Fast Computation of Stabilizing Predictive Control Laws

Copyright © IFAC Algorithms and Architectures for Real-Time Control, Ostend, Belgium, 1995

FAST COMPUTATION OF STABILIZING PREDICTIVE CONTROL LAWS L. Chisci, A. Garulli and G. Zappa Dipartimento di Sistemi e Informatica, Universitti di Firenze Via di Santa Marta 3, 50139 Firenze, Italy tel. +55.4796.263, fax. +55.4796.363, e-mail: [email protected]

Abstract. A fast algorithm for Linear Quadratic (LQ) control with linear equality constraints is derived and exploited for stabilizing predictive control synthesis. The algorithm requires only O(Nn) computations for an nth order plant and N-steps prediction horizon, and possesses a remarkable numerical accuracy. Keywords: Predictive control, linear quadratic regulators, control algorithms, fast parallel algorithms, fast Kalman algorithms, computational methods, adaptive control.

1. Introduction Optimal Linear Quadratic (LQ) methods are widely used for control design in adaptive control (Bitmead et al., 1990). While offering high performance even for open-loop unstable and nonminimum phase pl!lnts, they also tend to be computationally intensive. To enable their use in applications requiring high sampling rates, it is therefore necessary to develop algorithms that are: 1) fast and numerically efficient; and 2) can be readily implemented on paraH~1 processors. These two objectives have been pursued by the authors in a number of papers (Chisci and Zappa, 1991a, 1991b, 1993; Chisci et al., 1993, 1994a, 1994b) . In (Chisci and Zappa, 1991a, 1991b, 1993) it has been shown that, using certain square-root algorithms (Peterka, 1986) of O( N n 2 ) computational complexity, LQ control synthesis can be successfully implemented on systolic arrays of O(n 2 ) processors in O(N) processing time for an nth order plant and N -steps prediction horizon. The algorithms and architectures in (Chisci and Zappa, 1991a, 1991b, 1993) can cope with possibly timevarying models and/or penalties and can, therefore, be employed for both "receding-horizon" (Bitmead et al., 1990) and "iterations-spreadin-time" (Peterka, 1984) adaptive LQ schemes. Further, in (Chisci and Zappa, 1991b, 1993) it has been shown that linear-equality terminal constraints like those employed in GPC (Clarke et 487

al., 1987) and in the stabilizing predictive controller SIORHC/CRHPC/SGPC (Mosca et al., 1990; Clarke and Scattolini, 1991; Kouvaritakis et al., 1992) can be accomodated as well in the algorithms and their systolic implementations. More recent contributions (Chisci et al., 1993, 1994a, 1994b) have pointed out the remarkable fact that for quasi-stationary LQ control problems (i.e. problems with a few time variations in the model and/or penalties) it is possible to derive ''fast'' algorithms ofreduced O(Nn) complexity which can be implemented on arrays of O(n) processors in the same O(N) processing time. In particular, (Chisci et al., 1993) has considered the time-invariant LQ problem (no time variations in the model and penalties) which is the simplest case of LQ problem for which a fast algorithm exists, and the time-invariant LQ problem with terminal state weighting (one variation in the penalties). Subsequent references (Chisci et al., 1994a, 1994b) considered GPC, which consists of an LQ problem with constrained terminal inputs.

This paper will present more general fast algorithms capable of handling the more complicated SIORHC/CRHPC/SGPC controller which consists of an LQ controller with terminal constraints on both inputs and outputs. In SIORHC/CRHPC/SGPC (Mosca et al., 1990; Clarke e Scattolini, 1991; Kouvaritakis et al., 1992), output terminal constraints have in fact been added to GPC in order to guarantee stability

despite the finite prediction horizon. The fact that previously no fast algorithm derivation seemed possible for SIORHC/CRHPC/SGPC predictive controllers has puzzled the authors for a long time. Here it is shown that SIORHC/CRHPC/SGPC can be reduced to a slightly modified version of GPC wherein the constraint that a suitable number of terminal inputs be zero is replaced by the constraint that n terminal inputs (where n is the plant order) are given by a deadbeat dynamic output feedback control law. Hence a fast algorithm for SIORHC/CRHPC/SGPC can easily be obtained using a similar derivation as in (Chisci and Zappa, 1991a, 1991b, 1993) and via solution of a suitable Bezout equation .

