COMPARISON OF DIFFERENT POLYNOMIAL FACTORIZATION APPROACHES AS AN ALTERNATIVE TO FIR APPROXIMATION TO SOLVE THE `1 DESIGN PROBLEM P.-O. Malaterre∗ , M. Khammash† ∗
Cemagref, 361 rue J.-F. Breton, BP 5095, 34033 Montpellier Cedex 1, France e-mail:
[email protected] † Iowa State University, Department of Electrical and Computer Engineering, Ames, Iowa 50011-3060, USA e-mail:
[email protected] Keywords: polynomial factorization, FIR approximation, Youla parametrization, `1 control, numerical tools
ming approach [1]. This can be stated as solving: γ opt =
Abstract The aim of this paper is to present an application of polynomial factorization as an alternative to FIR approximation in the design of a `1 controller. The polynomial matrix fractions have been calculated for the H, U and/or V matrix transfer functions coming from the Youla parametrization of the plant to be controlled. Different approaches are possible depending on which one of the H, U and/or V matrix transfer functions are factorized. The objective of these approaches is to reduce the number of constraints, variables and/or non-zero coefficients in the Linear Programming problem to be solved for the `1 control design, and if possible, to accelerate the convergence of the upper and lower bounds of the `1 norm.
1 Introduction z y
w
P
u
inf
K stabilizing
kFl (P, K)k1
(1)
where P represents the LTI discrete-time generalized plant (Figure 1), K the LTI discrete-time controller, Fl (P, K) = Φ the lower linear fractional transformation of P by K. We assume the dimensions of w, z, u, and y are nw , nz , nu , and ny respectively. The number of states x of the plant P is denoted nx . It can be shown [1], that this problem can be formulated as that of finding: γ opt =
kH − U ∗ Q ∗ V k1 inf n ×n
Q∈`1 u
(2)
y
where ∗ denotes convolution, H ∈ `n1 z ×nw , U ∈ `1nz ×nu , and n ×n V ∈ `1 y w are fixed and depend on the problem data: P , nw , nz , nu , and ny . Remark: In order to simplify the notation, the convolution mark ∗ will be omitted in the following sections.
2 Scaled-Q method In [2] it is proved that upper and lower bounds for γ opt can be obtained by solving the two following finite linear programs:
K Figure 1: Standard framework In many control design problems, the objective is to find a stabilizing Linear Time-Invariant (LTI) discrete-time controller K which, in addition to stabilizing the closed loop system, minimizes some norm(s) of the transfer matrix Φ : w → z (Figure 1). If the considered norm is the H2 or H∞ norm of the transfer matrix Φ in the frequency domain, then a state-space solution exists for the controller. But in the case of the `1 norm of the impulse response, no state-space solution exists for the moment, and the problem must be solved by a Linear Program-
An upper bound for γ opt : ν N (β) = minQ∈`nu ×ny kH − Rk1 1 kQk1 ≤ β subject to R = U PN (Q)V
(3)
A lower bound for γ opt : ν N (β) = minQ∈`nu ×ny kH − Rk1 1 kQk1 ≤ β subject to PN R = PN (U QV )
(4)
where PN is the truncation operator and β sufficiently large. n ×n
In [2] it is proved that when an optimal solution Qopt ∈ `1 u y for the `1 problem (Equation (2)) exists (in particular it is the
time
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Figure 2: FIR approximation for H, U and V with ε = 10−3
The numerical approach used to solve the previous problems can lead to a large number of linear constraints in the LP problem if the FIR approximation of H (resp. U and V ) is very long. The length of the FIR approximation is adjusted so that the original system and the FIR approximation have a `1 norm difference less than ε (e.g.: ε = 10−8 ). This length depends on the magnitude of the larger pole of H (resp. U and V ). The closer to zero they can be placed, the shorter the FIR approximation is. Theoretically, for a stable and minimum phase system, they can be placed as close to zero as desired, which corresponds to a short FIR approximation. But in reality, for numerical reasons, the best maximum magnitude of the poles of H (resp. U and V ) can still be large (e.g.: 0.84, discrete time, z-transform).
