Hybrid Switched Gain-Scheduling Control for Missile

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 39, No. 10, October 2016

Hybrid Switched Gain-Scheduling Control for Missile Autopilot Design

Downloaded by CLARKSON UNIVERSITY on October 6, 2016 | http://arc.aiaa.org | DOI: 10.2514/1.G001791

Chengzhi Yuan∗ University of Rhode Island, Kingston, Rhode Island 02881 Yang Liu† Harbin Institute of Technology, Harbin 150001, China Fen Wu‡ North Carolina State University, Raleigh, North Carolina 27695 and Chang Duan§ Prairie View A&M University, Prairie View, Texas 77446 DOI: 10.2514/1.G001791 This paper presents a new hybrid switched gain-scheduling control method for missile autopilot design via dynamic output feedback. For controller design purpose, the nonlinear missile plant model is first converted to a switched linear fractional transformation system. Then, the new hybrid switched gain-scheduling autopilot is designed, which consists of a switching dynamic output–feedback linear fractional transformation controller and a supervisor enforcing a controller state reset at each switching time instant. The proposed hybrid control scheme is shown to provide a systematic yet simple framework for missile autopilot design. Specifically, the control synthesis conditions that guarantee weighted L2 stability performance are formulated in terms of a finite number of linear matrix inequalities, which can be solved effectively via convex optimization without parameter-space gridding. Furthermore, stringent controlled performance and strong robustness against parameter perturbations are achieved using this new control approach, whereas no parameter variation information is required for both controller synthesis and implementation. The advantages of the proposed design approach over existing methods will be shown through nonlinear simulations for the missile autopilot design over a wide range of operating conditions.

I.

autopilots with guaranteed reliability (e.g., [4–12] and references therein). Typically, the LPV-based gain-scheduling control techniques for missile autopilot designs can be classified into two categories with different representations of the missile dynamics. The first one is the quasi-LPV representation. As shown in Fig. 1a, the missile dynamics are described by a quasi-LPV model Gρt whose state space matrices depend on a set of time-varying parameters ρt. ρt may contain some plant state variables (e.g., the angle of attack variable α for the pitch-axis missile model considered in this paper), and it is assumed to be measurable in real time for scheduling the controller gains Kρt. The second one is the linear fractional transformation (LFT) representation. As seen in Fig. 1b, different from the former case, the missile dynamics is represented as a feedback interconnection of G and Θt. G constitutes the nominal linear time-invariant (LTI) dynamics of the missile model, while the Θt block contains all the time-varying/ scheduling parameters, serving similar roles as ρt in the quasiLPV case. Because of the space limitation, refer to [13–15] for more descriptions about the LPV/LFT gain-scheduling control techniques. Current researches of missile LPV control mainly focus on constructing different types of Lyapunov functions. Two types of Lyapunov functions frequently adopted in the field include constant Lyapunov functions (CLFs) [11,14,16,17] and parameterdependent Lyapunov functions (PDLFs) [12,18–22], which lead to two respective synthesis frameworks: the scaled small-gain framework and the dissipative systems framework. With the CLFbased scheme, the underlying missile LPV system is specified as an LFT form, and the LFT gain-scheduling controller is fully characterized by a finite number of LMIs [11]. When compared to the CLF-based scheme, the PDLF-based scheme provides a more general and flexible framework for missile control synthesis. However, there are several problems associated with this scheme, which could largely limit its applicability in practice. The most critical problem is that the synthesis conditions are often formulated as parameter-dependent LMIs involving infinite-dimensional LMIs,

Introduction

T

HE autopilot design for modern missiles is a well-known, challenging problem, not only because the missile itself is a highly complex nonlinear system that involves nonminimum phase dynamics and has wide parameter variations during the missile operation, but also because it must be capable of providing stringent controlled performance across the entire flight envelope. Gain scheduling, as one of the most popular nonlinear control design techniques, has been demonstrated to be successful for missile autopilot design [1–3]. In traditional gain scheduling, a fundamental guideline for missile autopilot design is to gain schedule local robust controllers that are designed for several linearized missile models at their respective equilibrium points, so as to yield a global controller. Satisfactory performance could be achieved by using such traditional gain-scheduling controllers. However, due to the local nature of traditional gain-scheduling design, the reliability of such controllers remains questionable for the entire flight envelope, especially during the rapid transitions in the missile endgame. With the development of linear parametervarying (LPV) control theory, LPV-based gain-scheduling techniques have been proposed to pursue high-performance missile

Received 16 October 2015; revision received 14 May 2016; accepted for publication 1 June 2016; published online 8 August 2016. Copyright © 2016 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal and internal use, on condition that the copier pay the per-copy fee to the Copyright Clearance Center (CCC). All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the ISSN 0731-5090 (print) or 1533-3884 (online) to initiate your request. *Department of Mechanical, Industrial and Systems Engineering; [email protected]. † Center of Control Theory and Guidance Technology; liuyang5264@163. com. ‡ Department of Mechanical and Aerospace Engineering; [email protected]. edu (Corresponding Author). § Department of Mechanical Engineering; [email protected]. 2352

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Fig. 1

Two gain-scheduling missile control design frameworks.

which are generally difficult to solve. A standard approach to reduce this difficulty is to apply the gridding technique for the entire value set of the gain-scheduling variables (in this case the angle of attack α and the Mach number M for missile problem) and then to solve the resulting finite-dimensional optimization problem to obtain an approximate solution of the original synthesis problem. This approximated solution could be risky for stable missile control and typically requires extensive postanalysis to verify its validity [15]. Moreover, due to the lack of a systematic way of constructing the PDLFs, finding a proper PDLF could be quite involved and time consuming. For the missile problem, it came with extensive simulation and a complicated design procedure. Another potential problem encountered by the PDLF-based scheme is the requirement of parameter variation information for controller synthesis and implementation. This is generally prohibitive, because parameter derivatives either are not available or are difficult to estimate during system operation, especially for missile systems with fast and wide parameter variations. As such, from a practical consideration, the CLF-based scheme might be more suitable for missile autopilots design. However, one drawback of such an approach is that using a single constant Lyapunov function might be too conservative and the resulting controller might not be able to meet the performance requirement, especially when a large parameter variation range is considered. To overcome the deficiencies mentioned above, this paper will propose a new hybrid missile autopilot design approach based on a switching LFT control technique. With this technique, the entire parameter region is partitioned into several small subregions such that a family of LFT controllers can be designed for multiple parameter subregions, and better controlled performance can be achieved by switching among these controllers instead of simply using a single gain-scheduling controller derived from the CLFbased scheme. More important, compared to the PDLF-based scheme, the proposed hybrid switching control scheme will not only significantly reduce computational complexity in controller synthesis by avoiding parameter-space gridding, but also facilitate controller implementation because no derivative information of the scheduling parameters is required. For switched control systems, which can be described by an interaction between continuous time dynamics and discrete switching events [23], the problem of stabilization is a great challenge because of the switching signal involved. Typical methods for analysis and design of switched systems include using a common Lyapunov function against arbitrary switching [23] and using multiple Lyapunov functions for controlled switching [24,25]. The methodology of incorporating switching techniques into the LPV/LFT control design, which defines the switching LPV/ LFT control technique, was firstly presented in [26] and further extended in [27] by introducing average dwell-time switching logic [28]. Further investigations along this line can be found in [29,30]. It has been demonstrated that the switching LPV/LFT control technique is effective in improving the performance of LPV control systems. However, there are several problems yet to be addressed adequately in most of the current work, such as eliminating bilinear matrix inequality constraints in controller synthesis