2. Problem formulation Hereafter Sk, k = ... , -1, 0,1, ... , will denote a scalar time sequence; d the unit delay operator, viz. dS k ~ Sk-l; and P(d) a polynomial in d of degree ap. Consider the following I-DOF (l-degree-offreedom) dynamic control design problem. Given the discrete-time SISO plant

A( d)Yk = B( d)Uk-l

(1)

with inputs Uk, outputs Yk, A(O) performance index

1, and the

N

A 'L...JYk " ' 2 + PU2k_l J = k=1

(2)

with N ~ n ~ max{aA, aB}, P ~ 0, find the input sequence {uo, Ul, .. . ,uN-d to the plant (1) which minimizes (2), subject to the constraints

Uk

=0

Yk

and

=0

for

k

~

N.

(3)

The problem (1)-(3) can be rewritten more compactly as the following dynamic constrained optimization problem

.

{'\"N L.Jk=1 Yk + PU k_ 1 2

2

A(d)Yk = B(d)Uk_l! 1 ~ k ~ N, Uk = Yk = 0, k ~ N}

In predictive control, it is customary to use the control law (5) with k = 0 in order to define a constant dynamic output feedback

Ro(d)Ut = -So(d)Yt

(6)

according to the so called "receding-horizon control" strategy. The receding-horizon controller defined via (4) has been first considered in (Mosca et al., 1990) and therein named SIORHC (acronym for Stabilizing Input-Output Receding-Horizon Control). It was also independently introduced by Clarke and Scattolini (1991) with the name of CRHPC (Constrained Receding-Horizon Predictive Control) . More recently, a seemingly different but related predictive controller named SGPC (Stable Generalized Predictive Control) has been developed by Kouvaritakis et al. (1992) . They also proved in a later reference (Rossiter and Kouvaritakis, 1994) the equivalence between SIORHC/CRHPC and SGPC in the SISO case. For ease ofreference, the receding-horizon controller defined via (4) will be referred to in the sequel simply as SIORHC. Under the stated assumption that the prediction horizon N be greater than or equal to the plant order n , SIORHC guarantees a stable closed-loop system and thus overcomes the well-known stability limitations of finite horizon LQ control and GPC . This property has made attractive the use of SIORHC in combination with suitable recursive parameter estimators - in adaptive control schemes. From a computational point of view, the SIORHC law can be obtained either by a static programming approach as in (Mosca et al., 1990; Clarke and Scattolini, 1991; Kouvaritakis et al., 1992), or by a dynamic programming approach as in (Chisci et al., 1991). The former approach requires at best O(N 2 n) computations, if suitable fast algorithms for inversion of close-to-Toeplitz matrices (Chun et al., 1987) are being used. Conversely the dynamic-programming-based algorithm of Chisci et al. (1991) has O(Nn2) complexity. It is worth pointing out that, being N larger (usually much larger) than n, the dynamic programming approach is compu tationally cheaper. Further, as computer examples clearly demonstrate, the dynamic programming approach is to be preferred also from a numerical point of view, expecially when unstable plant models are dealt with.

To ensure existence of sequences uo, ... , UN -1 that satisfy the constraints (1) and (3), it will be assumed throughout the paper that A(d) and B(d) are coprime polynomials.

uO'.~~~_l

output feedback

(4)

where the symbol" I" stands for "subject to the constraints". As it is well known, the solution of (4) can be found via dynamic programming arguments in the form of a time-varying dynamic

In the following section, a new algorithm for SIORHC based on dynamic programming and having O(Nn) complexity will be presented. Towards this goal, the following key fact will be used . 488

Lemma 1 - Any input sequence {Uo, .. . , UN - d which guarantees (3) for the plant (1) , must satisfy the constraint

Moreover, GPC defined by GPC:

X(d)Uk+Y(d)Yk=O, Ie=N-n, ... , N-l (7) where X(d) and Y(d) satisfy the Bezout equation

A(d)X(d) + dB(d)Y(d) = 1 Proof - Omitted due to lack of space.

