A NEW DILATED LMI CHARACTERIZATION AND ITERATIVE CONTROL SYSTEM SYNTHESIS

A NEW DILATED LMI CHARACTERIZATION AND ITERATIVE CONTROL SYSTEM SYNTHESIS Noboru Sebe ∗ ∗ Kyushu Institute of Technology Department of Artificial Intelligence 680-4, Kawazu, Iizuka, 820-8502 Japan [email protected]

Abstract: This paper presents a new iterative design procedure for multiobjective and structurally constrained feedback control. The proposed design procedure is based on new dilated LMI characterizations for the system properties, such as H2 , H∞ norms and D-stability. The key idea of the new dilated LMI characterizations is divisions of A and B matrices in descriptor representation of systems. This paper also proposed to linearize the products of controller parameter K and auxiliary variable G by assigning the auxiliary variables instead of the change of variables technique. This linearization technique is effective from the viewpoints c of computational complexity and accuracy. Copyright 2007 IFAC Keywords: multiobjective control, decentralized control, linear matrix inequalities, descriptor systems, dilation, Lyapunov variables.

1. INTRODUCTION The aim of this paper is to propose an iterative design procedure based on the dilated linear matrix inequalities (LMI) characterizations for the system properties. The dilated (or extended) LMI characterizations enable us to use parameter-dependent Lyapunov functions for robust system analysis and synthesis (de Oliveira et al., 1999; Peaucelle et al., 2000; Apkarian et al., 2001; de Oliveira et al., 2002) and independent Lyapunov functions for multiobjective control synthesis problems (Shimomura et al., 2001; Ebihara and Hagiwara, 2004). These results have promoted the great advance of the control theory. On the other hand, there have been many attempts to improve the performance by iterative designs (Shimomura and Fujii, 2005; Kami and Nobuyama, 2004). Although the iterative designs

are effective, the procedures are rather specific for each combination of objective functions. In this paper, new dilated LMI characterizations for iterative design of multiobjective and structurally constrained feedback control are derived. The key idea is dividing system and input matrices into some pieces. With the derived characterizations, an iterative design procedure is proposed. This paper also demonstrates the effectiveness of the proposed design procedure through numerical examples. We use the following notations. I and O denote the identity and zero matrix, respectively. For a matrix M , M −1 and M T are the inverse and transpose matrix of M , respectively. He{M } is a ¯ (M ) is the maxshorthand notation for M +M T . σ imum singular value of M . In some partitioned symmetric matrices, the symbol ’’ denotes each of its symmetric blocks.

2. ESSENTIAL IDEA AND DESIGN PROCEDURE

Ti (i = 0, 1) denote the transfer functions from wi to zi . Assume γ∞ is a given scalar. Then, the problem is to find a static feedback u = Ky which minimizes γ2 = T0 2 under T1 ∞ < γ∞ .

2.1 Essential idea Let us consider a system below: x˙ = Ax.

(1)

One of the descriptor representations of the above system can be given by     I O ˙ O A x ˜= x ˜, (2) O O I −I where x ˜ = [xT xT ]T . With this representation, Chen (2004) has pointed out that the dilated (or extended) LMI characterization for stability of the system (1) can be obtained by applying the stability characterization for descriptor systems. In this paper, new dilated LMI characterizations are proposed. The key idea is dividing matrices A and B into some pieces. Let a given system be x˙ = Ax + Bu, y = Cx + Du. Then, divide A and B matrices as follows: A = A¯11 + A¯12 A¯−1 A¯21 , 22

¯1 + A¯12 A¯−1 B ¯ B=B 22 2 .