and enhancing the implementability of LPV/LFT controllers [29,30]. In this paper, we present a new hybrid gain-scheduling control method for missile autopilot design by using the switching LFT control technique with the average dwell-time logic. For controller design purposes, the nonlinear missile dynamics will be first modeled as a switched LFT system by partitioning the entire operating region of the scheduling variables (the angle of attack α and the Mach number M) into several subregions. Then, based on this setup, the new hybrid switched gain-scheduling autopilot design scheme is proposed, which consists of a switching dynamic output–feedback LFT controller and a supervisor that monitors the switching signal and enforces a reset rule for the controller states at each switching time instant. The proposed hybrid control scheme provides a systematic yet simple framework for high-performance missile autopilot design. It is advantageous over existing missile control methods in the following important ways. First, the associated control synthesis conditions that guarantee weighted L2 stability performance are formulated in terms of a finite number of LMIs. As a result, the controller coefficient matrices can be readily solved through a convex optimization problem without the gridding and postanalysis procedures that are typically required in most existing LPV design techniques [12,15,29]. Second, stringent controlled performance over a wide range of operating conditions and strong robustness against perturbed system parameters are achieved without using scheduling parameter variations in both controller synthesis and implementation. On the other hand, it is worth mentioning that the current results have extended the methodologies of Yuan and Wu [30] from switched linear systems to switched parameter-dependent LFT systems with more general settings and have been successfully applied to solve the missile autopilot design problem. Furthermore, in contrast to Lu et al. [29], the new controller state reset mechanism proposed in this paper does not require full plant state information for controller state reset use, which greatly facilitates the controller’s implementation. The rest of this paper is organized as follows. Section II presents the hybrid switched gain-scheduling controller for a class of switched LFT systems with average dwell time. The controller synthesis conditions are also given in this section. Section III then applies the proposed approach to the missile benchmark problem. This includes a derivation of a switched LFT model for the missile, the control problem setup, and the discussion on controller synthesis results and nonlinear simulations. Finally, Sec. IV concludes the paper. The following notations are used in this article. R stands for the set of real numbers, and R is the set of positive real numbers. Rm×n is the set of real m × n matrices. The transpose of a real matrix M is denoted by MT. The hermitian operator Hef·g is defined as HefMg M MT for real matrices. The identity matrix of dimension n × n is denoted by In. Sn and Sn are used to denote the sets of real symmetric n × n matrices and positive definite matrices, respectively. If M ∈ Sn , then M > 0 M ≥ 0 indicates that M is a positive definite matrix (positive semidefinite) and M < 0 M ≤ 0 denotes a negative definite (negative semidefinite) matrix. A block diagonal matrix with matrices X 1 ; X2 ; · · · ; Xp on its main diagonal is denoted by diagfX1 ; X 2 ; · · · ; Xp g. Furthermore, we use the symbol ⋆ in LMIs to denote entries that follow from symmetry. For x ∈ Rn, its norm is defined as kxk ≔ xT x1∕2 . The space of square integrable functions T 1∕2 is denoted by L2 ; that is, for any u ∈ L2, kuk2 ≔ ∫ ∞ 0 u tut dt is finite. For two integers k1 < k2 , we denote Ik1 ; k2 fk1 ; k1 1; · · · ; k2 g.

II.

Hybrid Switched Gain-Scheduling Control of Switched LFT Systems

As shown in Fig. 2, we consider an open-loop switched LFT parameter-dependent system described by

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x_ p

2

3

Ap;σ

6 7 6 C 6 qp 7 6 7 6 p0;σ Gσ : 6 6 e 76 6 4 5 4 Cp1;σ Cp2;σ

y

Bp0;σ

Bp1;σ

Dp00;σ

Dp01;σ

Dp10;σ

Dp11;σ

Dp20;σ

Dp21;σ

32 3 xp 76 7 7 Dp02;σ 76 pp 7 7 76 6 d 7 Dp12;σ 7 4 5 5 Dp22;σ u Bp2;σ


pp Θ σ q p

(1)

in which xp ∈ Rn is the plant state, u ∈ Rnu is the control input, e ∈ Rne is the controlled/performance output, d ∈ Rnd is the disturbance, y ∈ Rny is the measurement output, and pp , qp ∈ Rnp are the pseudo-input and pseudo-output. σ is a piecewise constant function of time, called a switching signal, which takes its values in the finite set I1; N p ; N p > 1 is the number of subsystems. It is assumed that the LFT representation in Eq. (1) is well posed; that is, I np − Dp00;i Θi with i ∈ I1; N p are invertible for any allowable parameter values. Furthermore, the time-varying parameter Θi satisfies kΘi k ≤ 1, ∀ i ∈ I1; N p and obeys the following structure: Θi fdiagfθ1;i I r1;i ; θ2;i I r2;i ; · · · ; θs;i I rs;i g∶jθj;i g ≤ 1; j ∈ I1; sg; i ∈ I1; N p

(2)

P in which sj1 rj;i np represents the order of the corresponding LFT for all i ∈ I1; N p . The time-varying parameter Θi is assumed to be available in real-time for gain-scheduling control use. All of the state-space data are real and assumed to be known for control design. Two assumptions concerning the switched LFT plant in Eq. (1) are also given: Assumption (A1): The triple Ap;i ; Bp0;i