A(d)Yk = B(d)Uk_l' 1 $ le $ N , Uk=O, Ie~Nu}

(8)

•

Remark - The above lemma states that the last n elements UN-l, " " UN-n of the optimal solution of (4) are provided by a constant deadbeat dynamic~output feedback . Consequently, the constraints, {3) can be replaced by (7) and the number of degrees offreedom in (4) is reduced from N to N - n . In other words, the SIORHC problem can be equivalently stated as

M = N - n, X(d) and Y(d) satisfying (8) P(d), Q(d) , K(d), L(d) , >. as in (Ub) .

The dynamic weight polynomials P, K , L, Q which can be used for frequency weighting - as well as the exponential weight)" - which can be used to impose a prescribed degree of stability provide further design knobs for tuning the control performance.

1 $ le $ N , A(d)Yk = B(d)Uk_l ' X(d)Uk + Y(d)Yk = 0, N - n $ le $ N -I} (9)

•

{Ef=l ),,2(N-k) {[P(d)Yk + K(d)Uk_d 2 + [L(d)Yk-l

3. Fast GLQ algorithm In this section, a fast algorithm for the design of the GLQ controller defined by (10) is derived. For subsequent convenience, let us first rewrite the cost in (10) as

Lemma 1 and the consequent remark motivate the introduction of the following Generalized Linear Quadratic (GLQ) control problem

UO, ···, UM_l

M = Nu, X (d) = 1, Y (d) = 0 P(d), Q(d), K(d), L(d), )" as in (11b) . Finally, SIORHC defined by (9) corresponds to

SIORHC:

GLQ: mm

corresponds to the settings

J(N) =

Ef=l ),,2(N-k) {[PNYk + KNUk_d2 + [LNYk-l

+ Q(d)Uk_l]2} I

+ QN Uk_d 2}

+

+ E~l ),,2(N-k) {[LtYk-l + Qtuk_d 2 - [LNYk-l + QNUk_tl2}

A(d)Yk = B(d)Uk_l ' 1 $ le $ N, X(d)Uk = -Y(d)yA:. M $ le $ N - I}

(12)

(10) In (10): P , K, L, Q, X , Y are generic polynomials in d with the only requirement that P(O) ¥ 0 and X(O) ¥ 0; M $ N is the control horizon; and )" E (0,1].

where

and, for the sake of brevity, the polynomial argument d has been , and will be in the sequel , omitted. Following the same lines as in (Chisci et al., 1994a), the cost (12) will be recursively manipulated making use of the following elementary polynomial transformations.

Remark - Notice that GLQ encompasses standard LQ control, GPC and SIORHC. In fact , standard LQ control defined by LQ:

Linear Polynomial Rotation - Given polynomials U, V, W , Z, with V(O) ¥ 0, (U ; W) = LIN (U, V; W, Z) will denote the pair of polynomials such that is a special case of GLQ corresponding to

dU = U M=N, P(d) = 1, Q(d) =.fP, K(d) = L(d) = 0, )" = 1. (11)

et

V

W=W-etZ

with 489

et

= U(O)jV(O)

•

Circular Polynomial Rotation - Given polynomials U, V, W, Z, with U2(0) + V2(0) #= 0, (U, V; W,Z) CIR (U, V; W,Z) will denote the quadruple of polynomials such that

=

and using Lemma 2, it is possible - after straightforward but tedious calculations - to express the cost (12) as J(N)

= J(N +

with

c

= V(O)j JU2(0) + V2(0);