(3a) (3b) (4a) (4b)

With the above divisions, a descriptor representation of the system (3) can be given as follows:       ¯1 B I O ˙ A¯11 A¯12 x ˜ + ¯ u, (5a) x ˜= ¯ ¯ O O A21 A22 B2   y= C O x ˜ + D u, (5b) where x ˜ = [xT ξ T ]T and ξ is an auxiliary state. New dilated LMI characterizations can be obtained by applying LMI characterizations for descriptor systems, such as H∞ -norm characterization (Masubuchi et al., 1997), H2 -norm characterization (Takaba and Katayama, 1998), regional pole placement (Kawata, 2005). 2.2 Controller Synthesis In this section, let us apply the idea in previous subsection to the H2 /H∞ static feedback design. Note that fixed order dynamic controllers can be similarly designed with augmented matrices given in (Iwasaki and Skelton, 1994). Furthermore, the design procedure proposed here can also deal with multiobjective control problems with any combinations of objective functions and structurally constrained control problems. Let us consider a generalized plant x˙ = Ax + B0 w0 + B1 w1 + B2 u, z0 = C0 x + D01 w1 + D02 u, z1 = C1 x + D10 w0 + D11 w1 + D12 u, y = C2 x + D20 w0 + D21 w1 + D22 u.

(6a) (6b) (6c) (6d)

As this paper aims to propose an iterative design procedure, let us assume that a previously ˆ is given. With this designed feedback gain K ˆ designed K and the feedback K to be designed, the explicit descriptor representation of the closed loop system, which is used for new dilated LMI characterizations, is proposed as follows: ˜x ˜x + B ˜ w, E ˜˙ = A˜ ˜ ˜ ˜ y˜ = C x ˜ + Dw, ˜

(7a) (7b)

˜ = block diag{I, O}, where E ⎡ ⎤ ⎡ A B B   L 0 ˜ B ˜0 B ˜1 A ˜ ⎢ O −I O A˜ B ⎢ ˜ ˜ ⎥ ⎢ ˜ 01 ⎦ = ⎣ = ⎣ C0 D00 D ˜ C0 O O C˜ D ˜ 10 D ˜ 11 C˜1 D C1 O D10 ⎧⎡ ⎫ ⎤ ⎤ ⎡ B2 O ⎪ ⎪ ⎪ ⎪ ⎨⎢ ⎬ ⎥ ⎢ BR ⎥ O ˆ ˆ ⎥ ⎥ ⎢ ⎢ + ⎣ K + (K − K) ⎣ D02 ⎦ D02 ⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ D12 D12   × C2 O D20 D21 , B2 = BL BR , ξ = BR u,       x w0 y x ˜= , w ˜= , y˜ = 0 . ξ w1 y1

⎤ B1 O ⎥ ⎥ D01 ⎦ D11

(7c) (7d) (7e)

The decomposition of B2 in (7d) will be discussed later. Then, the H2 /H∞ control problem can be formulated as an optimization problem as below. Problem 1. Find symmetric matrices P2 , P∞ , Q, and matrices K, G21 , G22 , G∞1 , G∞2 , such that minimize γ2 subject to γ22 ≥ trace Q,   ˜ P˜2 B ˜0 He{P˜2 A} < O,  −I   Q C0 > O,  P2 ⎡ ⎤ ˜ P˜∞ B ˜1 He{P˜∞ A}  ⎣  ⎦ < O,  −γ∞ I ˜ ˜ C1 D11 −γ∞ I P∞ > O,   P2 G21 ˜ P2 = , O G22

 P˜∞ =

(8) (9) (10) (11) 

P∞ G∞1 . O G∞2

(12) (13)

Similar to the the conventional dilated LMI characterizations, there are no product terms of P∗ and K in the above conditions. On the other hand, the product terms of G∗2 and K exist. Thus, the linearization is required to solve the problem. Instead of the actual linearization which will be mentioned in the next subsection, the conceptual iterative design procedure is summarized here.

4

Algorithm 1. Let K (i) be the feedback gain which is design at the i-th design iteration.