As expected the number of variables, constraints and nonzero coefficients involved are larger for ε = 10−8 than for ε = 10−3 , since the length of the FIR approximations for H, U and V is larger.
4 Size of the minimization problem The original problem (Equation (2)) can be written in a more general form as:
(6)
Example from Paper [2]
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The number of variables (size(x)), constraints (size(b1 ) + size(b2 )) and non-zero coefficients (in A1 and A2 matrices) in Equation (5) are increasing functions of the length of Q (lenq = N ), as detailled in section 4. Figure 2 (resp. 3) displays these numbers for the upper and lower bounds for the example described in [2], for ε = 10−3 (resp. ε = 10−8 ). The value of the `1 norm for the upper bound and for the lower bound for different β is displayed, along with the computational time on a PC based on a Pentium III, 667 Mhz.
FIR (1e−008) option 150
# const dn
3 Incentive for polynomial factorization
γ opt = inf Q∈`nu ×ny kΦk1 1 subject to the linear constraints: Wl ΦWr = Wh − Wu QWv
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(5)
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# nzero dn
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A1 x ≤ b1 A2 x = b2
# nzero up
min c0 x , subject to
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It is then possible to translate all equations and constraints of the above problems (Equation(3) and (4)) into classical finite dimension linear programming problems of the usual form:
Example from Paper [2] 0.2
400 # var up
In order to transform this problem into a Finite Dimensional Linear Programming problem (FD LP) one solution is to compute Finite Impulse Response (FIR) approximations of H, U and V , and to look for the optimal Q with a finite support of length N .
FIR (0.001) option 150
# const up
ˆ and Vˆ have no zeros on the unit case when the λ-Transforms U opt circle), then ν N (β) % γ , ν N (β) & γ opt and QN → Qopt as N → ∞.
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Figure 3: FIR approximation of H, U and V with ε = 10−8
where Wl , Wr , Wh , Wu and Wv are FIR approximations or polynomials (matrices). Their lengths are noted lenWl , lenWr , lenh, lenu and lenv, respectively. These W(∗) objects will be computed in different cases in sections 5 to 8. The number of variables, constraints and non-zero coefficients of the LP problems defined by Equation (5) can be expressed as functions of these variables (lenWl , lenWr , lenh, lenu and lenv), and the number of inputs and outputs of the generalized plant P (nw , nu , nz and ny ). In the following subsections these functions are detailed for the upper bound (more memory consuming than the lower bound).
The number of variables x of the LP problem (Equation (2)) for the upper bound (Equation (3)), noted nvars can be expressed as: nvars = av lenq + bv with: av = 2(nz nw + nu ny ) bv = 1 − 3nz nw + 2nz nw (lenu + lenv)
In the following sections different alternatives to FIR approximations are described and compared on the same example as the one described in [2]. The av , bv , ac , bc , az and bz coefficients are summarized in Table 1.
5
The only way to reduce this number is to reduce lenu + lenv. This can be done either by reducing the precision ε of the FIR approximations (cf. Figures 2 and 3 for the options we call hereafter FIR3 and FIR8), or by using a polynomial factorization, if the length of the polynomials is shorter than the FIR approximations (cf. sections 5 to 8 for the options we call hereafter POLUV, POLHUV, POLPI and POLPHI). 4.2 Number of constraints The number of constraints (given by b1 and b2 ) of the LP problem for the upper bound (Equation (3)), noted nconst can be expressed as: nconst = ac lenq + bc with: ac = nz nw bc = nz + nu + nz nw (lenu + lenv + lenWl + lenWr − 3) As for the number of variables, the rate of increase ac of the number of constraints nconst as a function of lenq cannot be changed by polynomial factorizations. But, the initial number of constraints bc can be reduced if (lenu + lenv + lenWl + lenWr ) if globally reduced.