S

= U(O)j JU2(0) + V2(0)

Hyperbolic Polynomial Rotation - Given polynomials U, V, W and Z, with V2(O) > U2(O), (U, V; W,Z) HYP (U, V; W,Z) will denote the quadruple of polynomials such that

=

Z

+ [LN-lY-l + QN_lU_l]2}

where: J(N - 1) is the same as J(N) with N replaced by N -1, and (d-1QN_l;L N _ l ) LIN (Qt-l' X; Lt_I' Y). Proceeding in the same way backwards over the prediction horizon, for k N - 1, ... , M, one gets

= [-sh ch -Sh] [U W] ch V Z

J(N)

ch

= V(O)j JV2(0) -

U2(O);

sh

= U(O)j JV2(0) -

U2(0).

N 1 ~ 2• · 2~[PiYO + KiU-d 2 = J(M) + ~=M

+ [LiY-l + Qiu-d

.

(14) Notice that the last term in (14) does not depend on the input variables Uo, Ull .•. , UM-I which must be, therefore, optimally chosen so as to minimize J(M)

with

2

=

=

~]

KN_lU_d

=

(VjZl HC2!R (U, VjW,Z) will be written when • only V and Z are required.

[dU V

1) + .\2(N-l) {[PN-lYO

= L:~l ~2(M-J:) {[PMYJ: + KMUJ:_d

2

+ [LMYk-l + QM Uk_l]2 + [L!tYk-l + Q!tuk-d - [LMYJ:-I + QMuJ:-I] 2}

•

Use will also be made of the following lemma.

Lemma 2 - Given arbitrary polynomials Lt, Qt (i 1,2), X and Y, with X(O)::fi 0, the following equality holds

=

L:~=l (LtYJ: + QtuJ:)2 - (L,YJ: + Q,UJ:_l)2 = (L+YJ: + Q+uJ:)2 - (L-YJ: + Q-uJ:_d 2

=

(15) Next, applying the following sequence of polynomial transformations

= LIN (PM,A;KM,-B) (KM-l,QM-li PM-I, LM-l) = (PM-1iKM-l)

=CIR (KM-I, QMi PM-I,LM)

~

-

r+

-

(QM-l' QM-li L M _ l , LM-l)

=

+ + = CIR (QM,QM-l; LM,LM-d

where (Q, j L;) (Q+jL+) (Q-;L-)

=

LIN (Qt, X; Lt, Y)

i

= 1,2

= HCIR (Qt,Lt;Qt,Lt) = LIN(Q+,XjL+,Y)

Proof - This is nothing but a generalization to the case of a non zero polynomial Y of a result reported in (Chisci et al., 1994a). The proof follows in a similar way. • Applying the following sequence of elementary polynomial transformations

= LIN (PN,A;KN,-B) (KN-l, QN-lj PN-l, LN-d = (PN-l; KN-d

=CIR (KN-l,QN; P N(~-IQt_Ij ~-l Lt_I) =

lt

= LIN (Lr+M - l , A; QM-l' -B) (LM-I;QM-l) = LIN (L~_l' A;Q~_l' -B) ~

(L1i-I;Q1i-l)

+ + = HCIR (QN-I' QNj LN-I, L N )

= LIN (QN-l, X; LN-l, Y)

(LN-l;QN-l)

= LIN (LN-l,A;QN-l,-B)

J(M) = J(M - 1) + ~2(M-l) {[PM-IYO

+ KM_1 U _Il 2 2

+ [L1i_lY-l + Q!t_lu_d 2 - [L M_1Y-I + QM_I U _d } + (LM-lYM-l + QM_lUM_t}2 where: the first term J(M -1) is the same as J(M) with M replaced by M -1 and, therefore, does not depend on UM-l; the second term does not affect minimization, being independent of UQ, ... , UM-I; and, finally, the last term is the only one depending on UM -1. Hence, the best choice for UM -1 is given by (5) for k + M-I and

LN)

(QN-l; iN-I)

(16)

one gets

(13) 490

RM-l

= QM-l,

SM-l

=LM-l.