Cp0;i Bp2;i ; Cp2;i

is stabilizable and detectable for all Θi satisfying Eq. (2) and i ∈ I1; N p . Assumption (A2): Dp22;i 0 for all i ∈ I1; N p . Note that these two assumptions are quite standard in the context of LPV control. Assumption (A1) guarantees the existence of a stabilizing gain-scheduling output–feedback controller for each subsystem in Eq. (1), whereas assumption (A2) considerably simplifies the derivation. This technical assumption can be relaxed with more complicated formulas (see [15] for more details). The proposed hybrid gain-scheduling control scheme (as shown in Fig. 2) consists of a standard switching dynamic output–feedback controller Cσ and a supervisor enforcing a reset rule to the controller state at each switching time instant. σ is the switching signal governing the switching between the subsystems. It is assumed everywhere that σ is continuous from the right. For the missile control problem, the value of switching signal σ depends on the time-varying parameter Θσ (which will be clarified in Sec. III), and

thus governs the dynamic behaviors of the closed-loop system. As such, the switching signal σ will be incorporated into the hybrid controller for the missile autopilot design to achieve overall stability and the desired controlled performance. The switching gainscheduling controller Cσ has one-to-one correspondence with the switched LFT plant Gσ . Specifically, during the flowing time intervals, the switched LFT plant Gσ will be controlled using the switching gain-scheduling controller Cσ . Once switching occurs, the controller state will undergo a reset/jump action at the switching time instant. The introduction of a reset mechanism into the switching controller structure is motivated from [30]; interested readers are referred to this reference for more details. A switched Lyapunov function consisting of multiple quadratic Lyapunov functions [24], together with an average dwell-time (ADT) logic [28], will be used for stability analysis and control design. The switched Lyapunov function can be defined as V σ x xT Pσ x

in which Pσ is a positive definite matrix with σ ∈ I1; N p . Each Lyapunov function V i x, i ∈ I1; N p corresponds to one particular subsystem. For ADT switching control, the following definition is first recalled: Definition 1 ([28]): A switching signal σ is said to possess the property of ADT if there exist two positive numbers N 0 and τa such that N σ T; t ≤ N 0

T−t ; τa

The proposed hybrid control scheme.

0≤t≤T

(4)

in which N σ T; t denotes the number of switches over the time interval t; T, τa and N 0 are the average dwell time and the chatter bound, respectively. Remark 1: The concept of ADT switching has been extensively used in the literature for stability analysis and stabilization of switched systems [28,30,31]. The ADT constraint in Eq. (4) essentially provides an upper bound on the switching frequency that occurred during a finite interval t; T. The special case of N 0 1 reduces to a dwell-time condition in which any two consecutive switches must be separated by at least τa units of time. Also, as an extreme case, τa → 0 implies that the constraint on the switches almost disappears and arbitrary fast switching is allowed. Different from traditional Lyapunov-based analysis, to guarantee stability of a switched control system under the ADT switching logic, the value of the discontinuous Lyapunov function V σ is not necessarily decreasing along the entire system trajectory but instead is allowed to increase within a bound at each switching time instant. The possible increases of the Lyapunov function will be compensated by its decreasing during the dwell-time interval by limiting the switching frequency over a given time interval no greater than 1∕τa . The constraint on Lyapunov functions at each switching time instant, which is the so-called boundary condition, is formulated as 1 V ≤ V i ≤ μV j ; μ j

Fig. 2

(3)

∀ i; j ∈ I1; N p

(5)

in which μ > 1 is a prespecified scalar. In most previous studies of dwell-time switching output– feedback control, the controller synthesis conditions usually turn out to be nonconvex due to the existence of the boundary condition in Eq. (5). By embedding a reset loop into the switching control structure (as shown in Fig. 2), this problem will be resolved following the methodology from [30]. On the other hand, the boundary condition in Eq. (5) in dwell-time switching control also leads to another difference from other switching control techniques; that is, the controlled performance of the resulting closedloop system is measured by a weighted L2 gain γ in the sense of [31],

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Z

∞

e−λt eT tet dt ≤ βx0 γ 2


0

Z

∞

dT tdt dt

(6)

0

for some positive number λ and some class K∞ function βx. Now, we are in the position to consider the control synthesis problem for the switched LFT system in Eq. (1) using dynamic output– feedback controllers. Different from most existing switching control methods [29], we will assume that, in addition to the measurement output signal, partial plant states of Eq. (1) are also measurable online for feedback control use. This is particularly true for the missile control problem. It will be shown in Sec. III that the proposed hybrid controller will be implemented by directly using the state information from the robust performance weighting functions. Specifically, based on the switched LFT plant in Eq. (1), we assume without losing any generality that xp xTp1 xTp2 T , with xp2 being exactly measurable in real time without noise, xp1 ∈ Rn1 , xp2 ∈ Rn2 , and n1 n2 n. In such a case, we can partition the measurement output y of Eq. (1) as y ≔ yT1 yT2 T with y1 ∈ Rny1 , y2 ∈ Rn2 , and ny1 n2 ny , such that y2 xp2 . Accordingly, the output matrices Cp2;i ; Dp20;i ; Dp21;i can be rewritten as follows for all i ∈ I1; N p : C p2;i D p20;i D p21;i Cp2;i Dp20;i Dp21;i (7) 0 I n2 0 0 Then, based on the hybrid control scheme as shown in Fig. 2, we will construct the following hybrid switched gain-scheduling controller for the switched LFT plant (1).

2

x_ k

3

2

Ak;σ

Bk0;σ

Bk1;σ

6 7 6 Cσ : 4 qk 5 4 Ck0;σ Dk00;σ Dk01;σ u

Ck1;σ Dk10;σ Dk11;σ

pk Θ σ q k x k Δ1;ij Δ2;ij

y2 xk

;

2 3 3 xk Bk2;σ 6 7 6 pk 7 7 Dk02;σ 7 56 6y 7 4 15 Dk12;σ y2

when switching occurs

(8)

Here, xk ∈ Rn1 is the controller state and Δ1;ij ∈ Rn1 ×n2 and Δ2;ij ∈ Rn1 ×n1 are the reset matrices. The two subscripts of the reset

matrices i; j ∈ I1; N p , with i ≠ j, are used to denote the indices of the preswitching subsystem i and the postswitching subsystem j, respectively. pk , qk ∈ Rnp are the pseudo-input and pseudo-output connecting to the parameter block Θi . The controller gain matrices Ak;i ;Bk0;i ;Bk1;i ; Bk2;i ;Ck0;i ;Ck1;i ;Dk00;i ; Dk01;i ;Dk02;i ;Dk10;i ;Dk11;i ; Dk12;i of compatible dimensions and the reset matrices Δ1;ij , Δ2;ij are subject to design, whereas the gain-scheduled matrices Θi , ∀ i ∈ I1; N p are copied from the open-loop plant (1). With the open-loop LFT plant (1) and the hybrid controller (8), the resulting hybrid closed-loop system can be deduced as

