International Journal of Control Vol. 80, No. 4, April 2007, 555–568
Robust and gain-scheduling control of LFT systems through duality and conjugate Lyapunov functions K. DONG and F. WU* Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695 (Received 6 January 2006; in final form 23 October 2006) In this paper, we study stability and performance properties of linear fractional transformation (LFT) parameter-dependent systems using duality theory and tools from convex analysis. A pair of conjugate functions, the convex hull and the maximum of a family of quadratic functions, are used for analysis and synthesis of LFT systems. Sufficient synthesis conditions for both robust state feedback and gain-scheduling output feedback control problems are formulated as a set of linear matrix inequalities (LMIs) with linear search over scalar variables. Finally, a numerical example is used to demonstrate the advantages of the proposed approaches.
1. Introduction Originated from the gain scheduling control methodology (Shamma and Athans 1990, Rugh 1991), the study of linear fractional transformation (LFT) and linear parameter-varying (LPV) systems have received considerable attention in recent years (Rugh and Shamma 2000). For LFT system analysis and synthesis, the current research mainly focused on the construction of different types of Lyapunov functions. Using the scaled small-gain theorem, LFT control design technique was developed in Packard (1994) and Apkarian and Gahinet (1995) using constant Lyapunov functions. When the plant depends on the parameters in linear fractional transformations, the LFT gain-scheduled controller is fully characterized by a finite number of LMIs. Further ramifications along this line can be found in Scorletti and El Ghaoui (1995) and Scherer (1999). On the other hand, parameter-dependent Lyapunov functions were also exploited for LFT systems (Wang and Balakrishnan 2002, Wu and Dong 2006). However, the solution to LFT analysis and synthesis problems using parameterdependent Lyapunov functions were often formulated as parameter-dependent linear matrix inequalities (LMIs), which belong to a special type of convex optimization
*Corresponding author. Email:
[email protected]
problem with high computational complexity and is generally difficult to solve. Duality is a well established concept for linear time invariant (LTI) systems, for example, the dual system of x_ ¼ Ax is _ ¼ AT , the matrix pair (A, B) is stabilizable if and only if (AT, BT) is detectable, kD þ CðsI AÞ1 Bk1 ¼ kDT þ BT ðsI AT Þ1 CTk1 and so on. The system x_ ¼ Ax is stable if and only if there is a positive definite matrix P such that ATP þ PA < 0. Moreover, ATP þ PA < 0 is equivalent to AP1 þ P1AT < 0, which is the stability condition for the dual system _ ¼ AT . From convex analysis, it is well-known that 12T P1 is a conjugate function of 12xT Px. Two classes of conjugate functions, convex hull and maximum quadratic functions are particularly important for dynamic system analysis. The convex hull function was proposed to enlarge stability domain for saturated linear systems in Hu and Lin (2003). Alternative linear differential inclusions (LDIs) have also been derived for saturated control systems in Hu et al. (2005b). Moreover, in Goebel et al. (2004) and Hu et al. (2005a), the convex duality between these two classes of Lyapunov functions has been explored and used to enhance stability analysis of LDIs and saturated linear systems. When the stability condition seems difficult to solve for an original system, resorting to its dual formulation could be advantageous. Once a Lyapunov function was found in the dual space of an
International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online 2007 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/00207170601080213
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LFT system, then its conjugate function can be used for analysis purpose of the original LFT system. This provides an alternative approach to the search for a suitable Lyapunov function for stability and performance enhancement. These nice dual properties of LTI and LDI systems motivate us to further study duality properties of LFT systems and use the conjugate Lyapunov function for LFT analysis and synthesis problems. In this paper, we first state stability and performance properties of LFT systems from duality perspective. Built upon available analysis results (Hu et al. 2005b) from duality theory and conjugate functions, a new control synthesis approach will be developed for robust and gain-scheduling control of LFT systems. Using a convex hull Lyapunov function and a maximum function of a family of quadratic functions, the control synthesis conditions can be formulated as a set of LMIs plus scalar variables and solved by LMI optimizations combined with linear search. Different from previous LFT control techniques, our proposed approach results in non-linear parameter-dependent controllers for better controlled performance. The performance improvement of the new control design technique will be demonstrated using an example. After submitting the manuscript, the authors also found a similar result on state feedback control of LDI systems from an independent research in Hu (2006). The notation is fairly standard and will be defined when it is necessary. R stands for the set of real numbers and Rþ for the non-negative real numbers. Rmn is the set of real m n matrices. The transpose of a real matrix M is denoted by MT. A block diagonal matrix with matrices X1 , . . . , Xp on its main diagonal is denoted by diagfX1 , . . . , Xp g. For two integers k1 and k2, k1 < k2, we denote I½k1 , k2 ¼ fk1 , k1 þ 1, . . . , k2 g. We use Snn to denote real, symmetric n n matrices, and Snn for þ positive definite matrices. If M 2 Snn , then M > 0 (M 0) indicates that M is a positive definite (positive semi-definite) matrix and M < 0 (M 0) denotes a negative definite (negative semi-definite) matrix. For x 2 Rn , its norm is denned as kxk ¼ ðxT xÞ1=2 . The space of square integrable functions is denoted by L2 . The spaces of continuously differentiable functions will be denoted by C1 . @fðxÞ means the subdifferential of f(x), being the set of all subgradients. If f(x) is differentiable, @fðxÞ has only one element (derivative). The paper is organized as follows: x 2 provides some preliminary results for this research; the dual properties of LFT systems are studied in x 3. In x 4 and x 5, robust state feedback control and gain-scheduling output feedback control of LFT systems are addressed separately. Using convex hull and maximum quadratic functions as Lyapunov functions, the control synthesis conditions for these problems are formulated as LMIs
with a set of scalar variables. Section 6 uses a simple second-order example to demonstrate the proposed control design approach for LFT systems. Finally, the paper concludes in x 7.