2

Proceeding in the same way for k = M-I, . .. , 0 one gets all the optimal inputs UM-I, ... , Uo in (5) by setting RIt: = QIt: and Sit: = Lit:. Summing all the residual terms, one also gets the (constrained) minimum cost as

• Extension to 2-DOF problems. Although for simplicity of exposition only the I-DOF control problem has been considered in the paper, the proposed fast algorithms can be extended to the design of 2-DOF controllers in both cases where a dynamic model and, respectively, future values of the reference signal are available.

4. Numerical issues The resulting FGLQ (Fast GLQ) algorithm is reported in Table 1. Remarks • FSIORHC (Fast SIORHC) algorithm. The SIORHC controller can be designed using the following two-step FSIORHC algorithm: Step 1. Find the solution (X, Y) of (8) having minimum degree with respect to Y. Step 2. Use the FGLQ algorithm with X and Y as computed in Step 1, and M = N - n. • Computational complexity. Similarly to the FLQ (Fast LQ) and FGPC (Fast GPC) algorithms proposed in (Chisci et al., 1994a), the FGLQ algorithm consists of a sequence of O(N) recursions each consisting of a suitable sequence of linear transformations on polynomials of O( n) degree. The overall computational complexity is therefore O(Nn). More precisely, the FGLQ recursions involve eight polynomials: two of them, namely Lit: and QIt: , give the desired control law, while the · name"'IPK ot her SIX, Y It: , It: , Lit:+ , QIt:+ , L; and Q;, parametrize the minimum cost according to (17). Hence, in the input-output formulation of this paper, the polynomials Lit:, QIt: and, respectively, Pit:, Kit:, Lt, Qt, L;, Q; .play a role analogous to that of the feedback gain and, respectively, Riccati matrix in the state-space formulation. Table 2 reports a precise operation count of the FLQ, FGPC, FGLQ and FSIORHC algorithms in terms of elementary polynomial (linear, circular and hyperbolic) rotations. Note that the FSIORHC requires an extra O(n 2 ) computational cost, not reported in the table, for the solution of the Bezout equation (8). • Dynamic weighting . It is worth pointing out that the proposed fast algorithms handle in a straightforward way possible dynamic weights P(d) ::/; 1, K(d) ::/; 0, L(d) ::/; 0 and Q(d) ::/; .JP without any extra computational burden. In fact, the polynomials P, K, Land Q are only used in the initialization (cf. Table 1). Dynamic constraining via X and Y polynomials is also possible. As a matter of fact, SIORHC amounts to using specific dynamic constraints with X and Y given by (8). 491

A study on the numerical properties of various stabilizing predictive control algorithms has been carried out by Rossiter and Kouvaritakis (1994) . As a matter of fact, they only considered algorithms based on static programming. These have the peculiarity to require inversion of an appropriate matrix, which - expecially for unstable plants and long prediction horizons - can be highly ill-conditioned, thus yielding possible numerical problems. The analysis of Rossiter and Kouvaritakis (1994) points out that, despite their theoretical equivalence, static-programming implementations of SIORHC/CRHPC and SGPC have quite different numerical properties, more precisely SGPC is numerically superior to SIORHC. This difference is due to the fact that SIORHC and SGPC require inversion of differently conditioned matrices. However, it must be pointed out that algorithms based on dynamic programming like FSIORHC and the O(Nn2) algorithm in (Chisci et al., 1991) do not explicitly involve matrix inversion and, hence, do not suffer from such numerical problems, To demonstrate this, the following numerical example (Rossiter and Kouvaritakis ,1994) has been considered. Numerical example - Let the polynomials A(d) and B(d) in (1) be given by A(d) B(d)