2

x_ cl

3

2

Acl;σ

Bcl0;σ

Bcl1;σ

32

xcl

3

6 7 6C 4 qcl 5 4 cl0;σ

Dcl00;σ

6 7 Dcl01;σ 7 54 pcl 5

Ccl1;σ

Dcl10;σ

Dcl11;σ

e

d

pcl Θcl;σ qcl x cl Ar;ij xcl ;

when switching occurs

in which xcl xTp1 xTp2 xTk T , pcl pTp and

pTk T , qcl qTp

(9)

qTk T ,

Recall that σ ∈ I1; N p is the switching signal. Based on the above definitions of the two subscripts of Ar;ij , we have i σ and j σ . The superscript “” stands for the time instant immediately after the switching, because a switching occurs instantaneously. The synthesis conditions for the hybrid controller in Eq. (8) are summarized in the following theorem. Theorem 1: Consider the open-loop switched LFT system in Eq. (1). Given two positive scalars, λ0 ∈ R and μ > 1, if there exist positive n definite matrices Ri ∈ Sn , S1;i ∈ Sn1 , and Li ; Ji ∈ Sp , rectangular n1 ×n2 n ×n , A^ k;i ∈ R 1 2 , B^ k0;i B^ k1;i B^ k2;i ∈ matrices S2;i ∈ R n1 ×np ny1 n1 ^ R , Ck0;i ∈ Rnp ×n2 , C^ k1;i ∈ Rnu ×n2 , D^ k00;i D^ k01;i D^ k02;i ∈ np ×np ny1 n1 , D^ k10;i D^ k11;i D^ k12;i ∈ Rnu ×np ny1 n1 , and R ^ ^ Δ1;ij Δ2;ij ∈ Rn1 ×n1 n2 , and a positive scalar γ ∈ R, such that the following conditions hold for all i; j ∈ I1; N p with i ≠ j,

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2


6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

HefAp;i Ri Bp2;i D^ k12;i C^ k1;i g λ0 Ri I n1 0 ATp;i C Tp2;i D^ Tk11;i BTp2;i B^ k2;i A^ k;i λ0 I n1 0 Li BTp0;i D^ Tk10;i BTp2;i BTp0;i D Tp20;i D^ Tk11;i BTp2;i BTp1;i D Tp21;i D^ Tk11;i BTp2;i Cp0;i Ri Dp02;i D^ 12;i C^ k1;i D^ k02;i C^ k0;i Cp1;i Ri Dp12;i D^ k12;i C^ k1;i

I S2;i Ap;i B^ k1;i C p2;i n1 λ0 S1;i 0 T B^ k0;i T Bp0;i S1;i S2;i T D Tp20;i B^ Tk1;i BTp1;i S1;i S2;i T D Tp21;i B^ Tk1;i Cp0;i Dp02;i D^ k11;i C p2;i In1 0 T J i Cp0;i D^ k01;i C p2;i In1 0 T Cp1;i Dp12;i D^ k11;i C p2;i In1 0 T 3 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 7 7 ⋆ ⋆ ⋆ ⋆ ⋆ 7 7 −Ji ⋆ ⋆ ⋆ ⋆ 7 7 0 −γI nd ⋆ ⋆ ⋆ 7 7 7 Dp00;i Dp01;i 0

(12)

>0

(13)

so that M1;i ; N 1;i ∈ Rn1 ×n1 are invertible and M2;i ; N 2;i ∈ Rn2 ×n1 . 3. Compute the controller matrices Ak;i , Bk0;i , Bk1;i , Bk2;i , Ck0;i , Ck1;i , Dk00;i , Dk01;i , Dk02;i , Dk10;i , Dk11;i , Dk12;i and Δ1;ij , Δ2;ij for all i; j ∈ I1; N p and i ≠ j as

lnμ ≔ λ0

(14)

and the weighted L2 gain from the disturbance d to the controlled output e is less than γ. Moreover, the controller matrices are obtained through the following algorithm: 1. Partition matrices Ri

R1;i RT2;i

0

with R1;i , R1;i ∈ 0

R2;i R3;i

Sn1 ,

0

R2;i , R2;i ∈ R 0

R−1 i

and

T

n1 ×n2

0

R1;i

R2;i R3;i

0T

, R3;i ; R3;i ∈ −1

0

R2;i 0

0

0

Sn2

and let

Dk11;i D^ k11;i Dk12;i Ck1;i f D^ k12;i C^ k1;i − Dk11;i C p2;i Ri gΩ−1 i ^ Bk1;i N −1 1;i fBk1;i − S1;i S2;i Bp2;i Dk11;i g ^ ^ Bk2;i Ak;i N −1 1;i f Bk2;i Ak;i − S1;i S2;i Ap;i Ri Bp2;i Dk11;i C p2;i Ri Bp2;i Dk12;i Ck1;i Ωi − N 1;i Bk1;i C p2;i Ri gΩ−1 i Dk10;i D^ k10;i − Dk11;i D p20;i Li U−T i ^ Dk01;i W −1 i Dk01;i − J i Dp02;i Dk11;i T ^ Dk00;i W −1 i fDk00;i − J i Dp00;i Li Dp02;i Dk10;i U i

− J i Dp02;i Dk11;i W i Dk01;i D p20;i Li gU−T i ^ Bk0;i N −1 1;i fBk0;i − S1;i S2;i Bp0;i Li Bp2;i Dk11;i Dp20;i Li Bp2;i Dk10;i UTi − N 1;i Bk1;i D p20;i Li gU−T i

0

S3;i R3;i R2;i − S2;i S1;i − R1;i R2;i − S2;i S for all i ∈ I1; N p0. Then, we have Si ≔ 1;i ST2;i the matrix S1;i − R1;i is invertible [30].

I np − Li J i Ui W Ti

in which X i ∈ Rn1 ×n1 , and define Mi ≔ −Ri N i X i such that Mi , N i satisfy the identity Si Ri N i MTi In. Furthermore, we partition Mi , N i as M1;i N 1;i ; Ni Mi M2;i N 2;i

in which Li ; J i ; i ∈ I1; N p are block-diagonal matrices commutable with Θi , then the hybrid closed-loop system in Eq. (9) is exponentially stabilized by the hybrid controller in Eq. (8) for every switching signal σ with average dwell time¶ τa

(11)

2. Solve N i ∈ Rn×n1 , Ui ; W i ∈ Rnp ×np for all i ∈ I1; N p through the following factorizations, respectively:

⋆

Ri

I n1

⋆

He S1;i

S2;i > 0. Note that S3;i

Note that the average dwell time τa is explicitly determined by the values of λ0 and μ. For the convenience of presentation, these two parameters λ0 and μ will be called the dwell-time parameters throughout this paper.

^ ^ Dk02;i Ck0;i fW −1 i Dk02;i Ck0;i − J i Cp0;i Ri − Ji Dp02;i D^ k12;i C^ k1;i − Dk01;i C p2;i Ri gΩ−1 i −1 ^ ^ Δ1;ij Δ2;ij N −1 1;j f Δ1;ij Δ2;ij − S1;j S2;j Ri gΩi

¶

in which Ωi ≔

h RT

2;i

MT1;i

R3;i MT2;i

i .