With this iterative design procedure, let us define (i) some variables. Let γ2g be the guaranteed upper (i)

bound, i.e., the optimized value γ2 , and γ2a be the actually achieved H2 -norm of the closed loop sys(i) (i) tem with the designed K (i) . Evidently, γ2a ≤ γ2g . (i)

(i)

3

x0 y + xy0 = r + x0 y0

2.5

y

(i) Find a static feedback K which satisfies T1 ∞ < γ∞ . Set K (0) = K and i = 1. ˆ = K (i−1) in (7). Solve the optimiza(ii) Set K tion problem with the linearized conditions of (9) and (11). (We will mention the actual linearization in the next subsection.) Set K (i) = K where K is the solution. (iii) If a stopping criterion is satisfied, exit. Otherwise, set i = i + 1 and go to Step (ii).

actual boundary approximation by proposed method approximation by Shimomura

3.5



2 1.5

z0 +

√

z02 +4βr − z0 + , 2β

√

z02 +4βr 2



(x0 , y0 ) 

1

βx 20 − r z0

,

βr− y 02 z0



0.5 0

0.2

0.3

0.4

0.5

x

0.6

0.7

0.8

0.9

1

Fig. 1. Inner convex approximation of non-convex constraint xy < r. (r = 1, (x0 , y0 ) = (0.3, 1.5), β = 1, z0 = βx0 − y0 .) (i−1) , assign In addition to G∗1 = Pˆ := P∗

G∗2 = βI

(β > 0).

(15)

The matrices P2a and P∞a denote the Lyapunov variables which evaluate the H2 and H∞ -norms of the closed loop system with K (i) , respectively. Note that these matrices are different from the solutions P2 and P∞ at the i-th design iteration.

Then the condition corresponding to the stability characterization part in (9) or (11) becomes    ˆ 2 BL A + B2 KC P Pˆ BL < O. (16) He ˆ 2 −I O βI BR (K − K)C

2.3 Linearization by assignment of variables

Let us consider a sufficient condition for (16).      ˆ 2 BL O O A + B2 KC P Pˆ BL He + ˆ 2 −I O βI O βI BR (K − K)C

The dilated LMI conditions in the previous subsection utilize the previously designed K (i−1) . In this subsection, a linearization with the Lyapunov (i−1) (i−1) and P∞a is proposed. variables P2a As mentioned before, there exist the product terms of G∗ and K. Obviously, the problem becomes linear, if G∗ is fixed. Thus we now propose to linearize the conditions (9), (11) by assigning (i−1)

G21 = P2a

BL ,

(i−1) G∞1 = P∞a BL ,

(14)

and G22 and G∞2 as arbitrary positive definite matrices, e.g., identity matrices. Then, the next theorem holds. Theorem 2. Assume the conditions (9) and (11) be linearized by the variable assignments (14). (i) (i−1) holds. Then, the inequality γ2g ≤ γ2a ˆ = K (i−1) , PROOF. From (9) to (13), let K = K (i−1) (i−1) P2 = P2a , P∞ = P∞a , and G22 , G∞2 be arbitrary positive definite matrices. Then, the (i−1) optimal value is γ2a . 2

Schur complement formula reduces (17) to ˆ 2} He{P (A+B2 KC2 )} + βΦ{BR (K − K)C 1 + Φ{BLT (P − Pˆ )} < O, (18) β where Φ{M } = M T M . With β = 1, BL = B2 , BR = I, this condition is essentially equivalent to that proposed in (Shimomura and Fujii, 2005). It is clear that the condition (16) is less conservative than (18). The conditions (9) and (11) with the variables assignments (14) give another interpretation of the inner convex approximation proposed in (Shimomura and Fujii, 2005) and reduce the conservatism. Note that the condition (16) can be also represented by the linearized condition with the additional square term as follows, 1 ˆ 2 Φ{βBR (K − K)C He{P (A + B2 KC2 )} + 2β − BLT (P − Pˆ )} < O. (19) Saeki (2006) has independently proposed the same 1

2.4 Connections with other results For the sake of simplicity let us consider the stability characterization He{P (A + B2 KC2 )} < O. The H2 and H∞ -norm characterizations parallel to the stability characterization.

(17)

< O.

1

condition with β = 1, BL = UB ΣB2 , BR = ΣB2 VBT , where UB ΣB VBT is a singular value decomposition of B2 . This decomposition of B2 has an advantage if B2 is not full rank, such as the reliable controller design. (See Example 6). For an intuitive understanding of the linearization (14), a simple numerical example is given below.