2. Preliminaries In this section, we will provide some preliminary results relevant to our subsequent development. The first lemma states that a set of LFT matrices is solely characterized by its vertices, and its proof can be found in Boyd et al. (1994). Lemma 1: Given ? ¼ fdiagf1 , 2 , . . . , s g : i 2 C1 ðRþ , RÞg, we define two sets as ¼ fA þ BðI DÞ1 C: detðI DÞ 6¼ 0, 2?,jj 1g, ¼ fA þ BðI DÞ1 C: 2?,ji j ¼ 1, i 2I½1, sg: Then ¼ Co, where Co stands for convex hull. The second lemma can be shown through Schur complement and a simple argument. Lemma 2: For a positive definite function hðxÞ ¼ maxi 2 I½1, m hi ðxÞ and positive definite matrices X, Y > 0, the inequality 2@hi ðxÞAy þ yT CT XCy þ @hi ðxÞBYBT @T hi ðxÞ < 0 whenever hi ðxÞ hj ðxÞ, 8j 2 I½1, m, is a necessary and sufficient condition for 2@hðxÞAy þ yT CT XCy þ @hðxÞBYBT @T hðxÞ < 0 to hold for all x, y 2 Rn . The remaining materials are basically from convex analysis and can be found in Rockafellar (1970) and Boyd and Vandenbergh (2004). For any function f : Rn ! R, its conjugate function is defined as f ðÞ ¼ sup fT x fðxÞg x 2 Rn n
for 2 R . In this paper, we are mainly interested in those functions f that are convex, positive definite and homogeneous of degree r > 1 (f(x) ¼ rf(x) for 0). For this class of functions, we have following properties about its conjugate functions: I. II. III. IV. V.
f*() is finite for every 2 Rn ; f* is a convex, positive definite, and positively homogeneous of degree s > 1 where 1/r þ l/s ¼ 1; f*() ¼ sf*(); for any > 0, the conjugate of g(x) ¼ f(x) is g ðÞ ¼ f ð=Þ; T 2 @fðxÞ if and only if xT 2 @f ðÞ.
The usefulness of conjugate Lyapunov function has been shown in studying stabilization problem for LDI
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Robust and gain-scheduling control of LFT systems systems and saturated control systems in Goebel et al. (2004) and Hu et al. (2005a). Consider an LDI x_ Av Bv x 2 Co , ð1Þ e Cv Dv d where the subscript v stands for the vertices of the LDI and v 2 I½1, Nv . Its dual system is also an LDI system in the form of T Av CTv _ , 2 Co w BTv DTv z and dim(x) ¼ dim(), dim(e) ¼ dim(w), dim(d) ¼ dim(z). We then have the following theorem that establishes the equivalent performance analysis conditions in the original space and its dual space. Theorem 1: Given the LDI (1) and > 0. Let V : Rn ! R be a closed proper convex, positive definite, homogeneous function of degree p > 1, and V* is its conjugate function. Then, the condition 3 2 2@VðxÞAv x @VðxÞBv xT CTv 7 6 T T I DTv 5 < 0, 8x 2 Rn , 4 Bv @ VðxÞ Cv x
Dv
I
v 2 I½1, Nv
ð2Þ
is equivalent to 2 2@V ðÞATv @V ðÞCTv 6 I 4 Cv @T V ðÞ BTv
DTv
3 T Bv 7 Dv 5 < 0,
and assume D ¼ 0 to simplify the derivation. Note that condition (2) implies 2 3 2@VðxÞAx @VðxÞB xT CT 4 BT @T VðxÞ I 0 5 < 0: Cx 0 I By Schur complement, equivalent to
the
above
inequality
is
1 1 2@VðxÞAx þ @VðxÞBBT @T VðxÞ þ xT CT Cx < 0: Therefore 2@VðxÞx_ þ 1 eT e dT d ¼ 2@VðxÞðAx þ BdÞ þ 1 eT e dT d 1 1 ¼ 2@VðxÞAx þ @VðxÞBBT @T VðxÞ þ xT CT Cx jj 1=2 BT @T VðxÞ 1=2 djj2 < 0: If function V(x) is differentiable, the subdifferential @VðxÞ can be replaced by @VðxÞ=@x, which renders inequalities (2) and (3) to be the same as the wellknown bounded real lemma. Thus the LDI system is uniformly asymptotically stable with kek2 < kdk2. œ
8 2 Rn ,
I
v 2 I½1, Nv :
ð3Þ
Moreover, if one of the above conditions is satisfied and V(x) is differentiatiable, then kek2 < kdk2. Proof: Since V is a closed proper convex function, we have @V ¼ dom @V ¼ range @V ¼ dom @V ¼ Rn , which means any x 2 Rn can be represented by @V ðÞ for some . (¼)) Pick any x 6¼ 0 and T 2 @VðxÞ. Then 6¼ 0, since 0 2 @VðxÞ would imply x minimizes V(x). Thus 2 T 3 2 Av x T Bv xT CTv 4 BT ð4Þ I DTv 5 < 0: v Cv x Dv I From the property (V) of conjugate functions, T 2 @VðxÞ implies xT 2 @V ðÞ. Then (3) can be derived by replacing x in (4) with @VT ðÞ, since condition (4) is true for all xT 2 @V ðÞ. ((¼) vice versa. To prove the last claim, we define A B Av Bv ¼ Co C D Cv Dv
3. Duality properties of LFT systems Consider an LFT parameter-dependent system given by 2 3 2 32 3 A B0 x x_ B1 4 q 5 ¼ 4 C0 D00 D01 54 p 5, ð5Þ d e C1 D10 D11 p ¼ q,
ð6Þ
where x, x_ 2 Rnx , d 2 Rnd is the disturbance, e 2 Rne is the controlled output, p, q 2 Rnp are the pseudo-input and output. The time-varying parameter obeys the following structure: ? ¼ fdiagð1 , 2 , . . . , s g : i 2 C1 ðRþ , RÞ, ji j 1, i 2 I½1, sg:
ð7Þ
Note that the block size of each parameter i is one. The LFT representation is well-posed, i.e., (I D00) is invertible for any allowable parameter values. The L2 gain of the LFT systems (5)–(6) is defined as jjejj2 : 2 ?, jjdjj2 6¼0 jjdjj2 max
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By absorbing into its state space matrices, the LFT systems (5)–(6) can be rewritten as x x_ AðÞ B1 ðÞ : ð8Þ ¼ C1 ðÞ D11 ðÞ d e From Lemma 1, the LPV system (8) can be represented by x B1, v x_ Av , 2 Co C1, v D11, v d e which is an LDI specified at 2s vertices of . As shown in Lu et al. (1996), the dual form of original LFT systems (5)–(6) is 32 3 2 3 2 T CT0 CT1 A _ 7 6 7 6 T T T 76 p ð9Þ ¼ 4 q~ 5 4 B0 D00 D10 54 ~ 5, BT1
