Design of Smooth Switching LPV Controller Based on Coprime Factorization and H∞ Performance Realization

This paper discusses the problem of designing smooth switching control based on the coprime factorization method. Aiming at the instantaneous chattering phenomenon generated by the linear parameter varying (LPV) controller during the switching moment, the moving region of the gain-scheduling variables is divided into a specified number of local subregions with overlapped region. Riccati inequality is used to solve the central controller for $H_{\infty} $ performance. The Youla free parameters are designed using the coprime factorization for each parameter sub-region by considering the system’s global and local performance requirements. Youla free parameter switching is used to improve the smoothness of switching and suppress transient response disturbance. Finally, the effectiveness of the method is verified by a simulation example.


I. INTRODUCTION
Since Shamma [1] proposed the linear parameter varying (LPV) system, it has solved the shortcomings of traditional variable gain control technology. The LPV system can better describe the nonlinear and time-varying characteristics of complex physical systems. In recent decades, it has quickly become an interesting research topic in control theory [2]- [4]. Typically, a single controller is designed to control an LPV plant. When the time-varying parameters change in an extensive range, the single LPV controller usually causes conservative control performance. Lu and Wu [5] proposed a design method of switching the LPV controller to solve this problem. The method of switching LPV controllers was quickly applied to some work, such as ultra-high-speed aircraft [6], [7], Aero-Engine [8]- [10], satellite attitude control [11]. Postma and Nagamune [12] used a switching LPV controller to regulate the air-fuel ratio. By dividing The associate editor coordinating the review of this manuscript and approving it for publication was Choon Ki Ahn. the parameter range into smaller sub-regions, a separate LPV controller was designed for each sub-region and switched according to the operating point in [12]. The LPV controller was found by solving the convex optimization problem of linear matrix inequality (LMI). Besides, Zhao and Nagamune [13] proposed an output feedback switching LPV controller design method. The switching controller can ensure the stability and gain performance of the closed-loop system when the measurement scheduling parameters are inaccurate.
To overcome the transient response of the switching controller at the switching moment, Chen et al. [14] proposed a smooth switching LPV controller, which attracted the interest of researchers, and the research results have been applied to various engineering fields [7], [15]. A smooth switching gain scheduling controller was designed to solve the problem of large-scale offshore wind turbines operating within the full wind speed range in [16]. Also, Chen [17] considered the LPV control of the delayed switching state feedback. It linearly interpolated the controller variables on the switching surface to achieve smooth switching during the switching process and switching on the overlap region. However, this method cannot quantitatively evaluate the switching smoothness and obtain the relative stability in overlapped subregions. Besides following the idea of linear interpolation of the control matrix on the switching surface, Hanifzadegan and Nagamune [18] introduced the smoothness index and imposed constraints on the controller's derivative matrix to make up for the defects in [17].
Coprime factorization approach for control and design of systems can be traced back to the literature [19]. Both Youla parametrization and dual Youla parametrization are based on the doubly coprime factorization [20]. Quadrat [21], [22] proposed Q parameters through the double coprime factorization of the transfer matrix. The controller designed by Youla parameterization can improve the system's robust stability and reject the adverse effects caused by existing disturbances. In [23], a Youla parameterized adaptive controller was proposed for the mechanical systems to deal with the unknown vibration caused both by the deterministic disturbance and the random disturbance. Bianchi and Sánchez-Peña [24] proposed a switching LPV controller design method based on the idea of Youla parameterization. The controller design is decomposed into two steps, one focused on ensuring global stability and the other on achieving the desired performance in each subset. But it does not consider the smooth switching strategy. It is very complicate to calculate the Youla parameters.
This work presents a new smooth switching technology to design the switching of Youla parameters. Compared with the literature [25], the novelty of this method is that we introduce a parameter overlap division method to improve the smoothness of switching. Also, we use the coprime factorization technique to obtain the Youla parameters of each sub-area from the controller. Then, the Youla parameter of the overlapping area is obtained by linear interpolation of the Youla parameter of the two adjacent sub-regions. The contribution of this word is to use the Youla parameter switch Q instead of the switch controller to achieve smooth switching. The flexibility and freedom of the switching controller are increased, and the transient response disturbance during the switching is suppressed. This paper is organized as follows. Section II presents the definition and the problem statement of controller design. Section III details the process of the smooth switching LPV controller which designed by Youla parameterization. Conclusions and future research directions are discussed in Section IV.

