Fixed-Structure ℋ2 Controller Design: An LMI solution

Proceedings of the 7th IFAC Symposium on Robust Control Design The International Federation of Automatic Control Aalborg, Denmark, June 20-22, 2012

Fixed-Structure H2 Controller Design: An LMI solution Arash Sadeghzadeh ∗ ∗

Department of Electrical Engineering, Faculty of Engineering, Razi University, Kermanshah, Iran (e-mail: [email protected])

Abstract: In this paper sufficient conditions for fixed-structure H2 controller design in terms of solutions to a set of linear matrix inequalities are given. The iterative synthesis procedure converges to a suboptimal solution and must be initialized either with a Schur stable polynomial or with a stabilizing controller. The monotonic decreasing of the norm bound is established theoretically as well as demonstrated numerically with an example. Keywords: Fixed-order control, Fixed-structure control, H2 performance, Strictly positive real, Linear matrix inequality. 1. INTRODUCTION Fixed-structure controller design is a challenging problem in theory and in practice. A fixed-structure controller design problem arises when simplicity, hardware limitations, or reliability in the implementation of a controller dictates a low order controller. It is well known that fixed-order controller design leads to either a non-convex rank constraint or bilinear matrix inequalities which are computationally intractable. In the last few years, various innovative techniques for designing fixed-structure controllers have been proposed. A methodology to design fixed-order stabilizing controllers of single-input single-output (SISO) plants is given in Fujisaki et al. (2008). The approach is based on the idea of splitting the controller parameters into randomized parameters and deterministically designed parameters. An approach for fixed-structure controller design is given in Malik et al. (2008). The set of all stabilizing controllers is approximated by the union of the feasible sets of finite Linear Programs (LPs). The approximation can be made arbitrarily accurate by increasing the number of feasible sets of LPs. Several iterative methods for reduced-order control have been proposed over recent years; see, for instance, Apkarian et al. (2003) and Apkarian and Noll (2006), and references therein. In Apkarian et al. (2003), an iterative technique to compute solutions of the fixed-order H2 and H∞ synthesis problem is presented. These problems are reformulated as regular SDP programs with additional nonlinear equality constraints. A nonsmooth optimization technique to solve fixed-structure H∞ synthesis is developed in Apkarian and Noll (2006). The synthesis procedure converges to a local minimum and must be initialized with a stabilizing controller.This approach is basically a local search gradient-based method which has no guarantee to find a solution even if the problem itself is solvable. Linear Matrix Inequalities (LMIs) have been extensively used to provide suitable formulation for several system and 978-3-902823-03-8/12/$20.00 © 2012 IFAC

331

control problems. The solution to these problems can be calculated with the well-known LMI softwares. In Henrion et al. (2003) an LMI formulation is given for fixed-order controller design in a polynomial framework, based on polynomial positivity conditions. An extension for this strategy for fixed-order H∞ controller design is provided in Yang et al. (2007). For fixed-order H∞ controller design an LMI formulation is used in Khatibi and Karimi (2010). This formulation is based on the strict positive realness of two transfer functions with the same Lyapunov matrix in the matrix inequality of the KYP Lemma. An extension of this approach is considered in Sadeghzadeh (2011). The problem of robust fixed-order controller design for systems with ellipsoidal parametric uncertainty is considered in Sadeghzadeh et al. (2011), Sadeghzadeh and Momeni (2011). An LMI based convex optimization problem for robust pole placement with sensitivity function shaping in H2 norm, using a fixed-order controller, is proposed in Karimi et al. (2007). In these papers, a convex set of stabilizing controllers is parameterized such that the closedloop characteristic polynomial divided by a so-called central polynomial is a strictly positive real (SPR) transfer function. This convex set is an inner approximation of the non-convex set of all fixed-order stabilizing controllers and the quality of this approximation is related to the choice of the central polynomial (SPR-maker). To the best of our knowledge, no LMI-based condition for fixed-order H2 controller design is available in the literature. In the present paper, we have provided sufficient conditions for fixed-structure H2 controller design. The proposed conditions have been proved without taking into account the SPRness concept of transfer functions. Although, it is easy to see that these LMI conditions contain the KYP lemma constraints. The convergence property of the proposed approach to a local minimum is established theoretically. The quality of the obtained suboptimal solution depends on an initial central polynomial. The initial central polynomial can be any Schur stable polynomial or may be the closed-loop characteristic polynomial computed for an initial stabilizing controller. In contrast to

