilde {X}^{1}_j (\rho)} & 0 \\ 0 & {\tilde {X}^{1}_j (\rho)} \\ \end{array}} \right]  \left[{\begin{array}{*{20}c} {\tilde {X}^{1}_i (\rho)} & 0 \\ 0 & {\tilde {X}^{1}_i (\rho)} \\ \end{array}} \right] \ge 0 \end{align} $  (20) 
consequently, the matrix inequality (17) can be obtained.
Remark 5: The diagonal structure of
Remark 6: The notation
We will apply the designed method to a turbofan engine model to show the effectiveness of the designed scheme.
A. The Switched LPV Model of AeroEnginesThe turbofan engine model is corresponding to a large, highbypass ratio twospool turbofan engine similar to the GE90. It is based on the data from GE90K engine of commercial modular aeropropulsion system simulation (CMAPSS) [29]. The input is
A switched LPV model is established with the method of curvefitting and small deviation linearization. A turbofan model is modeled by two different scheduling parameter sets as a switched LPV system. The turbofan engine model data is from [29]. The altitude and the fan speed are normalized by 10 000 and 3000, respectively. Then according to the curvefitting method, based on a family of local linearized models, a switched LPV model of turbofan engine is given as
$ \begin{align*} \dot x(t) = A_{i}(\rho)x(t) + B_{i}(\rho)u(t) +\omega(t) \end{align*} $ 
where
For simplicity we only consider the Mach number as the gain scheduling parameter and let the altitude as
$ \begin{align*} &A_1 = \left[{\begin{array}{*{20}c} {{\rm{3}}.{\rm{3786}}} & {{\rm{1}}.{\rm{3844}}} \\ {0.{\rm{7288}}} & {{\rm{4}}.{\rm{3411}}} \\ \end{array}} \right] + \rho \left[{\begin{array}{*{20}c} {{\rm{1}}.{\rm{3835}}} & {0.0{\rm{91}}0} \\ {{\rm{1}}.{\rm{2388}}} & {0.{\rm{4899}}} \\ \end{array}} \right]\\ &B_1 = \left[{\begin{array}{*{20}c} {240.6075} \\ {668.8695} \\ \end{array}} \right] + \rho \left[{\begin{array}{*{20}c} {1} \\ {105.8} \\ \end{array}} \right] \\ &A_2 = \left[{\begin{array}{*{20}c} {1.2267} & {0.3977} \\ {0.8172} & {0.6659} \\ \end{array}} \right] + \rho \left[{\begin{array}{*{20}c} {{\rm{1}}.{\rm{3204}}} & {0.4585} \\ {1.7429} & {2.{\rm{5165}}} \\ \end{array}} \right]\end{align*} $ 
$ \begin{align*} &B_2 = \left[{\begin{array}{*{20}c} {259.4093} \\ {588.7365} \\ \end{array}} \right] + \rho \left[{\begin{array}{*{20}c} {23.8} \\ {186.2} \\ \end{array}} \right]. \end{align*} $ 
The reference model is given as
$ \begin{align*} \dot{x}_r(t)=A_{r}x_{r}(t)+r(t), \quad x_r(0)=0 \end{align*} $ 
where
$ \begin{align*} A_{r}=\left[{\begin{array}{*{20}c} {2.915} & {1.0362} \\ {0.7871} & {3.4432} \\ \end{array}} \right]. \end{align*} $ 
We conduct simulation for the turbofan engine model with a varying parameter. The timevarying Mach number trajectory is shown in Fig. 1. Choosing disturbance
Download:


Fig. 1 The gain scheduling parameter. 
To solve the optimization problem (16) and (17) in Theorem 2, we obtain
$ \begin{align*} & Y_1(\rho)=\left[{\begin{array}{*{20}c} {2.2844} & {0.711} \\ {0.711} & {3.4579} \\ \end{array}} \right]+\rho \left[{\begin{array}{*{20}c} {0.0059} & {0.006} \\ {0.006} & {0.0143} \\ \end{array}} \right]\\ &Y_2(\rho)=\left[{\begin{array}{*{20}c} {1.5191} & {0.234} \\ {0.234} & {3.1166} \\ \end{array}} \right]+\rho \left[{\begin{array}{*{20}c} {0.0102} & {0.008} \\ {0.008} & {0.0535} \\ \end{array}} \right] \quad\quad\\ &K_{1}(\rho) =[0.0021, 0.0028]+ \rho [0.0010, 0.0004] \\ &K_{2}(\rho) =[0.0026, 0.0024]+ \rho[0.0005, 0.0001]. \end{align*} $ 
According to Theorem 2, we solve the
The
Download:


Fig. 2 The switching signal. 
Download:


Fig. 3 The tracking control error. 
Download:


Fig. 4 The fan speed increment. 
Download:


Fig. 5 The fuel flow increment. 
In this paper, we have studied the
[1]  D. Henrion, L. Reberga, J. Bernussou, and F. Vary, "Linearization and identification of aircraft turbofan engine models, " in Proc. IFAC Symp. Automatic Control in Aerospace, St Petersburg, Russia, 2004. https://www.researchgate.net/publication/222421234_Identification_of_photontagged_jets_in_the_ALICE_experiment 
[2]  D. K. Frederick, S. Garg, and S. Adibhatla, "Turbofan engine control design using robust multivariable control technologies, " IEEE Trans. Control Syst. Technol. , vol. 8, no. 6, pp. 961970, Nov. 2000. https://www.researchgate.net/publication/3332208_Turbofan_engine_control_design_using_robust_multivariable_control_technologies 
[3]  H. Zhao, J. F. Liu, and D. R. Yu, "Approximate nonlinear modeling and feedback linearization control for aeroengines, " J. Eng. Gas Turbines Power, vol. 133, no. 11, pp. 111601, May 2011. http://www.researchgate.net/publication/277455933_Approximate_Nonlinear_Modeling_and_Feedback_Linearization_Control_for_Aeroengines?ev=auth_pub 
[4]  W. Gilbert, D. Henrion, J. Bernussou, and D. Boyer, "Polynomial LPV synthesis applied to turbofan engines, " Control Eng. Pract. , vol. 18, no. 9, pp. 10771083, Sep. 2010. http://www.sciencedirect.com/science/article/pii/S0967066108001913 
[5]  J. X. Mu, D. Rees, and G. P. Liu, "Advanced controller design for aircraft gas turbine engines, " Control Eng. Pract. , vol. 13, no. 8, pp. 10011015, Aug. 2005. http://www.researchgate.net/publication/223522275_Advanced_controller_design_for_aircraft_gas_turbine_engines 
[6]  M. Pakmehr, N. Fitzgerald, E. M. Feron, J. S. Shamma, and A. Behbahani, "Gain scheduling control of gas turbine engines: stability by computing a single quadratic Lyapunov function, " in Proc. of ASME Turbo Expo 2013: Turbine Tech. Conf. and Expo. , San Antonio, Texas, USA, 2013. http://proceedings.asmedigitalcollection.asme.org/mobile/proceeding.aspx?articleID=1776311 
[7]  G. Wolodkin, G. J. Balas, and W. L. Garrard, "Application of parameterdependent robust control synthesis to turbofan engines, " J. Guid. Control Dyn. , vol. 22, no. 6, pp. 833838, Nov. 1999. http://www.researchgate.net/publication/239066319_Application_to_Parameter_Dependent_Robust_Control_Synthesis_to_Turbofan_Engines 
[8]  W. J. Rugh and J. S. Shamma, "Research on gain scheduling, " Automatica, vol. 36, no. 10, pp. 14011425, Oct. 2000. http://www.sciencedirect.com/science/article/pii/S0005109800000583 
[9]  G. J. Balas, "Linear, parametervarying control and its application to a turbofan engine, " Int. J. Robust Nonlinear Control, vol. 12, no. 9, pp. 763796, Jul. 2002. http://www.researchgate.net/publication/229885079_Linear_parametervarying_control_and_its_application_to_a_turbofan_engine 
[10]  H. Lin and P. J. Antsaklis, "Stability and stabilizability of switched linear systems: a survey of recent results, " IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 308322, Feb. 2009. http://www.researchgate.net/publication/224384599_Stability_and_Stabilizability_of_Switched_Linear_Systems_A_Survey_of_Recent_Results 
[11]  R. C. Ma and J. Zhao, "Backstepping design for global stabilization of switched nonlinear systems in lower triangular form under arbitrary switchings, " Automatica, vol. 46, no. 11, pp. 18191823, Nov. 2010. http://www.sciencedirect.com/science/article/pii/S0005109810003006 