Example 3. For a given scalar r, let us consider a constraint xy < r, where x, y(> 0) are the variables. (Note that x and y correspond to P BL and BR KC2 , respectively). Figure 1 illustrates the inner approximated regions obtained by the linearizations. The proposed approximation always touches the boundary of the actual non-convex region. Also note that β = xy00 make the approximated region a half plane, whose boundary is parallel to the extension line shown in Fig. 1. 3. SOME REMARKS ON IMPLEMENTATION 3.1 Choice of G∗2

Actual boundary Convex approximation K(i-1) K(i)

Admissible region

Fig. 2. Acceleration by line search. where P , R, d1 and d2 are the variables to be optimized, Acl , Bcl , Ccl , Dcl are the state space data of the closed loop system and γopt is its H∞ norm. Note that d1 and d2 are the scaling of the input and output. Then assign Pˆ = d11 P . 3.3 Acceleration by line search

The choice of G∗2 corresponds to the division P B2 KC2 → P BL × BR KC2 . Let us assign G∗2 = β∗ I. Roughly speaking, the inequality (19) suggests that the linearized condition (19) might be ˆ 2 less restrictive if (P − Pˆ )BL and β∗ BR (K − K)C are of same magnitude. As we cannot know the solutions K and P a priori, the best strategy is choosing β∗ to balance the magnitude of Pˆ BL and ˆ 2 . For the scalar case, this selection of β∗ β∗ BR KC make the approximated region a half plane. Take into account of the direct terms in (9) and (11), the assignments below are proposed. (i−1)

G22 =

σ ¯ (P2a BL ) I, σ ¯ (BR K (i−1) [ C2 D20 ])

G∞2 =

σ ¯ (P∞a BL ) I. σ ¯ (BR K (i−1) [ C2 D21 ])

(20a)

(i−1)

With the linearization (14), we search for solutions to the optimization problem over the inner convex approximation of the admissible region. Generally, the solution is on the boundary of the convex approximation and its neighborhood is also admissible. (See Fig. 2). Accordingly, we propose to improve the performance by searching a controller K defined by K = K (i−1) + α(K (i) − K (i−1) ) (α ≥ 1), (24) where α is a line search parameter. As the evaluation of norms consumes little computation, this line search might accelerate the optimization procedure. This idea is proposed in Saeki (2006) as a decent method.

(20b) 4. NUMERICAL EXAMPLES

3.2 Choice of Pˆ∞ ˆ the Lyapunov matrix Pˆ∞ For given γ∞ and K, is not determined uniquely, even if the given γ∞ is exactly same as the H∞ norm of the closed loop system. Halder and Kailath (1999) proposed a design procedure similar to (Shimomura and Fujii, 2005) and also proposed to choose Pˆ∞ by maximizing det Pˆ . But this choice does not seem to be adequate. For the dual system, one of the Lyaˆ = Pˆ −1 . It is obvious punov matrices is given by R ˆ Take into that maximizing det Pˆ minimizes det R. account of the dual system and the input-output scaling, this paper proposes to choose Pˆ as a solution to the optimization problem below. minimize trace(P + R) subject to ⎤ ⎡  He{P Acl } P Bcl ⎦ < O, ⎣   −d1 γopt I Dcl −d2 γopt I Ccl ⎤ ⎡ He{Acl R} Bcl  ⎦ < O, ⎣   −d1 γopt I Dcl −d2 γopt I Ccl R     P I d1 1 > O, > O, I R 1 d2

Three examples demonstrate the efficiency of the proposed method. All the examples are carried out by Robust Control Toolbox in MATLAB (Release 2007a) on a PC (Pentium 4, 3.2GHz with 2GB RAM). For all examples, the decomposition of 1

1

B2 is BL = UB ΣB2 and BR = ΣB2 VBT , and the (i−1) (i) stopping criterion is |γ∗a − γ∗a | < 1 × 10−5 . Example 4. (H2 /H∞ control). Let us consider a two mass - one spring system (Shimomura and Fujii, 2005). The coefficient matrices are ⎤ ⎡ A

B0

B1

B2

⎢ C0 D00 D01 D02 ⎥ ⎣ C1 D10 D11 D12 ⎦ C2 D20 D21 D22

⎡

(21) = (22) (23)