II. DEFINITION AND PROBLEM STATEMENT
We consider the following LPV plant where θ := θ 1 , . . . , θ s T is the vector of scheduling parameters, x ∈ R n is the state vector, w ∈ R n w is the vector of exogenous inputs, u ∈ R n u is the control input vector, z ∈ R n z is the vector of output signals related to the performance of the control system, y ∈ R n y is the measured output vector. The system matrices are the continuous and bounded function of the parameter θ and can be measured in real time. Assumption 1: (A(θ ), B 2 (θ ), C 2 (θ )) triple is parameterdependent stabilizable and detectable for all θ .
and for the switching LPV controller design, we can divide the interval into N sub-region, including overlapped regions and non-overlapped regions, as shown in Figure 1, where i is defined by the time-varying parameters of subregions. The overlapped regions are composed of two adjacent subregions, which can be expressed as The non-overlapped region i,i is the ith subregion except for the overlapped subregion, which can be expressed by the formula (4) In this paper, we define σ (t) : [t 0 , ∞) → N J = {1, 2, · · · , J } as the switching signal of the system, which is determined by the time-varying parameters θ . Therefore, switching between controllers only occurs on the boundary of the time-varying parameter subregion. Then, some important lemmas are provided as follows: Lemma 1: Given an LPV plant P 22 (θ ) with state space realisation (5) let a coprime factorization of the system P 22 (θ ) from (5) and a stabilizing controller K (θ ) be given by where the matrices N (θ ), M (θ ),Ñ (θ ),M (θ ), U (θ ), V (θ ), U (θ),Ṽ (θ) satisfy the double Bezout equation (8).
Then, the set of all regular controllers that are stable in the system can be parameterized as

Proof:
Assume that the controller K (θ) is an observer-based feedback controller given by wherē One possible way to construct the eight stable coprime matrices in (6) is then Based on the above coprime factorization of the system P 22 (θ) and the controller K (θ ). The parameterized formulas of all stabilizing controllers are obtained so that the system's stability can be guaranteed by a stable parameter Q(θ ) [26] where or by using the left decomposition form whereŨ Using the Bezout equation (8), the controller (10) or (11) can be realized by the lower linear fractional transformation in the parameter Q(θ ), where J k (θ ) is given as Remark 1: The formula (12) is the standard state-space H ∞ solution suggested by Doyle et al. [27]. This shows that the standard state-space H ∞ solution with a free parameter can also be obtained using constrained coprime factorization. If the controller can be obtained by Riccati methods, we can parameterize all stabilizing H ∞ controllers with free parameter Q using the proposed doubly coprime factorization.
Lemma 2 [28]: For the system (1), a set of stabilizing controllers K i (θ), i = 1, 2, · · · , p, for the system can be implemented as where J K 0 is formed as similar to (12) and the stable parameter Q i (θ ) is given by (14) or
By solving the inequations (15) and (16), a central controller with H ∞ control performance can be obtained . Remark 2: In this paper, we need to use Schur complementary lemma to transform Riccati inequality into LMI to obtain LPV central controller when solving Riccati inequality. The detailed derivation process can be found in [25] and [27]. When the solutions of Riccati inequality X (θ ) and Y (θ ) exist, the left and right constrained coprime factors can be found in any cases.