z
DT01
DT11
where QðlÞ ¼
m X
li Qi ,
i¼1
(
m
¼ l2R :
ð10Þ
~ has the same structure as . Similarly, one can where express equations (9) and (10) as an LDI " # (" #) ATv CT1, v _ 2 Co T T w B1, v D11, v z ~ vertices. using the convex hull of ’s Next, we examine a control system with feedback structure shown in figure 1(a) where both the plant G and controller K are assumed to have the same LFT parameter dependency on a block structured . Its dual system is shown in figure 1(b) where both plant GT and controller KT are the dual systems of the original G and K. Then K stabilizes G with respect to the block structure cl ¼ diagf, g if and only if KT stabilizes GT with respect to the same block structure ~ g. ~ ~ cl ¼ diagf, For a set of positive definite matrices Qi 2 Rnx nx , i 2 I½1, m, we will define a convex hull quadratic function (or so-called composite quadratic function in Hu and Lin (2003)) as
) li ¼ 1, li 0, i 2 I½1, m :
i¼1
The convex hull quadratic function (11) has its level set as the convex hull of the level sets of each xT Q1 i x (Hu et al. 2005a). Also, we have the following technical lemma for this type of functions. Lemma 3: The function gðxÞ ¼ Q1 ðl Þx, where l ¼ arg minl 2 xT Q1 ðlÞx, is continuously differentiable and @g ¼ Q1 ðl ðxÞÞ: @x
w
~ q, ~ p~ ¼
m X
ð12Þ
Moreover, we have @V ¼ xT Q1 ðl ðxÞÞ: @x Proof: g(x) is continuous and differentiable because the function (11) is continuous and differentiate (Hu and Lin 2003). Next, we need to show ð@g=@xÞ ¼ Q1 ðl Þ. Suppose that the partial derivative of g(x) at x0 is ð@g=@xÞjx¼x0 and let k be given. Since g(kx) ¼ kg(x) for all x 2 Rnx , then we have x gðkx0 þ xÞ gðkx0 Þ ¼ k g x0 þ gðx0 Þ k ¼k
! ! x @g x þ o @x x¼x0 k k
@g ¼ @x
! x þ oðjxjÞ: x¼xo
Θ
~ Θ
It follows that ð@g=@xÞx¼kx0 ¼ ð@g=@xÞx¼x0 . So we only need to consider those x on the boundary of the level set LV(1), i.e. @LV ð1Þ. Pick up two points x0 , x 2 @LV ð1Þ) with x ¼ x0 þ x and x small. To simplify the notation, denote P0 as the optimal Q1(l*) at x0 and P as the optimal Q1(l*) at x. Assuming
G
GT
Px P0 x ¼ P^ 0 ðx0 Þx þ oðjxjÞ:
K Θ
KT ~ Θ
(a)
(b)
In the following proof, we only consider two quadratic functions, that is, Q(l) ¼ lQ1 þ (1 l)Q2. For the case of m > 2, it can be proved similarly. If l*(x0) ¼ 0 or 1, it is clear that P^ 0 ¼ 0 because only one quadratic function is involved in this case. When the optimizer l* of Q1(l(x0)) is in the range of [0, 1 ] for any 0 < < 1, we will perturb Q1
VðxÞ ¼ min 12xT Q1 ðlÞx, l2
ð11Þ
Figure 1. Duality of a closed-loop LFT system: (a) original LFT system; (b) dual LFT system.
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Robust and gain-scheduling control of LFT systems to Q^ 1 ¼ ð1 ÞQ1 þ Q2 and keep Q^ 2 ¼ Q2 . Then a ^ with l^ 2 ½0, 1 is obtained such ^ lÞ perturbed function Qð that ^ Q^ 2 ^ ¼ l^ Q^ 1 þ ð1 lÞ ^ lÞ Qð ^ ÞQ1 þ ð1 lð1 ^ ÞÞQ2 ¼ lð1 ¼ lQ1 þ ð1 lÞQ2 ¼ QðlÞ,
^ Þ 2 ½0, 1, : 8l ¼ lð1
^ and Q are This implies the following two sets Q identical, ^ Q^ 2 , ^ : Qð ^ ¼ l^ Q^ 1 þ ð1 lÞ ^ ¼ fQð ^ lÞ ^ lÞ Q
l^ 2 ½0, 1g,
Q ¼ fQðlÞ : QðlÞ ¼ lQ1 þ ð1 lÞQ2 ,
l 2 ½0, 1 g:
^ and Q will be the same, Thus the optimal points from Q ^ ^ i.e. Qðl ðx0 ÞÞ ¼ Qðl ðx0 ÞÞ. Using the perturbed function, it is concluded that P^ 0 ¼ 0 at l^ ¼ 1 from previous argument. Since the function value remains unchanged after perturbation for l 2 ½0, 1 , we have P^ 0 ¼ 0 for the original function Q(l*) at l ¼ 1 . So P^ 0 ¼ 0 for all l 2 ½0, 1. Finally Px P0 x0 ¼ ðPx P0 xÞ þ ðP0 x P0 x0 Þ
¼ P0 x þ oðjxjÞ, therefore the partial derivative of g(x) at x0 is P0 ¼ Q1 ðl ðx0 ÞÞ. Finally, we get from equation (12) @V 1 T @ðQ1 ðl ðxÞÞxÞ T 1 ¼ x þ x Q ðl ðxÞÞ @x 2 @x h
From the definition of conjugate functions, it can be shown that the conjugate of convex hull quadratic function (11) is a maximum quadratic function in the form of V ðÞ ¼ max 12T QðlÞ ¼ max 12T Qi : l2
i 2 I½1, m
for any i, j 2 I½1, m and v 2 I½1, 2s . Moreover, for analysis and control synthesis problems of LFT systems, one can often resort to its dual formulation with the help of conjugate Lyapunov functions. Generally speaking, the dual approach provides alternative solutions when the analysis and synthesis problems are hard to solve for original systems. The following theorem provides a dual analysis condition for the original system (8), which can be found in references (Goebel et al. 2004, Hu et al. 2005a, b). It is restated here for completeness. Theorem 2: Given > 0, suppose that there exist positive definite matrices Qi 2 Snþx nx and scalars ij, v 0 for i, j 2 I½1, m and v 2 I½1, 2s such that 2 3 m P Qi ATv þ Av Qi þ "ij, v ðQi Qj Þ Qi CT1, v B1, v 6 7 j¼1 6 7 6 7 < 0, C1, v Qi I D11, v 5 4 BT1, v DT11, v I 8 i, j 2 I½1, m, v 2 I½1, 2s :
ð15Þ
Then the LPV system (8) is stable for any 2 ? and kek2 < kdk2.