[12]  D. Liberzon, Switching in Systems and Control. Boston, USA: Birkhäuser, 2003: 2134. 
[13]  J. Fu, R. C. Ma, and T. Y. Chai, "Global finitetime stabilization of a class of switched nonlinear systems with the powers of positive odd rational numbers, " Automatica, vol. 54, pp. 360373, Apr. 2015. http://dl.acm.org/citation.cfm?id=2784044.2784135 
[14]  M. S. Branicky, "Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, " IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 475482, Apr. 1998. http://www.researchgate.net/publication/3023026_Multiple_Lyapunov_functions_and_other_analysis_tools_for_switchedand_hybrid_systems 
[15] 
X. M. Sun, J. Zhao, and D. J. Hill, "Stability and 
[16]  R. C. Ma, Y. Liu, S. Z. Zhao, M. Wang, and G. D. Zong, "Nonlinear adaptive control for power integrator triangular systems by switching linear controllers, " Int. J. Robust Nonlinear Control, vol. 25, no. 14, pp. 24432460, Sep. 2015. http://www.researchgate.net/publication/264243268_Nonlinear_adaptive_control_for_power_integrator_triangular_systems_by_switching_linear_controllers 
[17]  B. Lu and F. Wu, "Switching LPV control designs using multiple parameterdependent Lyapunov functions, " Automatica, vol. 40, no. 11, pp. 19731980, Nov. 2004. http://dl.acm.org/citation.cfm?id=2239644 
[18]  B. Lu, F. Wu, and S. W. Kim, "Switching LPV control of an F16 aircraft via controller state reset, " IEEE Trans. Control Syst. Technol. , vol. 14, no. 2, pp. 267277, Mar. 2006. http://www.researchgate.net/publication/3332759_Switching_LPV_control_of_an_F16_aircraft_via_controller_state_reset 
[19] 
X. He and J. Zhao, "Multiple Lyapunov functions with blending for induced 
[20] 
S. Lim and J. P. How, "Modeling and 
[21]  F. Blanchini, S. Miani, and C. Savorgnan, "Stability results for linear parameter varying and switching systems, " Automatica, vol. 43, no. 10, pp. 18171823, Oct. 2007. http://www.sciencedirect.com/science/article/pii/S0005109807001975 
[22]  Q. K. Li, J. Zhao, and G. M. Dimirovski, "Robust tracking control for switched linear systems with timevarying delays, " IET Control Theory Appl. , vol. 2, no. 6, pp. 449457, Jun. 2008. http://www.researchgate.net/publication/3478232_Robust_tracking_control_for_switched_linear_systems_with_timevarying_delays 
[23] 
L. L. Hou, G. D. Zong, Y. Q. Wu, and Y. C. Cao, "Exponential 
[24]  A. Abdullah and M. Zribi, "Model reference control of LPV systems, " J. Franklin Inst. , vol. 346, no. 9, pp. 854871, Nov. 2009. http://www.sciencedirect.com/science/article/pii/S0016003209000787 
[25]  Y. Q. Huang, C. Y. Sun, C. S. Qian, and L. Wang, "Nonfragile switching tracking control for a flexible airbreathing hypersonic vehicle based on polytopic LPV model, " Chin. J. Aeronaut. , vol. 26, no. 4, pp. 948959, Aug. 2013. http://www.sciencedirect.com/science/article/pii/S1000936113000873 
[26]  Y. Y. Cao, Y. X. Sun, and C. W. Cheng, "Delaydependent robust stabilization of uncertain systems with multiple state delays, " IEEE Trans. Autom. Control, vol. 43, no. 11, pp. 16081612, Nov. 1998. http://www.researchgate.net/publication/3023191_Delaydependent_robust_stabilization_of_uncertain_systems_withmultiple_state_delays 
[27] 
L. X. Zhang and P. Shi, " 
[28] 
Q. G. Lu, L. X. Zhang, H. R. Karimi, and Y. Yang, " 
[29]  H. Richter, Advanced Control of Turbofan Engines. New York, USA: Springer, 2012. 