(15)

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Proof: The proof is based on the scaled bounded real lemma and the congruence transformation technique [32]. For the closed-loop system in Eq. (9) with Eq. (10), we define the multiple quadratic Lyapunov functions as V i xcl xTcl Pi xcl , ∀ i ∈ I1; N p and partition the Lyapunov function matrices Pi and the scaling matrices Λi according to the dimensions of the plant and controller states as follows.

originally given in a quasi-LPV form [7]) into the switched LFT design framework, we first convert the missile model to a switched LFT system in the form of Eq. (1). Then, a hybrid missile autopilot is designed to meet the stability and performance specifications by solving LMI optimization problem (16). A. Switched LFT Modeling of Missile

The pitch-axis missile model taken from [15] is described in the quasi-LPV form, α_ q_

We specify 2


Z1;i

6 6 4

T 1;i

Ri

MT1;i Li

UTi

MT2;i I np ; 0

I n1

3

0

2

7 7; 5

Z2;i

0

1;i

7 ST2;i 5 N Ti

0

T 2;i

S 3

6 In 4

I np

Ji

0

W Ti

such that Pi Z1;i Z2;i and Λi T 1;i T 2;i . Then, we have Mi T T −T T MT2;i . −Ri N i Xi and Y −1 i −W i Li U i , in which Mi ≔ M1;i Conditions (12) and (13) confirm that Pi > 0 and Λi > 0. According to [32] and [31], to establish switching stability and weighted L2 -gain performance for the closed-loop system in Eq. (9), we need to further prove the following Lyapunov conditions: 1 V_ i qTcl Λi qcl − pTcl Λi pcl eT e − γdT d < 0 V j − μV i ≤ 0 γ for all i; j ∈ I1; N p and i ≠ j. These two conditions can be further converted into matrix inequalities by using the scaled bounded real lemma and Schur complement [32]; that is, 2 6 6 6 6 6 6 6 4

⋆

⋆

⋆

BTcl0;i Pi

−Λi

⋆

⋆

BTcl1;i Pi

0

−γI nd

⋆

Ccl0;i

Dcl00;i

Dcl01;i

−Λ−1 i

Ccl1;i

Dcl10;i

Dcl11;i

0 −γIne μP ⋆ i Pj Ar;ij

⋆

3

HefATcl;i Pi g λ0 Pi

7 ⋆ 7 7 ⋆ 7 7 < 0; 7 ⋆ 7 5

Pj

≥0

η q

f1

1

α

f2

0

q

f5

0

α

0

1

q

min γ Ri ; S1;i ; S2;i ; Li ; Ji ; A^ k;i ; B^ k0;i ; B^ k1;i ; B^ k2;i ; C^ k0;i ; C^ k1;i ; ^ 1;ij ; Δ ^ 2;ij D^ k00;i ; D^ k01;i ; D^ k02;i ; D^ k10;i ; D^ k11;i ; D^ k12;i ; Δ

(16)

subject to conditions (11–13).

III.

Missile Modeling and Autopilot Design

In this section, we will present the design procedure for missile pitch-axis autopilots using the proposed hybrid switched gainscheduling control scheme. To fit the missile dynamics (which is

f3 f4 f6 0

δ; δ

(17)

with the nonlinear functions given by M cosα; f1 K α M an α2 bn jαj cn 2 − 3 8M f2 K q M2 am α2 bm jαj cm −7 ; 3 f3 K α Mdn cosα; f4 K q M2 dm ; M ; f5 K z M2 an α2 bn jαj cn −2 3 f 6 K z M 2 dn

(18)

where αt is the angle of attack in radians, qt is the pitch rate in degrees per second, Mt is the Mach number, δc t is the commanded tail deflection angle in degrees, δt is the actual tail deflection angle in degrees, ηc t is the commanded normal acceleration in g, and ηt is the actual normal acceleration in g. The variables ηt and qt are measured and thus available for feedback use. Angle of attack αt and Mach number Mt are variables to be used for gain-scheduling control purpose. The input to the missile is the commanded tail deflection δc t. The numerical values of various constants in the plant model are as follows: Kα 1.18587; an 0.000103

K q 70.586; deg−3 ;

cn −0.1696 deg−1 ; Consequently, performing congruence transformations on the above two inequalities with matrices diagfZ1;i ; T 1;i ; I nd ; T 2;i ; I ne g and diagfZ1;i ; Z1;j g, respectively, and after some tedious matrix calculation, conditions (11–13) and the controller formulas in Eqs. (15) can be deduced, which is what had to be proven. Based on the synthesis conditions given in theorem 1, the following LMI optimization problem aiming to minimize the weighted L2 gain performance with a specified pair of dwell-time parameters λ0 ; μ immediately follows:

am 0.000215 deg−3 ; cm 0.051 deg−1 ;

K z 0.6661697

bn −0.00945 deg−2 dn −0.034 deg−1 bm −0.0195 deg−2 dm −0.206 deg−1

These coefficients are valid for the missile traveling between Mach numbers 2 and 4 at an altitude of 20,000 ft. The operating range of the missile specified by α; M is such that −π∕6 ≤ α ≤ π∕6 and 2 ≤ M ≤ 4. To convert the nonlinear missile model (17) into a switched LFT form, we need to partition the gain-scheduling parameter set denoted by P ≔ −π∕6; π∕6 × 2; 4 into a finite number of closed subsets fP i gi∈I1;Np by means of a family of switching surfaces Sij (i; j ∈ I1; N p ), so that in each parameter subset, the dynamic behavior of the missile is governed by an associated LFT system. To derive the LFT model with respect to each subregion P i for all i ∈ I1; N p , we first approximate cosα using a second-order polynomial 1 − α2 ∕2 with a maximum error of 0.36% over the range of jαj ≤ π∕6 rad. The resulting LPV missile model is therefore expressed in terms of polynomial functions of the gain-scheduling parameters αt and Mt, which can be readily converted into an

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YUAN ET AL.

LFT form. The state-space LFT model with respect to the parameter subregion P i can be described as 2

x_ M

3

2

AM;i

BM0;i

7 6 6 4 qp 5 4 CM0;i η

CM2;i pp Θ i q p ;

in which xM ≔ α

32

xM

3

DM00;i

7 6 DM02;i 7 54 pp 5

DM20;i

DM22;i

i ∈ I1; N p

144−0.05s 1 s2 2 × 0.8 × 12s 144 0.5s 34.642 W e s s 0.057735 s W δ_ s 250.005s 1

W ref s

δ (19)

q T , and Θi ≔


BM2;i

open-loop interconnection for synthesis is shown in Fig. 3, in which the grey dashed block H corresponds to the hybrid switched gainscheduling controller to be designed and the weighting functions are defined by

θ1;i I4 0

0

W n1 s W n2 s 0.001

θ2;i I5

(for all i ∈ I1; N p ) satisfy Eq. (2) with a block dimension of 9. θl;i for l 1; 2 and i ∈ I1; N p are the normalized scheduling parameters satisfying αi αi αi ∕2 α i − αi ∕2θ1;i and i Mi ∕2 M i − Mi ∕2θ2;i with αi ; Mi ∈ P i Mi M i . The other system matrices are given in αi ; α i × Mi ; M Appendix A. Therefore, the resulting switched LFT model for the LPV missile plant in Eq. (17) consists of N p number of subsystems in the form of Eq. (19). The associated switching signal σ depends on the scheduling parameters α; M and follows the switching rule of σ i;

if α; M ∈ P i

(20)