⎤

0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 ⎥ ⎢ 0 ⎢ −1.25 1.25 0 0 0 −0.25 1 1 ⎥ ⎥ ⎢ ⎢ 1.25 −1.25 0 0 1 0.25 0 0 ⎥ ⎥ ⎢ 0 0.2 0 0 0 0 0 0 ⎥. ⎢ ⎥ ⎢ 0 0 0 0 0 0 0 2 ⎥ ⎢ ⎢ 1 −1 0 0 0 0 0 0 ⎥ ⎦ ⎣ 0 0 0 0.2 0 0 0 0 0 1 0 0 0.1 0 0 0

The problem is to find a controller which minimizes T0 2 under the H∞ constraint T1 ∞ < 1.

Table 1. Achieved H2 norm and computational expense (Example 4). Method Scherer et al. Shimomura and Fujii proposed method

H2 norm 3.47512 2.34767 2.33471

iteration – 27 27

Table 2. Achieved H2 norm and computational expense (Example 5).

time [s] – 34.69 12.38

3.8

Method Oliveira et al. centralized proposed (eq.(20)) proposed (β = 1)

Shimomura & Fujii proposed

3.6

H2 norm 8.53026 1.11095 1.24508 1.23657

iteration – – 1638 52

9

time [s] – – 551.38 16.32

G : eq.(21)

8

3.2

7

3

6

H2 norm

H2 norm

22

3.4

2.8

4

2.4

3

5

10

15

20

25

22

5

2.6

2.2 0

G =I

2

30

number of iteration

1 0

10

20

30

40

50

60

number of iteration

Fig. 3. Achieved H2 norm (Example 4). Fig. 4. Achieved H2 norm (Example 5).

The initial controller K (0) (s) is selected as ⎡ ⎤ −0.6434 0.8808

0.4445 −0.0293

−1.1816

⎢ −7.7217 0.2027 0.2398 −0.2033 −17.2304 ⎥ ⎢ −9.6207 0.0825 −1.3514 0.9307 −22.0262 ⎥ ⎣ ⎦ −15.8507 1.2653 −4.3137 −1.8643 −36.2088 0.1104 0.1513 −0.4851 0.4345 0

which is designed by Scherer et al. (1997). After 27 iterations, we obtain a controller ⎤ ⎡ −0.8370 0.8532 0.5021 −0.0427 −1.5752 ⎢ −7.2332 0.1068 0.2531 0.0074 −16.1312 ⎥ ⎢ −7.7456 −0.0249 −0.7676 0.9700 −17.7390 ⎥ . ⎦ ⎣ −9.7861 0.9648 −3.0536 −0.8123 −22.2775 0.1617 0.2233 −0.2539 0.3717 0

Table 1 and Fig. 3 show the comparison between the proposed method and that in Shimomura and Fujii (2005). The convergence rate of the proposed method is much faster than that of Shimomura’s. Example 5. (Decentralized H2 control). The proposed method is applied to a decentralized controller design. The assignment of G22 is also considered. The generalized plant is borrowed from Veillette et al. (1992) and its data are ⎤ ⎡ −2



A B1 C1 D11 C2 D21

1

1

1 1 0 0 0 0

⎢ 3 0 0 2 0 0 0 1 0⎥ ⎥ ⎢  ⎢ −1 0 −2 −3 1 0 0 0 0 ⎥ ⎢ −2 −1 2 −1 0 0 0 0 1 ⎥ B2 ⎥ ⎢ D12 = ⎢ 1 0 −1 0 0 0 0 0 0 ⎥ . ⎢ 0 0 0 0 0 0 0 1 0⎥ D22 ⎥ ⎢ ⎢ 0 0 0 0 0 0 0 0 1⎥ ⎣ 1 0 0 0 0 1 0 0 0⎦ 0

0

1

0 0 0 1 0 0

The problem is to find a decentralized controller which minimizes T1 2 . Let the initial decentral(0) (0) ized controller diag{K1 (s), K2 (s)} be     −8.28 −6.447 8 4.00 0 0 −2.831 −3.086 0