10.3182/20120620-3-DK-2025.00149

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

+ ✲ ❥ – ✻

✲

K(z)

✲

G(z)

✲

H(z, θ) =

S(z, θ) sn z n + sn−1 z n−1 + · · · + s1 z + s0 , = L(z, θ) ln z n + ln−1 z n−1 + · · · + l1 z + l0 (3)

let κ = ( 1 x0 · · · xm y1 · · · ym ) can be parameterized as

T

∈ Rv , then H(z, θ)

Fig. 1. Standard negative feedback configuration the SPR-based conditions for fixed-order controller design in Karimi et al. (2007), Khatibi and Karimi (2010) and Sadeghzadeh (2011) , it is not necessary to use a state space canonical form realization for the transfer functions. It is a well-known fact that the canonical form realizations are numerically ill-conditioned, therefore, employing a balance realization may improve the numerical conditioning in the control design algorithm. The rest of the paper is structured as follows. The problem formulation and preliminaries can be found in Section 2. Section 3 shows the main contribution of the paper. That is, how to design fixed-structure H2 controller. Section 4 is devoted to simulation example. Finally, some conclusions are drawn in the last section. The notation is fairly standard. Rn×m is the set of n × m real matrices. In is an n × n identity matrix. 0n×m and 0n are n × m and n × n zero matrices, respectively. The subscript for the dimension may be dropped if the sizes of matrices are clear from the context. M T is the transpose of a matrix M . P > 0 (≥ 0) means that P is symmetric positive (semi)definite. The state space realization of a transfer matrix G(s) is shown as follows:   A B . G(s) ↔ C D 2. PROBLEM FORMULATION AND PRELIMINARIES Consider the transfer function of a discrete-time linear time-invariant SISO system G(z, θ) =

θ0 z p + θ1 z p−1 + · · · + θp z q + θp+1 + · · · + θr−1

(1)

H(z, θ) =



AT0 QA0 − Q AT0 QB0 B0T QA0 B0T QB0 − 1   Q 0 C0T  0 1 DT  > 0. 0 C0 D 0 γ

such that: • the closed-loop system is internally stable • the closed-loop system achieves the H2 performance 2 kH(z, θ)k2 < γ.

(5) (6)

The problem addressed here is to provide LMI conditions for fixed-structure H2 controller design. Consider the state space realization (A0 , B0 , C0 , D0 ) for the transfer function H(z, θ) given by (3). To be concise, we consider two cases. First suppose that D0 is independent of the controller parameters and the other is the case that D0 is dependent on the controller parameters. Suppose that a Schur stable polynomial E(z) is given with the same order as L(z, θ). Let (A, B, C, D) to be the state space realization of the following transfer matrix  ψ1s  ..   .       ψvs  −1 A B  ψ E ↔ C D  1l   .   ..  ψvl 

(7)

It is worthwhile to remind that matrices A, B, C and D are constant matrices. Obviously, it is easy to see that we have    S A B     E Ds  ,  L  ↔  Cs Dl Cl E 

Where H(z, θ) can be any of the weighted closed-loop transfer functions. We consider where 332

< 0,

3. FIXED-STRUCTURE H2 CONTROLLER DESIGN

T

(2)



Since the controller parameters appear in the state space matrix A0 , therefore condition (5) is not an LMI with respect to the controller parameters and cannot be used for controller design. In the sequel, sufficient conditions for controller synthesis are provided.