¼ P^ 0 ðx0 Þx þ oðjxjÞ þ P0 x
¼ xT Q1 ðl ðxÞÞ:
through the following matrix inequality (Goebel et al. 2004) 2 3 m P ATv Qi þ Qi Av þ ^ij, v ðQi Qj Þ Qi B1, v CT1, v 6 7 j¼1 6 7 < 0, T T 4 B1, v Qi I D11, v 5 C1, v D11, v I ð14Þ
ð13Þ
Note that V and V* are homogeneous functions of degree 2. Also, V(x) is convex and continuously differentiable and V* is strictly convex but not differentiable everywhere. Using maximum quadratic functions, the stability and L2 gain of the LPV system (8) can be determined
Proof: Given the convex hull quadratic Lyapunov function V(x) and its conjugate function V*(), the LPV system (8) is stable and has its L2 gain from d to e less than if 2 3 2@V ðÞATv @V ðÞCT1, v T B1, v 4 C1, v @T V ðÞ I D11, v 5 < 0 BT1, v DT11, v I from Theorem 1. Note that V ðÞ ¼ maxi 2 I½1, m 12T Qi . Then using Lemma 2, the above inequality can be rewritten as 3 2 3T 2 Qi ATv þ Av Qi Qi CT1, v B1, v 6 7 6 C1, v Qi I D11, v 7 I 5 4 5 4 I
BT1, v
DT11, v 2 6 4
I
3 7 5 < 0 ð16Þ
I I
with constraints T Qi T Qj 0, 8i, j 2 I½1, m. Finally, the sufficiency of (15) to guarantee equation (16) can be easily verified using S-procedure. œ
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Note that either (14) or (15) is only sufficient for stability and performance analysis of LFT systems. They are generally not equivalent to each other. Both conditions are in the form of bilinear matrix inequalities (BMIs), which have variables Qi and scalars ^ij, v ðij, v Þ involved. A feasible approach to solve this matrix inequality is to perform linear search by adaptively gridding scalar variables. The resulting condition with fixed ^ij, v (or ij, v ) then becomes LMIs and can be solved efficiently. Another effective approach to solving the problem is the path-following method (Hassibi et al. 1999). Although there is no guarantee that the global optimal solution can be found, the convergence of the algorithms has been shown satisfactory in past experience.
We assume that (A1) the triple (A(), B2(), C2()) is stabilizable and detectable for any 2 ?, (A2) D11() ¼ 0 and D22() ¼ 0. For simplicity, D11() is assumed to be zero in the following derivation. The difference between D11 ¼ 0 and D11 6¼ 0 is mainly in the controller construction phase. For robust state feedback control problem, is treated as an uncertainty and all state information is assumed available (y ¼ x) for feedback control use. We would like to design a non-linear state feedback controller in the form of u ¼ Fðl ðxÞÞx
to stabilize the closed-loop system and minimize its L2 gain from d to e. With the state feedback control law (20), the dynamic equation of the closed-loop system becomes #" # " # " x x_ AðÞ þ B2 ðÞFðl ðxÞÞ B1 ðÞ ¼ C1 ðÞ þ D12 ðÞFðl ðxÞÞ 0 d e (" # ) " # Av þ B2, v Fðl ðxÞÞ B1, v x 2 Co , C1, v þ D12, v Fðl ðxÞÞ 0 d
4. Robust state feedback control Given an open-loop LFT parameter-dependent system 2 3 2 32 3 x_ A B0 B1 B2 x 6q7 6C D 76 p 7 D D 00 01 02 76 7 6 7 6 0 ð17Þ 6 7¼6 76 7, 4 e 5 4 C1 D10 D11 D12 54 d 5 y C2 D20 D21 D22 u p ¼ q,
ð18Þ
8v 2 I½1, 2s
where is defined as (7) with scalar parameter blocks. With the well-posedness assumption, the LFT system can also be transformed to its equivalent form 2 3 2 32 3 x_ AðÞ B1 ðÞ B2 ðÞ x 6 7 6 76 7 6 e 7 ¼ 6 C1 ðÞ D11 ðÞ D12 ðÞ 76 d 7 ð19Þ 4 5 4 54 5 y C2 ðÞ D21 ðÞ D22 ðÞ u 9 28 Ri ATv þ YTi BT2, v þ Av Ri þ B2, v Yi > > > > = 6< m 6 X 6> > þ ij, v ðRi Rj Þ > 6> ; 6: j¼1 6 6 C1, v Ri þ D12, v Yi 4 BT1, v
2
ð21Þ
then we have the following condition for synthesizing robust state feedback control law through conjugate Lyapunov functions. Theorem 3: If there exist positive definite matrices Ri 2 Snþx þnx , rectangular matrices Yi 2 Rnu nx , and scalars ij, v 0 for i, j 2 I½1, m and v 2 I½1, 2s such that 3
Ri CT1, v þ YTi DT12, v I 0
with
ð20Þ
7 B1, v 7 7 7 7 < 0, 7 7 0 5 I
8i, j 2 I½1, m, v 2 I½1, 2s :
ð22Þ
Define AðÞ
B1 ðÞ
6 6 C1 ðÞ D11 ðÞ 4
B2 ðÞ
3
A
7 6 6 D12 ðÞ 7 5 ¼ 4 C1
C2 ðÞ D21 ðÞ D22 ðÞ 2 3 B0 6 7 1 7 þ6 4 D10 5ðI D00 Þ C0 D20
2
C2
B1
B
3
D11
7 D12 7 5
D21
D22
Yðl Þ ¼
D02 :
li Yi ,
i¼1
Rðl Þ ¼
m X
li Ri ,
i¼1
in which l* is determined by the optimization problem min l
D01
m X
s:t:
P xT 0, m x i¼1 li Ri m P li ¼ 1, li 0:
i¼1
ð23Þ
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Robust and gain-scheduling control of LFT systems Then the closed loop system (21) is stable 8 2 ? and its L2 gain is less than under the state feedback law u ¼ Fðl ðxÞÞx ¼ Yðl ÞR1 ðl Þx: Moreover, if the vector function l*() is continuous, then u ¼ Fðl ðxÞÞx is a continuous feedback control law. Proof: From the optimization problem (23), it is clear that l* is the minimizer of convex hull quadratic function VðxÞ ¼ minl 2 12xT R1 ðlÞx, i.e. l ¼ arg min xT R1 ðlÞx ¼ arg min VðxÞ: l2
l2
By definition, the conjugate of V(xi) for xi 2 Xi will be mapped into V*(i), where i 2 i . As show in figure 2 for a two-dimensional example with two basis ellipsoids, segment A0 B0 belongs to the extreme point set X1 and segment AB is part of the set 1 in the dual space. Given V(x) as a closed proper convex function, then range @Vðxi Þ ¼ dom@V ði Þ ¼ i . From the property V of conjugate functions, we have xi 2 @V ði Þ , i 2 @Vðxi Þ. From Hu and Lin any x 2 @LV can be P (2004), i l x and ð@V=@xÞ ¼ xiT R1 represented by x ¼ m i ¼ i¼1 i T 1 x R ðl Þ for i 2 I½1, m0 . Define Fi ¼ Yi R1 , we also i have the following observation
Define the level-1 set of V as
Fðl Þx ¼ Yðl ÞR1 ðl Þx ¼
ns
LV :¼ fx 2 R : VðxÞ 1g:
m X
li Yi R1 ðl Þx
i¼1
For simplicity and without loss of generality, it is assumed that li > 0 for i 2 I½1, m0 and li ¼ 0 for i 2 I½m0 , m. Denote the level-1 set of the ellipsoid xT R1 i x as
nx 1 T 1 i 2 I½1, m: EðR1 i Þ :¼ x 2 R : 2 x Ri x 1 , Then we have an extreme point set in dom V
nx 1 T 1 Xi ¼ @LV \ @EðR1 i Þ ¼ x 2 R : VðxÞ ¼ 2 x Ri x ¼ 1 , i 2 I½1, m, where [m i¼1 Xi contains all the extreme points of LV. On the other hand, we define the set i in the dual space dom V* to be
i :¼ 2 Rnx : V ðÞ ¼ 12 T Ri ¼ 1 , i 2 I½1, m: 2 Then [m i¼1 i ¼ @LV . Since V*(k) ¼ k V*() for any k, it is sufficient to consider only those at which V*() ¼ 1.