Option POLUV
A first alternative can be to compute a right (resp. left) poly−1 nomial factorization of U (resp. V ): U = Nur Dur (resp. −1 V = Dvl Nvl ). In this method, H is still replaced by its FIR approximation. The original problem is then transformed into:
γ opt
= =
inf
n ×ny
Q∈`1 u
−1 −1 Nvl k1 kH − Nur Dur QDvl
e vl k inf×n kH − Nur QN 1 n
u e Q∈` 1
e = D−1 QD−1 with Q ur vl b vl b ur and D Proof : U and V are stable ⇒ the λ-Transform D have no unstable zeros −1 b ur b −1 ∈ `1 ⇒D and D vl (Weiner Theorem) e ∈ `1 , which means that searching and therefore Q ∈ `1 ⇔ Q e over Q is equivalent to searching over Q. POLUV option
Example from Paper [2]
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if Wl = Wr = 1 x=1 x = nz nw (lenWl + lenWr − 1) otherwise
(7)
If the introduction of polynomials Wl and Wr , through polynomial factorization, allows to reduce sufficiently (lenu + lenv), then the number of nonzero coefficients can be reduced. This reduction also depends on the relative importance of the variables nz nw and nu ny .
# var dn
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# nzero dn
nnzero = az lenq + bz with: az = 2nu ny (1 + nz nw (lenu + lenv − 1)) + 2nz nw (1 + x) bz = 2nz nw + nz + 2nz nw (1 + x)(lenu + lenv − 2)
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# nzero up
For memory reasons, this is probably the most important variable to check. The number of nonzero coefficients (in A1 and A2 ) of the LP problem for the upper bound (Equation (3)), noted nnzero can be expressed as:
# var up
300
4.3 Number of nonzero coefficients
(8)
y
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Figure 4: polynomial factorization (POLUV) of U and V (and FIR for H) This alternative is interesting and potentially better than the FIR approximation method as soon as the length of the polynomials Nur and Nvl is shorter than the length of the FIR approximations of U and V (lenu+lenv is reduced without increasing lenWl + lenWr , since we still have Wl = Wr = 1). In our
POLPI option
case this option proved to be the best (cf. Table 1 and Figure 4).
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In fact this is true if and only if the rate of the convergence of the upper and lower bounds as a function of lenq is not modified. This is the case in this example and with other examples we tested, but this is not true in general, since the optimizae instead of Q. This issue must be further tion is made on Q investigated and in particular the correlation between the caracteristics of the W(∗) objects and the convergence rate of the upper and lower bounds as a function of lenq.
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We can go one step further by computing a right polynomial −1 factorization of H: H = Nhr .Dhr . Then:
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6 Option POLHUV
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Figure 5: polynomial factorization (POLPI) of U and V , FIR of H and linear weights on Φ
Fl (P, K) = Φ= H − U QV −1 e vl = Nhr .Dhr − Nur QN e vl Dhr ⇔ ΦDhr = Nhr − Nur QN
8 Option POLPHI
The original problem is then transformed into: γ opt = inf Q∈` e nu ×ny kΦk1 1 subject to the linear constraints: e vl Dhr ΦWr = Nhr − Nur QN
Example from Paper [2]
150
(9)
with Wr = Dhr
The advantage of this option is to be based only on polynomial factorization, and does not depend on any FIR approximation. Therefore, the size of the optimization problem to be solved is only linked to the order of the system, and not to the pole location of H, U and V . For small order system, for which the pole placement is difficult, this is probably an interesting option. But, in our case, where we could place the poles close to 0.2, this was not the best option.
7 Option POLPI Another approach can be to compute a left (resp. right) poly−1 nomial factorization of U (resp. V ): U = Dul .Nul (resp. −1 V = Nvr .Dvr ). In this method the FIR approximation for H is kept. Then: Fl (P, K) = Φ= H − U QV −1 −1 = H − Dul .Nul QNvr .Dvr ⇔ Dul ΦDvr = Dul HDvr − Nul QNvr But, in our example, the reduction of (lenu + lenv) does not compensate for the increase of (lenWl + lenWr ) and globally this option implies less variables, but more constraints and more nonzero coefficients than the previous POLUV options. With this option, the optimization is still made on Q and therefore the convergence rate is not modified, compared to the FIR approximation option.