θ = [ θ0 θ1 · · · θr−1 ] ∈ Rr is a vector that parameterizes G. We consider a standard negative feedback configuration shown in Figure 1. The goal is to design a fixed-structure controller x0 z m + x1 z m−1 + · · · + xm , z m + y1 z m−1 + · · · + ym

(4)

where, ψ1s , · · · , ψvs , ψ1l , · · · , ψvl are constant polynomials. The following lemma gives the necessary and sufficient conditions for evaluating the H2 performance. Lemma 1. (Scherer and Weiland (2000)) Consider a SISO discrete-time transfer function H with the state space 2 realization (A0 , B0 , C0 , D0 ). Then kHk2 < γ if and only T if there exists Q = Q > 0 such that

where parameter

K(z) =

( ψ1s ψ2s · · · ψvs ) κ ( ψ1l ψ2l · · · ψvl ) κ,

(8)

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

Cs = κT ( Iv 0 ) C, Cl = κT ( 0 Iv ) C,

Ds = κT ( Iv 0 ) D, Dl = κT ( 0 Iv ) D.

(9) (10)

and condition (12) is equivalent to  P 0 CiT  0 1 DT  > 0. i Ci D i γ 

3.1 D0 is independent of the controller parameters In what follows, we consider the case that D0 is independent of the controller parameters. This happens e.g when the controller to be designed is assumed to be strictly proper. Note that here Ds and Dl are constant scalars. Theorem 2. Given a schur stable polynomial E(z), consider a SISO discrete-time transfer function H(z, θ), given 2 by (3). Then kHk2 < γ if there exists P = P T > 0 such that  AT P A − P AT P B − ClT 0  < 0,  B T P A − Cl B T P B − Dl − DlT DlT 0 Dl −(Dl + DlT )/2 (11)   P 0 CsT − D0 ClT   > 0, (12) 0 (Dl + DlT )/2 DsT T Cs − D 0 Cl Ds γ(Dl + Dl )/2     Ds Cs ) is the state space realization , where (A, B, Dl Cl T  S L and D0 = Ds Dl−1 . of the transfer matrix E E 

Proof. See Appendix A. Clearly, if we consider the state space realization given by (8)-(10), conditions (11) and (12) would be LMIs with respect to the controller parameters, thus, may be used for the controller synthesis. It is worthwhile to mention that the first two rows and two columns of condition (11) is the same condition as in the KYP lemma (Landau et al. (1997)), this implies that the stable polynomial E(z) is an SPR-maker for the denominator of the transfer function H(z, θ). Choice of the central polynomial E(z) is the main source of conservatism for fixed-structure H2 controller design. However, the upper bound γ on the H2 norm may be monotonically decreased by some iterations. Suppose that in iteration i − 1, with a central polynomial Ei−1 (z), a controller Ki−1 is resulted from conditions (11) and (12), with γ = γi−1 . Now, for the next iteration, consider Ei (z) = L(z, θ)|K=Ki−1 . In what follows, we have shown that Ki−1 (z) is a feasible solution for the LMIs of Theorem 2 with E(z) = Ei (z). Based on the following state space realization    S|K=Ki−1 Ai Bi       Di  ,  ↔  Ci  L| Ei K=Ki−1 0 1 Ei 

(13)

condition (11), using the Schur complement formula (Boyd et al. (1994)), can be written as 

ATi P Ai − P ATi P Bi BiT P Ai BiT P Bi − 1



< 0,

(14) 333

(15)

Taking into account that in this case a state space realization of H(z, θ)|K=Ki−1 can be considered (Ai , Bi , Ci , Di ). Obviously, conditions (14) and (15) with respect to Lemma 1 imply that Ki−1 (z) is a feasible solution for LMIs of Theorem 2 with γ = γi−1 . This means that it may possible to reach to a smaller γ with another controller whereas an upper bound for γ is γi−1 . However, these iterations do not guarantee to find an optimum solution, but they may cause a suboptimal one.