¼
m X
i li Yi R1 i x ¼
i¼1
m X
li Fi xi :
i¼1 i
i
In addition, for all 2 @Vðx Þ, we have Yi i 2 Fi xi . Using Lemma 2, it can be seen that inequality (22) implies for any i 2 i , ) 3 2( Ri ATv þ YTi BT2,v 2 i 3T Ri CT1,v þ YTi DT12,v B1,v 7 6 7 6 7 6 þA R þ B Y v i 2,v i 7 6 7 6 7 6 6 I 7 6 7 4 5 6 C1,v Ri þ D12, v Yi I 0 7 5 4 I BT1,v 0 I 2 i 3 6 7 6 7 6 I 7 < 0, ð24Þ 4 5 I where i 2 I½1, m, v 2 I½1, 2s . Note that [m i¼1 i ¼ @LV , therefore equation (24) holds for all 2 @VðxÞ.
R1
ξ2
x2 A' B'
A
R2
x1
ξ1
B
R2−1
R1−1 (a)
(b)
Figure 2. Level set of two conjugate Lyapunov functions: (a) Level set of convex hull Lyapunov function; (b) Level set of maximum Lyapunov function.
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Following the same procedure in proofing Theorem 1, (24) is equivalent to 2( 6 6 6 6 6 4
i iT T T 1 i xiT ATv R1 i x þ x Fi B2, v Ri x
)
3 x
i iT 1 i þxiT R1 i Av x þ x Ri B2, v Fi x
6 6 4
3T
x I I
2(
6 7 6 7 6 5 6 6 4
iT
CT1, v
þ
xiT FTi DT12, v
iT
x
R1 i B1, v
C1, v xi þ D12, v Fi xi
I
0
i BT1, v R1 i x
0
I
ðAv þ B2, v Fðl ÞÞT R1 ðl Þ 1
þR ðl ÞðAv þ B2, v Fðl ÞÞ
7 7 7 7 < 0: 7 5
will be
i 1 Replacing R1 i x by R ðl Þx and summing above inequalities over coefficient li , we get
2
function l* indirectly. Using the non-linear gainscheduled control law (25), the closed-loop system
x_ cl e
Acl ð, l Þ ¼ Ccl ð, l Þ
)
3 2 R1 ðl ÞB1, v 7 x 76 76 74 7 0 5 I
T
C1, v þ D12, v Fðl Þ
I
BT1, v R1 ðl Þ
0
Since V(x) is differentiable and ð@V=@xÞ ¼ xT R1 ðl Þ, the closed loop system (21) is indeed stable and has its œ L2 gain less than by invoking Theorem 1. The controller gain in equation (20) depends on l*, which can be solved from the optimization problem (23) either online or offline. Online solving LMIs is computationally more expensive and have better accuracy. On the other hand, for offline computation, one can grid the entire state space and solve (23) at each gridding point. Then a look-up table can be constructed for functional relation between the state x and l*. When implementing the controller, l* will be chosen from the look-up table through interpolation. The offline method requires much less online computational effort but is less accurate. 5. Gain-scheduling output feedback control When parameter is time-varying and measurable in real time, one can design a non-linear outputfeedback controller in the form of Ac ð, l Þ Bc ð, l Þ xc x_ c ¼ ð25Þ u y Cc ð, l Þ Dc ð, l Þ for the open-loop LFT system (19). Note that the non-linear controller gains depend on the parameter directly, as well as the controller state through
Acl ð, l Þ Ccl ð, l Þ
xcl d
ð26Þ
7 7 < 0: 5
I I
xTc T and the state-space data given by
with xcl ¼ ½ xT
3
ðC1, v þ D12, v Fðl ÞÞ
Bcl ð, l Þ Dcl ð, l Þ
2 3 AðÞ 0 B1 ðÞ Bcl ð, l Þ ¼4 0 0 0 5 Dcl ð, l Þ C1 ðÞ 0 0 2 3 0 B2 ðÞ þ4I 0 5 0 D12 ðÞ Ac ð, l Þ Bc ð, l Þ Cc ð, l Þ Dc ð, l Þ 0 I 0 : C ðÞ 0 D ðÞ 2
21
For gain-scheduling control synthesis purpose, we will employ a convex hull quadratic function and a maximum quadratic function VðxÞ ¼ 12xT R1 ðlV ðxc ÞÞx, Uðxc Þ ¼
1 T 2xc SðlU ðxc ÞÞxc ,
lV ¼ arg min xTc R1 ðlV Þxc , lV 2
lU
¼ arg max xTc SðlU Þxc lU 2
¼ arg max xTc Sk xc , k 2 I½1, r
in which RðlV Þ ¼
m X i¼1
lVi Ri ,
SðlU Þ ¼
r X k¼1
lUk Sk :
ð27Þ
563
Robust and gain-scheduling control of LFT systems As a continuous function of the controller states xc, lV can be calculated by the optimization algorithm min
Bc ðxc , Þ ¼ ðSðlU Þ R1 ðlV ÞÞ1 SðlU ÞLðxc , Þ, ð32Þ
lV
P xTc 0, m xc i¼1 lVi Ri m P lVi ¼ 1, lVi 0:
s:t:
Cc ðxc , Þ ¼ Fðxc , Þ,
ð33Þ
Dc ðxc , Þ ¼ 0,
ð34Þ
where
i¼1
Fðxc , Þ ¼ YðlV ÞR1 ðlV Þ,
lU
On the other hand, has its kth element equal to 1 and others to be zero when kth quadratic function xTc Sk xc is active. Therefore, the function lU ðxc Þ is piecewise constant with discontinuity at switching instant. The following theorem provides a sufficient condition for the existence of the non-linear control law (25) in a switched LPV form. Theorem 4: If there exist positive definite matrices Ri 2 Snþx nx and Sk 2 Snþx nx , rectangular matrices Yi 2 Rnu nx and Zk 2 Rnx ny , and scalars ij, v 0, ^kl, v 0 for i, j 2 I½1, m and k, l 2 I½1, r and v 2 I½1, 2s such that
Lðxc , Þ ¼ S1 ðlU ÞZðlU Þ: Proof: Given the controller gain (31)–(34), we will apply a similar transformation " xcl ¼ Tx~ cl ¼
I
0
I
I
# x~ cl
T to the closed-loop states, where x~ cl ¼ xT ðxc xÞT .