The weighting functions W ref and W e are concerned with the tracking performance of the actual normal acceleration ηt to its reference ηref t with small steady state error, while a larger error at high frequency range is allowed to avoid dynamic overload. W δ_ is used for penalizing the control effort such that reasonable tail deflection is achieved during the control process. W n1 and W n2 are applied to attenuate the effects of sensor noises dn1 and dn2 . Act represents the actuator dynamics describing the tail deflection mechanism δt, which has been chosen to be 1 for simplicity. Realizing the weighting transfer functions in Eq. (21) in state-space representation, we have W ref s:

x_1

Aref

x_2

Based on this switched LFT model, we will be able to design a hybrid switched gain-scheduling autopilot for the missile plant (17) by using the technique proposed in Sec. II. Remark 2: It should be pointed out that the above transformation of the LPV missile dynamics to a switched LFT model involves a polynomial approximation of cosα. This leads to a potential drawback of the LFT-based control method. However, as argued in [11], better approximation accuracy can be achieved by using higherorder polynomial functions. Furthermore, once a satisfactory LFT model is obtained, the associated hybrid switched gain-scheduling controller can be readily synthesized via convex LMI optimization without resorting to the computationally involved gridding and postanalysis design procedures (e.g., [10,15]). B. Control Problem Setup

The control objectives of the missile autopilot are to track step commands of the normal acceleration ηt and meet the desired controlled performance for the overall closed-loop system for all αt; Mt ∈ P. Specifically, the performance goals for the closedloop system are 1) track step commands input ηc t with time constant no greater than 0.35 s, maximum overshoot no greater than 10%, and steady-state error no greater than 1%; 2) maximum tail deflection rate for 1g step command input ηc t does not exceed 25 deg ∕s. To characterize these performance specifications, we will augment the missile plant with rational weighting functions. The resulting

(21)

ηref

x1

Bref ηc

x2 0

1

x1

0 ηc ; 1 x1

−144 −19.2 x2 x1 Cref 144 −7.2 x2 x2

W e s: x_3 Ae x3 Be eη −0.057735x3 17.321eη ; er Ce x3 De eη x3 0.5eη W δ_ s: x_4 Aδ_ x4 Bδ_ δ −200x4 − 1600δ; eδ_ Cδ_ x4 Dδ_ δ x4 8δ Combining the above weighting functions with the switched LFT missile model derived in Sec. III.A, one can obtain the weighted open-loop switched LFT system in the form of Eq. (1) for controller synthesis. This resulting switched LFT system contains six states, in which two are from the missile plant and the remaining four are from weighting functions. The system has four inputs including one control input, one command input, and two disturbance inputs. There are four system outputs, including two measurement outputs and two error/performance outputs. Specifically, the signals of the weighted switched LFT missile model are defined by xp ≔ α

q x1

e ≔ er

T ;

y1 ≔ y

eδ_

x2

x3

x 4 T ;

err T ;

d ≔ dn1

dn2

η c T ;

u ≔ δc δ

Fig. 3

Weighted open-loop interconnection of the missile plant.

The associated pseudo-input and pseudo-output signals pp ; qp , time-varying parameters Θi (i ∈ I1; N p ), and switching logic σ have the same definitions as for system (19). The weighted open-loop system matrices are summarized in Appendix B. For controller design, because four states from the weighting functions (i.e., x1 , x2 , x3 , and x4 ) could be readily computed online for feedback control use, the weighted open-loop switched LFT system can be further partitioned accordingly to be with Eq. (7). In such a case, we have the associated signals as follows:

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YUAN ET AL.

Table 1 Effects of the partitioning of P on optimized weighted L2 performance (with fixed λ0 ;μ 0.1;1.2) Partitioning of P Case I Case II Case III Case IV

γ # of variables CPU time, s 5.0881 853 5.893 3.3966 1801 29.968 2.6817 3697 246.733 2.4285 5593 803.431

Table 2 Effects of the dwell-time parameters λ0 ;μ on optimized weighted L2 performance (with fixed partition of P as in case III)


λ0 0.1 0.1 0.1 0.11 0.12 0.2

μ

τa ln μ∕λ0 , s

γ

1.2 1.5 1.8 1.8 1.8 1.8

1.8232 4.0547 5.8779 5.3435 4.8982 2.9389

2.6817 2.4620 2.3494 2.3529 2.3565 2.3828

xp1 ≔ α

q T ;

xp2 ≔ x1

y1 ≔ y

err T ;

y2 ≔ x1

x2 x2

x3 x3

x4 T ; x 4 T

Remark 3: Note that the actuator dynamics Act is assumed to be 1 for simplicity of discussion and streamlining the key methodology. From a practical point of view, the proposed approach can be readily modified to handle more realistic control scenarios by taking into account the actual actuator’s dynamics. In that case, the x4 state could still be reconstructed online by using the actuator’s output signal δc as the input to the weighting function W δ_ s. Remark 4: From Appendix B, we note that the Dp22;i (i ∈ I1; 2) matrices of the associated weighted open-loop switched LFT plant are nonzero, which violates assumption (A2). Nevertheless, this problem can be overcome by transforming the measurement output y to y~ with y~ y − Dp22;i u, so that the resulting system with a new measurement output y~ for synthesis will satisfy assumption (A2). Accordingly, for controller implementation, the control input u can be computed using u I Dk11;i Dp22;i −1 Ck1;i xk Dk10;i pk Dk11;i y1 Dk12;i y2 . The invertibility of I Dk11;i Dp22;i is deducible from the control synthesis.

In addition, the synthesis results in Table 2 are obtained under a fixed partition of P as in Case III of Fig. 4. It is observed that changes caused by either parameter λ0 or μ would affect the ADT constraint τa as well as the optimized weighted L2 gain γ. Specifically, if λ0 is fixed at 0.1, while μ varies from 1.2 to 1.8, τa increases with improved γ. On the other hand, when λ0 increases with fixed μ, τa decreases while γ becomes larger. This experimental data reflects that larger μ would result in improved disturbance attenuation level with tighter constraints on the average dwell time. In contrast, variations of λ0 would produce opposite effects on τa and γ as those of μ. This phenomenon can be explained by the fact that increasing λ0 is essentially imposing a more restrictive (faster) convergency rate on the closed-loop system, whereas larger μ would provide more freedom in synthesizing the multiple Lyapunov functions. In controller synthesis, in order to overcome the numerical issue caused by ill-conditioning matrices, we constrained the closed-loop poles within a circle jc 300j ≤ 300. This pole location constraint was formulated in terms of LMIs by using the method of [33] and incorporated it into the proposed control synthesis process. Selecting a pair of dwell-time parameters λ0 ; μ 0.1; 1.1 with a partitioning of P, as in Case I of Fig. 4, it yields a switched LFT missile model with two subsystems and an ADT constraint τa 0.9531 s. With this setting, we obtain the optimal weighted L2 performance level γ 5.0927. The associated controller gain matrices can be further obtained by using the algorithm in theorem 1. To implement the gain-scheduling autopilots, we will specify the Mach number Mt in the following form [6]:

1 _ Mt −jηtj sinjαtj Ax M2 t cosαt; vs

M0 4 (22)

C. Synthesis and Simulation Results

Based on the weighted open-loop switched LFT model, a hybrid switched gain-scheduling controller will be designed using the proposed synthesis conditions in Theorem 1. The associated controller matrices can be obtained by solving LMI optimization problem (16). For switching gain-scheduling control with ADT, because the scheduling parameter space partitioning on P and the dwell-time parameters λ0 ; μ are both involved in the synthesis process, we would like to study their influence on the optimized performance, that is, the weighted L2 gain γ. Tables 1 and 2 present the optimized γ values associated with different partitions of P and specifications of λ0 ; μ by solving problem (16). Table 1 compares four different scenarios with different partitions of P. As seen from Fig. 4, the four scenarios correspond to the cases in which P is partitioned evenly into 2, 4, 8, and 12 subregions. The synthesis results under fixed dwell-time parameters λ0 ; μ 0.1; 1.2 in Table 1 show that better-weighted L2 gains will be obtained as the parameter space P is partitioned into more subregions. However, this will also lead to increased optimization variables, which, in turn, demand more computational efforts.** **All computations were performed on an Intel 4 Core i7 1.8 GHz PC.

Fig. 4

Partitions of the scheduling parameter space P.

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YUAN ET AL.

40

10

η

c

30

η

5

20

0

10 −5

0

−10

−10 −20 0

0.5

a) Acceleration

1

1.5

2 Time (sec)

2.5

3

3.5

4

−15 0

0.5

b) Angle of attack

(g)

1

1.5

2 Time (sec)

2.5

3

3.5

4

1

1.5

2 Time (sec)

2.5

3

3.5

4

1

1.5

2 Time (sec)

2.5

3

3.5

4

(deg)

4 2 3.9

1.8


1.6 3.8 1.4 3.7

1.2 1

3.6 0

0.5

1

1.5

2 Time (sec)

2.5

3

3.5

4

0

0.5

d) Switching signal

c) Mach number M 10

500 200

5

−100 0 −400 −5 −10 0

−700

0.5

1

1.5

2 Time (sec)

2.5

3

3.5

4

−1000 0

0.5

f ) Tail-deflection rate (deg/sec)

e) Tail-deflectionangle (deg)

Fig. 5 Missile dynamic response to a sequence of step acceleration commands.

in which vs 1036.4 ft∕s represents the speed of sound at 20,000 ft and Ax 0.7P0 SCa ∕m, with P0 973.3 lbs∕ft2 being the static pressure at 20,000 ft, S 0.44 ft2 being the surface area, and Ca −0.3, m 13.98 slugs being the drag coefficient and mass of the missile, respectively. Another gain-scheduling parameter, the angle of attack αt, is not measurable in real time for this missile plant. We thus must estimate αt in terms of ηt, δt, and Mt from the output of Eq. (17). To this end, we will adopt the nonlinear static estimator from [10,15], which is a polynomial approximation of an inverse of the output equation in Eq. (17). Detailed analysis about its approximation accuracy can be found in [10,15]. Based on the above system setup, nonlinear simulations will be performed to verify the performance of the designed hybrid controllers. The missile dynamic responses to a series of step commanded acceleration are plotted in Fig. 5. In particular, from Figs. 5a and 5f, it is clearly seen that the performance goals are satisfied under the proposed hybrid control strategy. The angle of attack αt, the Mach number Mt, and the associated switching signals are displayed in Figs. 5b–5d, respectively. It is shown that the values of αt; Mt remain in the range of −π∕6; π∕6 × 2; 4 over the entire time history, which therefore validates the switched LFT model established in Sec. III.A. Furthermore, the average dwell time over the operating time interval [0, 4] s can be calculated by τa 4∕3 1.3333 > τa 0.9531, in which 3 is the number of switching times over [0, 4] s as can be obtained from Fig. 5d. This implies satisfaction of ADT constraint (14), in turn, guaranteeing switching stability.

To demonstrate the effectiveness of the proposed approach, the simulation results are compared with those obtained by using the PDLF method in [10]. By comparing Fig. 5 with fig. 2 of Wu et al. [10], it is observed that the hybrid controller is capable of rendering comparable controlled performance. On the other hand, we also conduct a similar simulation as in [10] to examine the robustness property of the hybrid controller by perturbing the aerodynamic coefficients Cn ≔ an ; bn ; cn ; dn and Cm ≔ am ; bm ; cm ; dm . We will carry out the perturbation independently between Cn and Cm . For Cn , we simultaneously change an , bn , and cn from their nominal values by 10%, while separately changing dn by 10% from its nominal value. Similarly, for Cm, the perturbations are carried out independently with 25% over the parameters am , bm , and cm and 25% over dm. As such, a total of 16 plots result from the combination of all of these variations, and the corresponding responses are shown in Fig. 6. It is observed that all cases present stable dynamic behaviors with only very slight degradation of missile performance under perturbations. Comparing these results with those in fig. 4 of Wu et al. [10], it is interesting to note that much larger performance degradation is witnessed under the same perturbations for the PDLF-based missile autopilot in [10]. This implies a very strong robustness property for the proposed hybrid controller, due to the avoidance of using parameter variation information. Apart from the robustness comparisons, we stress that the proposed hybrid switched gain-scheduling control scheme provides a systematic yet simple framework that overcomes deficiencies of the PDLF approaches.

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YUAN ET AL. 40

10

30

5

20

0

10

−5

0

−10

−10

−15

−20 0

0.5


a) Acceleration

1

1.5

2 Time (sec)

2.5

3

3.5

4

−20 0

500

5

200

0

−100

−5

−400

−10

−700

0.5

1

1.5

2 Time (sec)

c) Tail-deflection angle (deg)

IV.