3.2 D0 is dependent on the controller parameters In this case either Ds or Dl or both of them are dependent on the controller parameters. The following theorem may be used for fixed-structure controller design. Theorem 3. Given a schur stable polynomial E(z), consider a SISO discrete-time transfer function H(z, θ), given 2 by (3). Then kHk2 < γ if there exists P = P T > 0 such that   T A PA − P AT P B − ClT 0  < 0,  B T P A − Cl B T P B − Dl − DlT DlT T 0 Dl −(Dl + Dl )/2 (16)   T T P 0 Cs CL T  0 (Dl + DT )/2  0 −D l l   > 0 (17) T T  Cs  0 γ(Dl + Dl )/2 Ds T Cl −Dl Ds Dl + Dl    Ds ) is the state space realization , Dl T  S L . of the transfer matrix E E where (A, B,



Cs Cl

Proof. See Appendix B. Note that against the conditions of Theorem 2, the above conditions are LMIs with respect to Ds and Dl . Therefor, utilizing the state space realization given by (8)-(10), a fixed-structure controller can be designed even if D0 = Ds Dl−1 depends on the controller parameters. In the case that Dl is dependent on the controller parameters, the smallest feasible γ is obtained by a line search. Similar to the section 3.1, employ some iterations to choose the central polynomial will improve the performance of the designed controller. Note that condition (16), with the state space realization (13), is equivalent to (14). Moreover, by employing twice the Schur complement formula, it is easy to see that the LMI constraint (17) with the state space realization (13) is equivalent to (15). Therefore, in the iteration i, with the central polynomial Ei (z) = L(z, θ)|K=Ki−1 the resulted upper bound γi is equal or less than γi−1 .

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

4. NUMERICAL EXAMPLE

11.5

11

This section provides an example that demonstrates how effective the proposed approach is and that is an improvement over existing methods. Consider the following system

10.5

G(z) =

z − 0.2 z 3 − 1.2z 2 − 3.55z + 8.18

(18)

The goal is to design a fixed-structure stabilizing controller, with the minimum upper bound over the H2 norm of the following weighted closed-loop transfer function

H2 Norm

10

9.5

9

8.5

8

H(z) = W (z)

K(z) , 1 + G(z)K(z)

(19)

7.5

7 1

2

3

4

where, Wn (z) 0.9204z 2 − 1.7270z + 0.8097 W (z) = = . Wd (z) 35(z 2 − 1.9623z + 0.9626)

(20)

In this section, the optimization problems are solved by YALMIP (L¨ofberg (2004)) interface for the LMI solver SDPT3 (Toh et al. (1999)). 4.1 Full-order controller design Before dealing with the fixed-order controller design, we illustrate one of the advantages of our proposed approach. That is, the monotonic decreasing of the H2 norm bound. Using the command h2syn, the following strictly proper controller is resulted such that kH(z)k2 = 7.4538. K1 (z) =

5

6

7

8

Iteration

−0.7646z 4 + 10.95z 3 − 52.64z 2 + 74.56z − 32.11 z 5 − 1.544z 4 + 4.1z 3 − 8.172z 2 + 5.402z − 0.7836 (21)

Using Theorem 2, we design a controller with the same structure (fifth-order strictly proper) as the controller K1 (z). We consider an initial central polynomial E(z) = Wd (z − 0.5)8 . Figure 2 shows that the proposed method converges rapidly to the same norm bound kH(z)k2 = 7.4538. Now, consider the design of a fifth-order proper controller. Based on Theorem 3, the following controller is resulted. K2 (z) = 4.075z 5 − 18.62z 4 + 30.87z 3 − 24.02z 2 + 8.841z − 1.157 , z 5 − 2.764z 4 + 2.95z 3 − 1.501z 2 + 0.3449z − 0.02819 where, kH(z)k2 = 2.0146 is much less than that of the strictly proper controller K1 (z). 4.2 Low-order controller design It is worthwhile to mention that the designed controller K1 (z) has two unstable poles. Therefore, traditional order reduction methods are not able to provide a controller with order less than 2. Table 1 shows the H2 norm of the weighted closed-loop transfer function (19) computed for 334

Fig. 2. H2 norm of the weighted transfer function H(z) versus the central polynomial updates. some controllers obtained by Hankel norm approximation method (Glover (1984)). Now, consider the design of a first-order controller. Using Theorem 3, after 5 iterations of the central polynomial updates the upper bound γ = 2.2431 is obtained with the following controller: K1 (z) =