B. SMOOTH SWITCHING DESIGN OF YOULA PARAMETERS
To avoid an uneven transient response at the switching moment and improve the control system's performance, a smooth switching controller is designed by utilizing Youla parameterization. By using Youla parameter switching instead of controller switching, the control system structure is shown in Figure 2. The time-varying parameter set of the LPV system is divided into i subregions with overlaps. Each subregion is assigned a gain-scheduled LPV controller, and it is designed with the Youla parameter based on the coprime factorization. The following Theorem can be obtained by combining formula (12) and formula (14). (1), within the range of all parameters, Riccati inequalities (15) and (16) have symmetric non-negative stable solutions, if the time-varying parameter set of the LPV system is divided into i subregions with overlaps. By using coprime factorization, the set of all stabilization controllers can be represented by the Youla free parameter Q i (θ )

Theorem 1: Given an LPV controlled object
so that the Youla free parameter of smooth switching controller can be designed by Eq. (18) where the state space expression of Q i (θ ) is Proof: Since the central controller K ∞ (θ ) is solved by Riccati inequalities (15) and (16), then it is a stable controller that satisfies H ∞ control performance. According to the Lemma 1, it can be parameterized with coprime factorization.
The variation range of time-varying parameters is divided into several sub-intervals, and the LPV controller is designed for each parameter sub-region. Since the set of all stabilizing controllers can be represented by a Youla parameter Q, the controller can be represented by K i (θ) = F l (J k (θ ), Q i (θ )), where F l is the lower fraction transformation.
The region is divided into the overlapped regions and non-overlapped regions. As shown in Figure 1, non-overlapped sub-interval Youla parameter is Q i,i (θ ), overlapped sub-interval Youla parameter is Q i,i +1 .
According to the formula (14), Q i (θ ) is defined as To guarantee the Bezout characteristics similar to the formula (8), define the coprime factor as shown in the formula (19), by constructing Q i (θ ) ∈ RH ∞ , and each Q i (θ ) is stable, the Youla parameter of the non-overlapped region can be constructed as we can get where, 11 = A(θ ) + L i (θ )C 2 (θ ), 12 = B 2 (θ )F 0 (θ ) − L i (θ )C 2 (θ ), F 0 (θ ) is the feedback gain of the central solution, and A(θ) + B 2 (θ )F 0 (θ ) is stable. L 0 (θ ) is the observer gain with stable central solution, A(θ ) + L 0 (θ )C 2 (θ ) is stable. F i (θ ) is the feedback gain corresponding to each subregion of the parameter interval, L i (θ) is the observer gain corresponding to each subregion. The Youla parameter Q i,i+1 (θ) for overlapped subintervals is obtained by linear interpolation of the parameters Q i (θ ) and Q i+1 (θ ) of two adjacent subintervals. We construct the Youla parameter of the overlapped region through a convex combination, then the Youla parameter of the overlapped region is where α 1 + α 2 = 1. It is converted into a state-space expression that can be described as Then, the state space of the Youla parameter of the smooth switch controller in the overlapped region can be realized as follows Therefore, the Youla free parameter of smooth switching can be expressed as where Algorithm Given an LPV Plant Satisfying Assumptions (1-3), the Following Steps Lead to a Parameterized Controller 1. Solving central controller with Riccati inequalities (15) and (16). 2. According to Figure 1, the variation range of parameters is divided into subregions with overlaps. 3. Getting the parameter Q of Youla by the coprime factorization method. 4. According to the change of parameters, the switch of Youla parameter Q is performed.
The framework for parameterization of switching LPV controller can be summarized in the following.
Remark 3: The advantage of this design method is that we design the Youla parameters of each sub-regions from the controller based on the coprime factorization technology, and introduce the parameter overlap division method to improve the smoothness of the switching. It can increase the flexibility and freedom of the switching controller and effectively suppress the disturbance during the switching. However, this method has the disadvantage that the controller's parameterization will increase the complexity of the controller and increase the amount of calculation.