= 1 + 4.4911d - 2.6401d2 - 7.1034d3 =

-2.4506d4 0.318d - 0.6953~ - 1.0474d3

-

0.0043d4

where A( d) has an unstable root at z = d- 1 = -4 .75, and let the control weight in (2) be p = 1. Three different SIORHC algorithms, viz. SIORHC 1 based on static programming , SIORHC 2 (Chisci et al., 1991) based on dynamic programming and FSIORHC, have been compared. The control laws computed by SIORHC 2 and FSIORHC agree to within four decimal digits for all prediction horizons N ~ n = 5, and reach the asymptotic (oo-horizon LQ) values (to within four decimal digits) at N = 32. Conversely, SIORHC 1 exhibits numerical inaccuracies already 5. For N 11, the control law comfor N puted by SIORHCI looses its theoretically guaranteed stabilizing property, and for subsequent values of N things get worse until an overflow occurs .

=

=

The control laws and closed-loop pole locations for N = 10 (before SIORHC 1 turns to instability) are shown in Table 3.

Chisci, L., A. Garulli and G. Zappa (1994a). Fast algorithms for generalized predictive control. Systems and Control Letters, 23 , pp.339-348 .

Summing up, FSIORHC - despite its computational complexity lower of a factor O( N -1) than SIORHC 1 and of a factor O(n-l) than SIORHC 2 - is numerically as accurate as SIORHC 2 and by far more accurate than SIORHC 1 .

Chisci, L., A. Garulli and G. Zappa (1994b) . Fast parallel algorithms for predictive control. In : Proc. lEE Int. Con/. CONTROL 94, pp . 1248-1253. Coventry, UK.

5. Conclusion A new fast algorithm for stabilizing predictive control, named FSIORHC, has been presented. For an nth order plant and N-steps prediction horizon, FSIORHC exhibits a reduced O(Nn) computational complexity as compared to either O(N2) or O(N n 2 ) complexity of preexisting algorithms based on either static or dynamic programming. In addition, FSIORHC shares the same numerical accuracy of the dynamic-programming based O(Nn2) algorithms and thus overcomes the numerical difficulties of algorithms based on static programming. It is expected that FSIORHC can be implemented on linear arrays of O( n) processors in order to further reduce processing time to O(N) . It is the authors' opinion that the results of this work can be profitably exploited in order to both speedup and robustify the control design task for real-time adaptive control.

Chisci, L., E. Mosca and S. Zengl (1991) . A fast algorithm for a stabilizing receding-horizon controller. In: Proc. 30-th IEEE CDC, pp . 517-518 . Brighton, UK. Chun, J ., H. Lev-Ari and T. Kailath (1987) Fast parallel algorithms for QR and triangular factorization. SIAM J. Sci. Stat. Comput., 8 , pp. 899-913 . Clarke, D.W., C. Mohtadi and P.S. Thffs (1987) . Generalized predictive control - Parts I and 11. Automatica, 23, pp . 137-160. Clarke, D.W. and R. Scattolini (1991). Constrained receding horizon predictive control. lEE Proc. D, 138 , pp . 347-354. Kouvaritakis, B., J .A. Rossiter and A.O.T. Chang (1992) . Stable generalized predictive control. lEE Proc. D, 139, pp . 349-362. Mosca, E., J.M. Lemos and J . Zhang (1990) . Stabilizing I/O receding horizon control. In: Proc. 29th IEEE Control and Decision Con/. . Honolulu, HA . Peterka, V. (1984). Predictor-based self-tuning control. Automatica, 20 , pp. 39-50.

REFERENCES

Peterka, V. (1986) . Control of uncertain processes: applied theory and algorithms. Kybernetika, 22 (suppl.) , pp. 1-102.

Bitmead, R.R., M. Gevers and V. Wertz (1990) . Adaptive optimal conirol - The thinking man's GPC. Prentice Hall, Englewood Cliffs, NJ.

Rossiter, J .A. and B. Kouvaritakis (1994). Robustness and efficiency of generalized predictive control algorithms with guaranteed stability. In: Proc. lEE Int . Con/. CONTROL 94, pp . 1017-1022. Coventry, UK .