3 28 9 Ri ATv þ YTi BT2, v þ Av Ri þ B2, v Yi > > = 7 6< 6 m Ri CT1, v þ YTi DT12, v B1, v 7 P 7 6> > þ ij, v ðRi Rj Þ ; 7 6: 7 < 0, 8i, j 2 I½1, m 6 j¼1 7 6 7 6 6 C1, v Ri þ D12, v Yi I 0 7 5 4 T 0 I B1, v 9 8 3 2 T T T > = < Av Sk þ C2, v Zk þ Sk Av þ Zk C2, v > 6 r Sk B1, v þ Zk D21, v CT1, v 7 7 P 6> > 7 6: ^ þ ðS S Þ ; kl, v k l 7 6 l¼1 7 < 0, 8k, l 2 I½1, r 6 7 6 6 T T T B1, v Sk þ D21, v Zk I 0 7 5 4 0
C1, v
Ri I
I Sk
> 0,
8i 2 I½1, m,
k 2 I½1, r
ð30Þ
and for all v 2 I½1, 2s . Define m r X X YðlV Þ ¼ lVi Yi , ZðlU Þ ¼ lUk Zk ; i¼1
k¼1
Then the closed loop system (26) is stable for all 2 ? and its L2 gain is less than with an nth-order non-linear output feedback control gain as Ac ðxc , Þ ¼ ðSðlU Þ R1 ðlV ÞÞ1 fAT ðÞR1 ðlV Þ þ SðlU ÞðAðÞ þ B2 ðÞFðxc , Þ þ Lðxc , ÞC2 ðÞÞ þ 1 SðlU ÞðB1 ðÞ þ Lðxc , ÞD21 ðÞÞBT1 ðÞR1 ðlV Þ þ 1 CT1 ðÞðC1 ðÞ þ D12 ðÞFðxc , ÞÞg,
ð31Þ
ð28Þ
ð29Þ
I
Then " # T1 Acl T1 Bcl Ccl T Dcl " # T1 ¼ I 2 3 A B2 F B1 6 7 4 WSLC2 A þ B2 F þ WSLC2 þ M WSLD21 5 C1 D12 F 0 T I " # A~ cl B~ cl ¼: , C~ cl 0
564
K. Dong and F. Wu
where W ¼ (S R1)1 and A þ B2 F B2 F ~ Acl ¼ , M A þ WSLC2 þ M B1 B~ cl ¼ , ðB1 þ WSLD21 Þ C~ cl ¼ ðC1 þ D12 FÞ D12 F ,
Note that x~ Tcl ðXcl A~ cl þA~ Tcl Xcl þ 1 Xcl B~ cl B~ Tcl Xcl þ 1 C~ Tcl C~ cl Þx~ cl I I I T 1 T I ¼ x~ cl x~ cl 0 I 2 0 I ¼ xTc 1 xc þ ðxc xÞT 2 ðxc xÞ, where
M ¼ ðWS IÞðA þ B2 FÞ þ WAT R1 þ 1 WSðB1 þ LD21 ÞBT1 R1 þ 1 WCT1 ðC1 þ D12 F Þ: To prove the sufficiency, it is enough to show that A~ cl is stable for any 2 ?, and kek2 < kdk2 when conditions (28)–(30) hold. To this end, we consider a Lyapunov function in the form of Vcl ðx~ cl Þ ¼ 12x~ Tcl Xcl ðxc Þx~ cl in which 1 R ðlV Þ Xcl ðxc Þ ¼ > 0, SðlU Þ R1 ðlV Þ
2 4
xc
ðAv þ B2, v FÞT R1 1 6 ðAv þ B2, v FÞ 5 6 þR I 4 R1 ðC1, v þ D12, v FÞ I BT1, p 3T
2
dVcl þ 1 eT e dT d dt ¼2
dððxc xÞT Sðxc xÞÞ dt
þ2
dðxT R1 x ðxc xÞT R1 ðxc xÞÞ dt
þ
1 T
T
e e d d
_ T SðlU Þðxc xÞ ¼ 2ðx_ c xÞ 1 T 1 T @ðR ðlV Þxc Þ þ 4 x_ R ðlV Þxc þ x x_ c @xc 2x_ Tc R1 ðlV Þxc þ 1 eT e dT d ¼ x_~ Tcl Xcl x~ cl þ x~ Tcl Xcl x_~ cl þ 1 eT e dT d ¼ x~ Tcl ðXcl A~ cl þ A~ Tcl Xcl þ 1 C~ Tcl C~ cl Þx~ cl þ 2dT B~ Tcl Xcl x~ cl dT d ¼ x~ Tcl ðXcl A~ cl þ A~ Tcl Xcl þ 1 Xcl B~ cl B~ Tcl Xcl þ 1 C~ Tcl C~ cl Þx~ cl k 1=2 B~ Tcl Xcl x~ cl 1=2 dk2
1 ¼ ðA þ B2 FÞT R1 þ R1 ðA þ B2 FÞ þ 1 R1 B1 BT1 R1 þ 1 ðC1 þ D12 FÞT ðC1 þ D12 FÞ, 2 ¼ ðA þ LC2 ÞT S þ SðA þ LC2 Þ þ 1 SðB1 þ LD21 ÞðB1 þ LD21 ÞT S þ 1 CT1 C1 : By Schur complement and xTc 1 xc < 0 is equivalent to
3 2 B1, v 7 xc 74 0 5 I
ðC1, v þ D12, v FÞT R1
and RðlV Þ, SðlU Þ as defined in (27). Clearly, Xcl is positive definite and bounded from below and above because of the coupling condition (30). Using Lemma 3, we have 2
ð35Þ
I 0
vertex
properties,
3 5 < 0:
I
ð36Þ
I
Following the proofing procedure of Theorem 3, it can be shown that (28) is a sufficient condition to guarantee equation (36). Similarly, equation (29) implies (xc x)T 2(xc x) < 0 from Lemma 2 and S-procedure. This confirms that (35) is negative and 2
dVcl þ 1 eT e dT d < 0: dt
Therefore, the closed loop system (26) is stable and kek2 < kdk2. œ Note that the number m of quadratic functions involved in V(x), and the number r in U(x) is not necessary to be same. Although the synthesis condition in Theorem 4 is sufficient, it is less conservative than the single quadratic Lyapunov function case. When m ¼ r ¼ 1, Theorem 4 recovers the single quadratic Lyapunov function result in Becker and Packard (1994) as a special case. Since the output feedback law depends on both the current controller state and scheduling parameter, it can be thought as a non-linear gain-scheduling controller. Moreover, the parameter lV is a continuous function of state xc and the parameter value of lU is discontinuous about xc. This non-linear controller thus has the form of a switched LPV controller with lU serving as the switching signal. Unlike parameter-dependent Lyapunov function cases (Wu et al. 1996, Wu and Dong 2006), the controller gain in our proposed approach
565
Robust and gain-scheduling control of LFT systems does not require parameter variation rates information, which is desirable from control implementation point of view. As a result, the performance improvement is mainly due to the nonlinear nature of the controller. Same as robust state feedback case, lV and lU can be computed either online or offline.