IV. EXAMPLE
In this section, two examples are given to verify the effectiveness of the method in this article.
Example 1: we apply the technique of smooth switching LPV controllers to the system model of Firebee aircraft. The Firebee aircraft is a morphing aircraft, which is a complex parameter variable system that depends on the shape and structure. It can be modeled as an LPV system, and its scheduling parameter is the rate of change of the wing sweep angle. The parameters of this example can be found in [29]. where The state variable x = [ α, q] T represents the angle of attack and the pitch rate. The control input u = δ e denotes the elevator deflection.
The robust LPV control system structure of the Firebee aircraft is shown in Figure 3. where W e is the tracking error weight, W ω is the measurement noise weight, W u is the control weight, each weighting function is chosen as It is assumed that the variation rate of the wing sweep angle is as shown in Fig.4. When the smooth switching controller is used, there are four times of switching in 21.6s, 25.6s, 38s and 42.6s respectively, also marked with a small circles in Fig.4. Then, we perform simulations on the nonlinear model of morphing aircraft using the switching Youla parameter Q and switching controllers K, respectively. The simulation results are shown in Fig.5 and Fig.6. Fig.5 shows the controller output response. The blue solid line represents switching Youla parameter Q. The red dotted line represents switching controller K without Youla parameterization optimization. We can see that the control input with the interference of switching Youla parameter Q is smaller than switching controller K when switching occurs.    6 shows the system output response. It can be observed that using the switching controllers K. The system output has sudden undesirable transient behavior at the switching point. In contrast, using the switching Youla parameter Q, the anti-interference ability of the system output response from the switching Q is better than the switching K.
Example 2: To further demonstrate the effectiveness of the algorithm proposed above, another example is given.
where θ is a real parameter satisfying −2 θ 2 and can be measured online, w(t)is external input signal. The parameter region is divided into two overlapped subsets: [−2, 0.7], [−0.2, 2], the overlapping region is [−0.2, 0.7]. The time-varying parameter trajectory is shown in Fig. 7.   Switching occurs when the parameter trajectories cross the switching surface at 14.5s, 16.75s, 43.25s and 45.5s. The reference input select the step signal to enter at 5 seconds.
The simulation results are shown in Fig.8 and Fig.9. The dotted line represents the method of non-smooth switching in literature [24], and the solid line represents the method in this paper. It is obvious that the anti-disturbance capability of smooth switching is better than non-smooth switching.

V. CONCLUSION
This paper has investigated the problem with designing a smooth switching LPV controller based on Youla parameterization. The design of the switched controller mainly involves two steps. Firstly, the Riccati inequality is used to solve the central controller that satisfies the H ∞ control performance. The range of parameters is divided into several overlapping subregions, and an LPV controller is designed for each subregion. Secondly, each sub-area controller is parameterized by using the coprime factorization method to obtain the corresponding Youla parameters. The switch of Youla parameter Q is used to replace the switch controller to achieve smooth switching. This method can suppress the disturbances during switching, which satisfies the higher degree of freedom and flexibility to achieve better control performance. To verify the effectiveness of the proposed method, a morphing aircraft model is used as a simulation example. By applying the proposed smooth switching controller to the morphing aircraft, the flight system exhibits excellent stability and robustness and can switch smoothly. The method proposed in this paper can be extended to nonlinear systems, adaptive control systems, and other fields. Meanwhile, these results can be applied to the lathe vibration system, magnetic fluid deformable mirror system, and intelligent transportation system (ITS). TOSHIO EISAKA received the Ph.D. degree from Hokkaido University, in 1991. He was an Assistant Professor with Hokkaido University. He is currently a Professor with the Division of Information and Communication Engineering, and the Vice President of Kitami Institute of Technology. His research interests include the areas of control system design and its application, robust control, data science, and human-robot interaction.
LIEJUN LI received the B.S. degree in iron and steel metallurgy from Wuhan University of Science and Technology, China, in 1983, the M.S. degree from the University of Science and Technology Beijing, China, in 1990, and the Ph.D. degree in iron and steel metallurgy from Shanghai University, China, in 2005. He has presided over 32 scientific and technological projects at all levels, including the National 863 and 973. He has published 83 articles in academic journals and owned ten national invention patents. Sixteen of his achievements have won national, provincial and municipal science and technology awards, including two second prizes for national science and technology progress, six first prizes for provincial and ministerial science and technology progress, one National Excellent Patent Award, and one Guangdong Patent Gold Award. VOLUME 9, 2021