Chisci, L. and G . Zappa (1991a) . A systolic architecture for iterative LQ optimization. A utomatica, 27, pp. 799-810 . Chisci , L. and G. Zappa (1991b) . Systolic architecture for predictive control. In: Proc. IFAC Workshop AARTC, pp. 37-42. Bangor, UK . Chisci, L. and G. Zappa (1993). Mapping LQ control design on fixed-size array processors. In: Mutual Impact 01 Computing Power and Control (M. Karny and K. Warwick, (Ed.)) , pp. 181-193. Plenum Press, New York . Chisci, L., A. Garulli and G. Zappa (1993). Fast algorithms for LQ control design of timeinvariant systems. In: Proc. S2nd IEEE Control and Decision Coni., pp. 116-121. San Antonio, TX. 492

Initialization: PN P; KN K; LN L; QN = Q; Lt = 0; Qt = O. For k = N - 1, ... , M: (Pk;Kk) = LIN(PH1,A;Kk+l,-B) (Kk,Qk;Pk,L k ) = CIR(Kk,Qk+l;Pk,LHd

=

=

=

= HCIR(Qk' Qt+l;Lk, Lt+l) (Qk;Lk) = LIN(Qk,X;Lk, Y)

P-1Qt;A-1Lt)

=

(Lk;Qk)

LIN(Lk,A;Qk,-B) 11

11

Fork = M -1, ... ,0: (Pk;K k ) = (Kk,Qk;Pk,Lk) =

=+ -

=

LIN(Pk+1,A; K H1 , -B) CIR(Kk,Qk+l; Pk, Lk+d + + CIR(QH1' Qk;L k+1, Lk)

Lk)

=

HYP(Qk+l' Qk; L k+1 , Lk)

(Lt;Qt)

=

LIN(LklA;Qk ,-B)

-;-+ -

(Qk ,Qk;Lk ,Lk) --

(Qk,A

-1

--1

Qk;Lk,A

-;-+

=+

(L;;Q;) = LIN(L~, A; Q~, -B) Optimal control law: k = O,I, ... ,M - 1 Rk = Qk and Sk = Lk, Minimum cost: N- 1 A2i [L N - 1 A2i [P. - E i=O Jmin iYO + K iU-1 ]2 + E i=M iY-1 + Q]2 iU-1 + M 1 - Y-1 + Q-i U-1 ]2} i U-1 - [ Li + E i=O- A2i {[L+i Y-1 + Q+]2 Table 1. FGLQ Algorithm

" Algorithm FLQ FGPC FGLQ FSIORHC

I

#

LIN

N 1.5N +0.5M 3N 3N

#CIR N N 1.5N + 0.5M 2N - 0.5n

# HYP 0 M M N-n

I

# polynomials 4

11

6 8 8

Table 2. Computational complexity of fast algorithms Asymptotic N=oo SIORHC 1 N = 10 SIORCH 2 FSIORHC N=lO

Ro(d) So(d) poles Ro(d) So(d) poles Ro(d) So (d) poles

= =

=

--:

1 - 0.5581d - 1.7352d2 - O.OO71dJ -9.6637 + 9.7592d + 10.2574d2 - 6.2566d3 - 4.0658d4 0.7562 ± jO.0403, -0.6562, -0.5352, -0.2107 1 - 0.5522d - 1.762Od:l - 0.0072d3 -9.6948 + 9.7625d + 10.3500d2 - 6.3902d3 - 4.1285<[4 0.8375, 0.7125, -0.6562, -0.5351, -0.2146 1 - 0.4829d - 1.6860d:l - 0.0069d3 -9.6034 + 9.2402d + 10.1412d2 - 6.0088d3 - 3.9509d4 0.7203 ± jO.0803, -0.6477, -0.5365, -0.2107

Table 3. Control laws and closed-loop pole locations relative to the numerical example

493