4.249z − 8.299 . z − 0.1828

(22)

In order to reveal the impact of initial central polynomial on the result, the design procedure has been carried out with different initial central polynomials. Figure 3 shows the monotonic decreasing of the norm bound and convergence to a suboptimal solution. Table 1. H2 norm of weighted closed-loop transfer function for different controllers obtained by Hankel norm approximation method Controller order H2 norm

1st unstable

2nd unstable

3rd 8.6016

4th 7.4554

5. CONCLUSION We have presented LMI-based conditions for fixed-structure H2 controller design. The conditions can be easily extended for the robust control design for systems with parametric uncertainty. Moreover, they are applicable for linear parameter varying (LPV) systems. We are also planing to formulate similar LMI design conditions for MIMO systems. A numerical example clearly demonstrated the effectiveness and the advantages of our approach. REFERENCES Apkarian, P. and Noll, D. (2006). Nonsmooth H∞ synthesis. IEEE Transactions on Automatic Control, 51(1), 71–86. Apkarian, P., Noll, D., Thevenet, J.B., and Tuan, H.D. (2003). A spectral quadratic-SDP method with applications to fixed-order H2 and H∞ synthesis. European Journal of Control, 10(6), 527–538.

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

lipsoidal parametric uncertainty. International Journal of Control, 84(1), 57–65. Scherer, C. and Weiland, S. (2000). Linear Matrix Inequalities in Control. available at: http://www.isa.ewi.tudelf.nl/∼roos/courses /W14218/lmi052.pdf. Toh, K.C., Todd, M.J., and Tutuncu, R.H. (1999). SDPT3: a MATLAB software package for semidefinite programming. Optimization Methods and Software, 11, 545–581. Yang, F., Gani, M., and Henrion, D. (2007). Fixed-order robust H∞ controller design with regional pole assignment. IEEE Transactions on Automatic Control, 52(10), 1959–1963.

3

2.9

2.8

H2 Norm

2.7

2.6

2.5

2.4

2.3

2.2 1

Appendix A. PROOF OF THEOREM 2 2

3

4

5

6

Iteration

Fig. 3. The monotonic decreasing of the H2 norm bound for (19) with first order controllers designed with different initial central polynomials Boyd, S., El Ghaoui, L., Feron, E., and Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia. Fujisaki, Y., Oishi, Y., and Tempo, R. (2008). Mixed deterministic/randomized methods for fixed order controller design. IEEE Transactions on Automatic Control, 53(9), 2033–2047. Glover, K. (1984). All optimal Hankel-norm approximations of linear multivariable systems and their L∞ error bounds. International Journal of Control, 39(6). Henrion, D., Sebek, M., and Kucera, V. (2003). Positive polynomials and robust stabilization with fixed-order controllers. IEEE Transactions on Automatic Control, 48(7), 1178–1186. Karimi, A., Khatibi, H., and Longchamp, R. (2007). Robust control of polytopic systems by convex optimization. Automatica, 43(6), 1395–1402. Khatibi, H. and Karimi, A. (2010). H∞ controller design using an alternative to Youla parameterization. IEEE Transactions on Automatic Control, 55(9), 2119– 2123. Landau, I.D., Lozano, R., and M’Saad, M. (1997). Adaptive Control. Springer-Verlag, London. L¨ ofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In CACSD Conference. URL http://control.ee.ethz.ch/ joloef/yalmip.php. Malik, W.A., Darbha, S., and Bhattacharyya, S.P. (2008). A linear programming approach to the synthesis of fixed-structure controllers. IEEE Transactions on Automatic Control, 53(6), 1341–1352. Sadeghzadeh, A. (2011). Fixed-order H∞ controller design for systems with polytopic uncertainty. In 18th IFAC World Congress. Milan, Italy. Sadeghzadeh, A. and Momeni, H. (2011). Fixed-order robust H∞ control and control-oriented uncertainty set shaping for systems with ellipsoidal parametric uncertainty. International Journal of Robust and Nonlinear Control, 21, 648–665. Sadeghzadeh, A., Momeni, H., and Karimi, A. (2011). Fixed-order H∞ controller design for systems with el335

Consider (Ac , Bc , Cc , Dc ) to be the controllable canonical S(z,θ) , form realization of the transfer function H(z, θ) = L(z,θ) given by 

  Ac =   

0 0 .. .