Table 1.
Optimal performance of robust state feedback control.
Method QLF with scaling matrix Single QLF Convex hull Lyapunov function
L2 gain
CPU time (sec)
3.922 3.921 3.343
1.453 0.266 0.375
6. Example In this section, a second-order LFT plant will be used to demonstrate the proposed approach. Both robust state feedback and gain-scheduling output feedback control laws will be designed for this LFT system. For robust state feedback case, a disturbance rejection problem will be considered. For gain-scheduled output feedback case, we consider a reference tracking problem. The LFT plant is in the form of 3 2 3 2 3 2 1 0:8 1 0 0 x_ 1 x1 6 x_ 2 7 6 1 1:6 0 3 2 7 6 x2 7 7 6 7 6 7 6 6 q 7¼6 0 6 7 1 0:5 0 1 7 7 6 p 7, ð37Þ 6 7 6 4 e 5 4 1 0 0 0 0 5 4d5 y u 1 0 0 0 0 p ¼ q:
ð38Þ
The state space matrices have LFT dependency of the scalar time-varying parameter 2 ½1, 1. First, robust state feedback control problem is solved, where is treated as an uncertainty. We assume all the states are measurable with no noise, therefore the measurement y will be ignored. The error output is the first state. The state feedback control law only depends on the plant states and the optimizer value l*. The control design objectives include minimizing the error output under disturbances with reasonable control force. They are quantified by rational weighting functions on the error output and control input channels. The weighting functions are chosen to be We ðsÞ ¼
0:1s þ 4 , s þ 0:4
Wu ðsÞ ¼
100s þ 100 : s þ 125
Three different robust controllers will be examined: (1) quadratic Lyapunov function (QLF) with scaling matrix (Apkarian and Gahinet 1995) (2) single quadratic Lyapunov function using condition (22) (Becker and Packard 1994); (3) convex hull Lyapunov function with two quadratic functions VðxÞ ¼ 12 xT ½l R1 þ ð1 l ÞR2 1 x;
The L2 gains calculated from these approaches are listed in table 1. Case 3 is solved by optimizing the synthesis condition (22) with 12, 1 ¼ 1:2, 21, 2 ¼ 5:4 and all other ij, v ¼ 0. The linear search range of ij, v is [0,10] with gridding density of 0.1. Although the linear search method provides a suboptimal result only, it is still much better than existing approaches. As shown in table 1, the robust control performance can be improved using our proposed approach. The CPU time of case 3 is based on a known sub optimal ,~which excludes the computational time for linear search. Depending on the search range, the linear search may consume much more computational effort. For simulation purposes, we choose the disturbance input as an impulse, and the parameter trajectory as in figure 3. The time domain simulation of robust controlled response is shown in figure 4. The dashed line corresponds to the controller using quadratic Lyapunov function with scaling matrix (Apkarian and Gahinet 1995), and the solid line is for our approach. Case 2 has identical simulation result as case 1. Note that online implementation of the proposed controller is used to get better comparison with other approaches. Although the response of control input has slightly larger magnitude, the error output using proposed robust controller is much smaller. Figure 5 shows the trajectory of optimal l*, which varies between 0 and 1. Optimal l* can be either calculated online or looked up from a pre-calculated table. Online calculation may need large computational effort when implementing controller, especially for large dimensional systems. Next, we will design a gain-scheduled output feedback controller for the LFT system. For this study, serves as a scheduling parameter, which is measurable in realtime and can be used in the controller gain adaptation. We want the measured output y to track a reference signal ref. The closed loop system configuration is given in figure 6. The weighting functions have been modified to
l ¼ arg min xT ½lR1 þ ð1 lÞR2 1 x, 0l1
using condition (22).
We ðsÞ ¼
0:003s þ 1:2 , s þ 0:004
Wu ðsÞ ¼
sþ1 : s þ 12500
566
K. Dong and F. Wu
Four gain-scheduling control approaches will be studied:
The L2 gains of different gain-scheduling control approaches are compared in table 2. In case 4, the gainscheduling control is designed by solving conditions (28)–(30) with 12, 1 ¼ 1:0, 21, 2 ¼ 7:0 and all other ij, v ¼ 0. The sub-optimal value of is calculated by linear search in the range of [0, 10] with gridding density of 0.1. Since U(x) is a single quadratic function, scalars "^ij, v are not necessary for the solution. From table 2, it can be seen that the L2 gain of gainscheduling control performance can also be improved by our proposed approach. The CPU time of case 4 which excludes the computais based on a known , tional time of linear search as well. All controllers are synthesized based on the same weighting functions. For the time-domain simulation, we use step input as reference and the same parameter trajectory in figure 3 as before. The time domain step response of different controllers are provided in figure 7 with dashed line for
(1) quadratic Lyapunov function with scaling matrix (Apkarian and Gahinet 1995); (2) quadratic Lyapunov function with full-block multiplier (Scherer 1999); (3) single quadratic Lyapunov function using conditions (28)–(30) (Becker and Packard 1994); (4) convex hull Lyapunov function V(x) with two quadratic functions and single quadratic function U(x) 1 VðxÞ ¼ xT ½lV R1 þ ð1 lV ÞR2 1 x, 2 lV ¼ arg min xTc ½lV R1 þ ð1 lV ÞR2 1 xc , 0lV 1
1 Uðxc Þ ¼ xTc Sxc , 2 using conditions (28)–(30). 1
2
0.8
1.5
0.6 0.4
1 λ2
Θ
0.2 0
0.5
0.2
0
0.4 0.6
0.5
0.8
1
1 0
5
10
15
20
25
0
30
2
4
Figure 3.
Parameter trajectory.
Figure 5.
0.1
1.2
0
1
−0.1 Control force
Error
8
Trajectory of l*.
Case 3 Case 1
0.8
−0.2 −0.3 −0.4
0.6 0.4 0.2
−0.5 −0.6 −0.7
6 t (sec)
t (sec)
0
Case 3 Case 1 0
2
Figure 4.
4
6
8
10
−0.2
0
2
4
6
t (sec)
t (sec)
(a)
(b)
8
10
Impulse response and control input of different robust controllers: (a) Error output; (b) Control force.