,

−2287 −70.67 2 128 0 0 −0.995 −0.03148 0

,

which is designed by Oliveira et al. (2000). Table 2 and Fig. 4 show the computation results with G22

defined by (20) and G22 = I. For this example, it is much better fixing G22 instead of defining by (20). This example suggests that not only the difficulty of choosing G22 , but also the ability of G22 to improve the convergence. With fixed G22 = I, we obtained a decentralized controller     −1.828 −5.634 2.857 0.059 −0.674 0.220 −0.098 −1.896 0

,

−2.044 −0.075 0.002 3.937 −2.240 0.054 134.126 −8.886 0

Example 6. (Reliable H∞ control). Let us design a reliable H∞ controller for the same generalized plant in Example 5. The design problem is called a ‘1-out-of-4 integrity problem’ (G¨ unde¸s, 1992). One normal operation mode and four operation modes with 1 failure among the 4 devices (2 actuators and 2 sensors) are considered, where failures are modeled by 0 transfer functions. The problem is to find a controller which minimizes worst H∞ norm of the above 5 modes. The initial controller is designed by the method shown in (Sebe and Mochimaru, 2005) and is ⎤ ⎡ −1.117

2.248 −0.718 −0.634 −0.349

⎢ −2.901 −1.171 1.669 −0.098 −0.635 ⎥ ⎢ 0.470 −1.369 −1.489 0.272 −0.285 ⎥ . ⎦ ⎣ −0.129 −0.628 0.299 −0.763 −0.353 −0.712 0.136 −0.256 −1.312 −1.021

The Lyapunov parameter Pˆ is selected as a solution to (1) the sub-optimization problem T1 ∞ < (i−1) γ∞a , (2) the optimization problem minT1 ∞ for each operation mode, (3) the optimization problem proposed in subsection 3.2. For each selection of Pˆ , the comparisons of the results with and without the acceleration are shown in Table 3 and Fig. 5. During the line search, we restrict n α as 2 2 (n = 0, 1, 2, . . .) and increase n while the achieved norm is decreasing. The efficiency of

.

Table 3. Achieved H∞ norm and computational expense (Example 6). Pˆ∞ initial K (0) (1) (2) (3) (1) with accel. (2) with accel. (3) with accel.

H∞ norm 5.94139 4.45079 4.45586 4.44955 4.45070 4.45119 4.44957

iteration – 160 240 77 66 66 32

time [s] – 307.65 555.50 285.53 167.68 194.26 128.32

6 Method 1 Method 2 Method 3 Method 1 with accel. Method 2 with accel. Method 3 with accel.

5.8 5.6

H∞ norm

5.4 5.2 5 4.8 4.6 4.4 0

50

100

150

200

time [sec]

Fig. 5. Achieved H∞ norm (Example 6). the proposed choice of Pˆ and the acceleration is confirmed. The obtained controller K (32) (s) is ⎡ ⎤ 7.915

8.129 −0.664

0.035

2.488

⎢ −9.799 −11.167 −0.720 −0.245 −3.271 ⎥ ⎢ 23.384 19.237 −3.867 2.884 6.057 ⎥ . ⎣ ⎦ −3.388 −6.349

−4.665 −0.684 −0.772 −1.605 −4.648 0.919 −2.479 −2.649

5. CONCLUSIONS This paper introduces new dilated LMI characterizations which are derived by divisions of system matrices in descriptor representation. This paper also proposes to linearize conditions by variable assignments. Based on the characterizations and linearizations, a new iterative design procedure is proposed for multiobjective control and structurally constrained controller designs. The main contribution of the paper is that the paper clarifies the connection between the dilated LMI characterizations and the inner convex approximation approaches. The other contribution is that the paper provides a unified framework for iterative control system synthesis.

ACKNOWLEDGEMENTS The author would like to thank Prof. Eitaku Nobuyama and Prof. Masami Saeki for the helpful discussions. This research is supported in part by the Ministry of Education, Culture, Sports, Science and Technology, Japan, under the Grantin-Aid for Scientific Research (B) No.19360192.

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