We can go one step further by computing a right polynomial −1 factorization of H: H = Nhr .Dhr . Then: Fl (P, K) = Φ = H − U QV −1 −1 −1 − Dul .Nul QNvr .Dvr = Nhr .Dhr −1 ⇔ Dul ΦDvr = Dul Nhr Dhr Dvr − Nul QNvr −1 = Dul Nhr (Dhr Dvr ) − Nul QNvr = Dul Nhr A − Nul QNvr −1 where A = Dhr Dvr is computed using for example the TC04AD Slicot subroutine [3]. Then a right polynomial fac−1 torization of A is computed: A = Nar Dar −1 ⇔ Dul ΦDvr = Dul Nhr Nar Dar − Nul QNvr ⇔ Dul ΦDvr Dar = Dul Nhr Nar − Nul QNvr Dar
The original problem is then transformed into: γ opt = inf Q∈`nu ×ny kΦk1 1 subject to the linear constraints: Wl ΦWr = Dul Nhr Nar − Nul QNvr Dar
(10)
with Wl = Dul and Wr = Dvr Dar The advantage of this option is to be based only on polynomial factorizations, as for the POLHUV option. In addition, the optimization is still made on Q and therefore the convergence rate is not modified, compared to the FIR approximation option. But for high order systems, the polynomials Dul and Dvr Dar may add too many linear constraints on Φ and nonzero coefficients. In this case the previous option POLPI is probably better.
FIR3 FIR8 POLUV POLHUV POLPI POLPHI
av 10 10 10 10 10 10
bv 141 245 61 101 37 117
ac 4 4 4 4 4 4
bc 75 127 35 75 39 119
az 162 266 82 306 210 610
bz 282 490 122 2410 682 6842
Table 1: Parameters for the calculation of the number of variables, constraints and nonzero coefficients
9
Conclusion
Different alternatives to the classical FIR approximations of the Youla parameters H, U and V have been presented and compared. These alternatives are based on the polynomial factorization of some or all of these Youla parameters. e the method minimizing the For a given length of Q (or Q), number of constraints and nonzero coefficients is POLUV. The second best option is POLPI, but the gain is not very significant compared to the classical FIR option.
The number of variables, constraints and nonzero coefficients involved in the LP problems to solve can be easily computed before running the optimizations, and therefore the best option can be selected a priori. Although this was not the case in our example, the convergence rate of the Scaled-Q method can also be affected by the choice of the option. The link between this convergence rate and the different options must be further investigated in order to draw definite conclusions on the best approach.
Acknowledgment The material presented in this note was studied during my stay at ISU as a Visiting Scientist. I would like to express deep gratitude to Mustafa Khammash who kindly welcomed me in his research group. I would also like to thank Cemagref, Montpellier, France for its financial support and my colleagues there that accepted to do part of my share of work during this period. We also acknowledge support by NSF grant ECS 9457485.
References [1] M. A. Dahleh and I. J. Diaz-Bobillo, Control of uncertain systems: a linear programming approach. Prentice-Hall, 1995. [2] M. Khammash, “A new approach to the solution of the `1 control problem: the Scaled-Q method,” IEEE Transaction on Automatic Control, vol. 45, pp. 180–187, February 2000. [3] E. Barth, T. Beelen, P. Benner, C. Benson, R. Byers, R. Dekeyser, F. Delebecque, M. Denham, F. Dumortier,
A. Emami-Naeini, D.-W. Gu, A. Geurts, S. Hammarling, G. Van Den Hurk, B. Kgstrm, C. Kliman, M. Konstantinov, D. Kressner, A. Laub, C. Oara, C. Paige, T. Penzl, P. Petkov, V. Sima, S. Steer, F. Svaricek, M. Vanbegin, P. Van Dooren, S. Van Huffel, A. Varga, M. Verhaegen, L. Westin, H. Willemsen, T. Williams, and X. Hongguo, “The control and systems library SLICOT.” http://www.win.tue.nl/niconet/NIC2/slicot.html, 2000.