10
567
Robust and gain-scheduling control of LFT systems the gain-scheduled control using quadratic Lyapunov function with scaling matrix (Apkarian and Gahinet 1995), and solid line for our proposed approach. Note that cases 1, 2 and 3 provide exactly the same simulation results. Again, the proposed gain-scheduling controller provides better tracking capability with slightly larger control force. The proposed controller achieves not only faster transient response, but also less steady state tracking error. For a fair comparison, it should be pointed out that the computational time of proposed robust and gain-scheduling control approach is close to other
eu d
Wu ref Controller
approaches for the scalar parameter case as shown in tables 1 and 2. However, if matrix contains more parameters i, the vertex points of the set ? will increase exponentially. This will lead to large number of LMIs and require heavier computational cost. In addition, increasing the number of quadratic functions in convex hull Lyapunov function and the maximum function also leads to increased number of LMIs. Figure 8 provides the comparison of two different Lyapunov function level set. Because the close loop system is eighth-order, it is difficult to visualize a 8-dimensional level set. Figure 8 is the intersection of the close loop level-1 set in the plant state plane. Convex hull and maximum quadratic functions are more powerful and versatile than a single quadratic function.
2
–
u
Plant e
y
We
1
dn
System
structure
of
gain-scheduled
output
0.5 x2
Figure 6. feedback.
Case 1 Case 4
1.5
0 −0.5
Table 2.
Optimal performance of gain-scheduled output feedback control.
Method QLF with scaling matrix QLF with full-block multiplier Single QLF Convex hull Lyapunov function
−1
L2 gain
CPU time (sec)
3.523 3.522 3.521 3.000
1.391 0.906 0.328 0.406
−1.5 −2 −0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
x1
Figure 8.
Level set comparison.
1.8
1.2
1.6 1 1.4 Control input
y
0.8 0.6 0.4
1.2 1 0.8 0.6 0.4
0.2
Case 4 Case 1
Case 4 Case 1
0.2
0
0 0
Figure 7.
5
10
15
20
0
5
10
t (sec)
t (sec)
(a)
(b)
15
20
Step response and control input of different gain-scheduled controllers; (a) Error output; (b) Control force.
0.2
568
K. Dong and F. Wu
Therefore, they provide the possibility to expand the level set closer to the optimal one. As can be seen, the Lyapunov function level set of the proposed method is much larger than those from other methods, which is the reason why our new controller can perform better.
7. Conclusion In this paper, we first studied LFT parameter-dependent systems using duality theory. It has been shown that conjugate Lyapunov functions provide attractive alternative for LFT stability and performance analysis purpose. Then we employed a convex hull Lyapunov function and a maximum function of multiple quadratic functions for control synthesis problems. Both robust state feedback and gain-scheduling output feedback control of LFT systems have been considered. The control synthesis conditions were formulated and solved using LMI optimizations combined with linear search over scalar variables. The resulting nonlinear controllers depend on plant/controller states and the scheduling parameters, but not on the parameter variation rates. As demonstrated on a numerical example, the newly proposed control design approach provides better controlled performance than existing LPV controllers.
Acknowledgements Work supported in part by NSF Grant CMS-0324397.
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S.P. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, Philadelphia, PA: SIAM, 1994. S. Boyd and L. Vandenberghe, Convex optimization, Cambridge: Cambridge Univ. Press, 2004. R. Goebel, A.R. Teel, T. Hu and Z. Lin, ‘‘Dissipativity for dual linear differential inclusions through conjugate storage functions’’, in Proc. 43rd IEEE Conf. Dec. Contr., Paradise Island, Bahamas, pp. 2700–2705, 2004. A. Hassibi, J. How and S. Boyd, ‘‘A path-following method for solving BMI problems in control’’, in Proc. Amer. Contr. Conf., San Diego, CA, pp. 1385–1389, 1999. T. Hu, ‘‘Nonlinear control design for linear differential inclusions via convex hull quadratic Lyapunov functions’’, in Proc. 2006 Amer. Contr. Conf., Minneapolis, MN, pp. 3818–3823, 2006. T. Hu and Z. Lin, ‘‘Composite quadratic Lyapunov functions for constrained control systems’’, IEEE Trans. Automat. Contr., 48, pp. 440–450, 2003. T. Hu and Z. Lin, ‘‘Properties of composite quadratic Lyapunov functions’’, IEEE Trans. Automat. Contr., 49, pp. 1162–1167, 2004. T. Hu, R. Goebel, A.R. Teel and Z. Lin, ‘‘Conjugate Lyapunov functions for saturated linear systems’’, Automatica, 41, pp. 1949–1956, 2005a. T. Hu, A.R. Teel and L. Zaccarian, ‘‘Performance analysis of saturated systems via two forms of differential inclusions,’’ in Proc. 44th IEEE Conf. Dec. Contr., pp. 8100–8105, 2005b. W. Lu, K. Zhou and J.C. Doyle, ‘‘Stabilization of uncertain linear systems: an LFT approach’’, IEEE Trans. Automat. Contr., 41, pp. 50–65, 1996. A.K. Packard, ‘‘Gain scheduling via linear fractional transformations’’, Syst. Contr. Letts., 22, pp. 79–92, 1994. R.T. Rockafellar, Convex analysis, Princeton: Princeton Univ. Press, 1970. W.J. Rugh, ‘‘Analytical framework for gain scheduling’’, IEEE Contr. Sys. Mag., 11, pp. 74–84, 1991. W.J. Rugh and J.S. Shamma, ‘‘Research on gain scheduling’’, Automatica, 36, pp. 1401–1425, 2000. J.S. Shamma and M. Athans, ‘‘Analysis of gain scheduled control for nonlinear plants’’, IEEE Trans. Automat. Contr., AC-35, pp. 898–907, 1990. C.W. Scherer, ‘‘Robust mixed control and LPV control with full block scalings’’, in Recent Advances of LMI Methods in Control, L. El Ghaoui and S. Niculescu, Eds: Philadepha SIAM, 1999. G. Scorletti and L. El Ghaoui, ‘‘Improved Linear Matrix Inequality Conditions for Gain Scheduling,’’ in Proc. 34th IEEE Conf. Dec. Contr., New Orleans, LA, pp. 3626–3631, 1995. F. Wang and K. Balakrishnan, ‘‘Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems’’, IEEE Trans. Automat. Contr., 47, pp. 720–734, 2002. F. Wu and K. Dong, ‘‘Gain-scheduling control of LFT systems using parameter-dependent Lyapunov functions’’, Automatica, 42, pp. 39–50, 2006. F. Wu, X.H. Yang, A.K. Packard and G. Becker, ‘‘Induced L2 norm control for LPV systems with bounded parameter variation rates’’, Int. J. Robust Non. Contr., 6, pp. 983–998, 1996.