1 0 .. .

0 1 .. .

··· ··· .. .

0 0 .. .

  0  0     Bc =  ..   .  0 

0 0 0 ··· 1 1 −l0 ln−1 −l1 ln−1 −l2 ln−1 · · · −ln−1 ln−1 −1 sn−1 − Dc ln−1 ] , C c = l n [ s 0 − D c l0 s 1 − D c l 1 · · · −1 D c = s n ln . (A.1) Additionally, consider the similarity transformation with matrix T which converts the state space model     Ds Cs ) , (A, B, Dl Cl to a controllable canonical form realization 

  −1 ¯ A = T AT =   

0 0 .. .

1 0 .. .

0 1 .. .

··· ··· .. .

0 0 .. .



0 0 0 ··· 1 −1 −1 −1 −e0 e−1 −e e −e e · · · −e 1 n 2 n n−1 en n

¯ = T B = [ 0 0 · · · 0 1 ]T , B     Cs C¯s T −1 = Cl C¯l  s 0 − D s e0 s 1 − D s e1 · · · = e−1 n l0 − D l e 0 l1 − D l e 1 · · ·       sn e−1 Ds D¯s n ¯l = Dl = ln e−1 . D n

sn−1 − Ds en−1 ln−1 − Dl en−1

    



It is easy to see that Cc = Dl−1 (Cs − Dc Cl )T −1 , Dc =

Ds Dl−1 .

The inequality constraint (11) can be written as

(A.2) (A.3)

7th IFAC Symposium on Robust Control Design Aalborg, Denmark, June 20-22, 2012

 −P −ClT 0   −Cl −Dl − DlT DlT T 0 Dl −(Dl + Dl )/2  T A +  B T  P ( A B 0 ) < 0. 0 

(A.4)

We pre- and post-multiply (A.4) by matrix 



−1 e0 e−1 n − l0 ln −1 e1 en − l1 ln−1 .. .

   −T   T  0 0 I 0 0 I 0     0 0 I   en−1 e−1 − ln−1 l−1 n n 0 1 1

(A.5)

and its transpose, respectively. We obtain  where,

ATc QAc − Q ATc QBc BcT QAc BcT QBc − 1



< 0,

Q = 2(Dl + DlT )−1 T −T P T −1 .

(A.6) (A.7)

Since the matrix given by (A.5) is a full row rank matrix, LMI (A.6) holds if the LMI (11) holds. Now, we pre- and post-multiply condition (12) by matrix 

Dl + 2

DlT

− 12



T −T 0  0 I 0 0

 0 0 I

(A.8)

and its transpose, respectively. Taking into account (A.2) and (A.3), we obtain the following condition which is equivalent to (12).  Q 0 CcT  0 1 DT  > 0. c Cc Dc γ 

(A.9)

Now, using Lemma 1 and based on conditions (A.9) and (A.6), we conclude that LMIs (11) and (12) are sufficient conditions for satisfaction of the H2 performance. Appendix B. PROOF OF THEOREM 3 The proof follows the same pattern as that of Theorem 2. Thus, we shall give an outline of the proof only. As we mentioned in the proof of Theorem 2, LMI (A.6) holds if LMI (16) holds. Note that Q is given by (A.7). We pre- and post-multiply condition (17) by the full row rank matrix 

Dl + 2

DlT

− 21



T −T 0  0 1 0 0

 0 0  0 0 1 −Ds Dl−1

(B.1)

and its transpose, respectively. We obtain LMI (A.9) with the same Q as in (A.6). Therefor, similar to the proof of Theorem 2, this ends the proof. 336