2.5

3

3.5

4

BM0;i

−Kα

−Kα α i

0

0

−1000 0

2 Time (sec)

2.5

3

3.5

4

1.5

2 Time (sec)

2.5

3

3.5

4

(deg)

0.5

2

Conclusions

α 2 i 1 − 2i Kα an M 2 i Kq am M

1−

α 2i 2

AM;i 4

i an α i bn Signαi Kα M

1

CM0;i

The system matrices for the missile LFT model (19) with respect to the parameter subregion P i for any i ∈ I1; N p are given by

0

αi

60 6 6 60 6 60 6 60 6 60 6 60 6 40

0 0 0 0 0 0 0

0

0

α 2i 2

1−

i an αi α i M 2 i an αi M 2

0 0 an Mi 0 i am Mi M am Mi i an Mi M

i an α i bn Signαi αi α i M 2 i an α i bn Signαi αi M 2

α 2i 2

6 6 6 6 6 6 6 6 6 6 6 6 4

αi 0 Mi an α i bn Signαi 0 i am α i bm Signαi Mi M Mi am α i bm Signαi i an α i bn Signαi Mi M

2

1 2

3

i ϒn;i αi α i M i ϒn;i αi M αi α i αi Mi ϒn;i Mi i ϒm;i Mi M Mi ϒm;i i ϒn;i Mi M

αi α i 2 αi 2

0 0 0 0 0 0 Mi

2 3 α 2 i 1 − 2i Kα dn M 5 BM2;i 4 2i K q dm M

α 2 i − 13 1 − 2i Kα cn M 2 8 i K q cm M

Kα

0

21

Appendix A: System Matrices of the Missile LFT Model

2

3 i ϒn;i 1 Kα M 5; 2i ϒm;i 0 Kq M

1−

2i am α i bm Signαi Kq M

dynamics are modeled as a switched LFT system, based upon which a hybrid LFT controller is then synthesized by solving a finitedimensional LMI optimization problem. This new hybrid control scheme advances existing LPV control methods for missile autopilot design in two important ways: 1) simplified design procedure via convex optimization with reduced computational complexity by avoiding parameter-space gridding and 2) eased controller implementation without information of scheduling parameter variations. Promising controlled performances and strong robustness of the hybrid autopilot have been witnessed through nonlinear simulations.

DM00;i

1.5

d) Tail-deflection rate (deg/sec) Fig. 6 Perturbed response to a sequence of step acceleration commands.

A systematic hybrid switched gain-scheduling control scheme has been proposed for missile autopilot design by using the average dwell-time switching technique. First, the original nonlinear missile "

1

b) Angle of attack

(g)

10

−15 0

0.5

i αi α i M 6 − cn α6i Mi

− cn

0 0 − cn 3Mi 0

i 8cm Mi M 3 8cm Mi 3 − cn M3 i Mi

3 0 07 7 07 7 07 7 07 7; 07 7 07 7 05

# 0

0

0

Kq

i Kq M

0

21

i Mi 2 d n αi α 6 1 dn αi M i 6 2

DM02;i

0

0

3

0

0

0 0 0 0 0 0 0

0 0 0 0 0 Mi 0

07 7 7 07 7 07 7 07 7 07 7 07 7 05

0

0

0

6 0 6 6 0 6 6 6 dn Mi 6 0 6 6 d MM 6 m i i 4 d M m i i dn M i M

3 7 7 7 7 7 7 7 7 7 7 7 7 5

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YUAN ET AL.

DM20;i 0 0

2i ϒn;i CM2;i K z M 2i K z an M

2i an α i Kz M

in which 1 i Signαi an α 2i ; ϒn;i 2cn − cn M i bn α 3 8c M ϒm;i −7cm m i bm α i Signαi am α 2i 3


Appendix B: Weighted Open-Loop System Matrices The weighted open-loop system matrices for controller synthesis are given as follows for all i ∈ I1; N p .

0 ;

2i DM22;i K z dn M

i bn Signαi K z M

[9]

References [1] Lawrence, D. A., and Rugh, W. J., “Gain Scheduling Dynamic Linear Controllers for a Nonlinear Plant,” Automatica, Vol. 31, No. 3, 1995, pp. 381–390. doi:10.1016/0005-1098(94)00113-W [2] Rugh, W. J., “Analytical Framework for Gain Scheduling,” IEEE Control System, Vol. 11, No. 1, 1991, pp. 79–84. doi:10.1109/37.103361 [3] Shamma, J. S., and Athans, M., “Analysis of Gain Scheduled Control for Nonlinear Plants,” IEEE Transactions of Automatic Control, Vol. 35, No. 8, 1990, pp. 898–907. doi:10.1109/9.58498 [4] Apkarian, P., Gahinet, P., and Becker, G., “Self-Scheduled H∞ Control of Linear Parameter Varying Systems,” Proceedings of American Control Conference, Baltimore, MD, 1994, pp. 856–860. [5] Biannic, J.-M., and Apkarian, P., “Missile Autopilot Design via a Modified LPV Synthesis Technique,” Aerospace Science and

2 K z cn M i 3

0 0 Kz

Technology, Vol. 3, No. 3, 1999, pp. 153–160. doi:10.1016/S1270-9638(99)80039-X [6] Nichols, R., Reichert, R., and Rugh, W., “Gain Scheduling for H∞ Controllers: A Flight Control Example,” IEEE Transactions on Control Systems Technology, Vol. 1, No. 2, 1993, pp. 69–79. doi:10.1109/87.238400 [7] Reichert, R. T., “Dynamic Scheduling of Modern-Robust-Control Autopilot Designs for Missiles,” IEEE Control Systems, Vol. 12, No. 5, 1992, pp. 35–42. doi:10.1109/37.158896 [8] Schumacher, C., and Khargonekar, P. P., “Missile Autopilot Designs Using H∞ Control with Gain Scheduling and Dynamic Inversion,” Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1998,

Acknowledgment This work is supported in part by National Science Foundation grant CMMI-1200242.

−

[10]

[11]

[12]

[13]

pp. 234–243. doi:10.2514/2.4248 Shamma, J. S., and Cloutier, J. R., “Gain-Scheduled Missile Autopilot Design Using Linear Parameter Varying Transformations and μ-Synthesis,” Journal of Guidance, Control, and Dynamics, Vol. 16, No. 2, 1993, pp. 256–263. doi:10.2514/3.20997 Wu, F., Packard, A., and Balas, G., “Systematic Gain-Scheduling Control Design: A Missile Autopilot Example,” Asian Journal of Control, Vol. 4, No. 3, 2002, pp. 341–347. doi:10.1111/asjc.2002.4.issue-3 Pellanda, P. C., Apkarian, P., and Tuan, H. D., “Missile Autopilot Design via a Multi-Channel LFT/LPV Control Method,” International Journal of Robust and Nonlinear Control, Vol. 12, No. 1, 2002, pp. 1–20. doi:10.1002/(ISSN)1099-1239 Lim, S., and How, J. P., “Modeling and H∞ Control for Switched Linear Parameter-Varying Missile Autopilot,” IEEE Transactions on Control Systems Technology, Vol. 11, No. 6, 2003, pp. 830–838. doi:10.1109/TCST.2003.815610 Apkarian, P., Pellanda, P. C., and Tuan, H. D., “Mixed H2 ∕H∞ MultiChannel Linear Parameter-Varying Control in Discrete Time,” Systems & Control Letters, Vol. 41, No. 5, 2000, pp. 333–346. doi:10.1016/S0167-6911(00)00076-1


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