LTR Design of Discrete-Time Integral Controllers based on Disturbance Cancellation

2a-092

Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA

LTR DESIGN OF DISCRETE-TIME INTEGRAL CONTROLLERS BASED ON DISTURBANCE CANCELLATION Hai-Jiao Guo·, Tadashi Ishihara·· and Hiroshi Takeda··· 'Department of Electrical Engineering, Tohoku University, SelUini, Japan "Graduate Sclwol of Information Sciences, Toho/cu University, SelUini, Japan ··'Department of Electrical Engineering, Tohoku Gakuin University, Tagajo, Japan Abstract: For discrete-time models witb direct feedthrough terms, we consider a design of integral controllers wbicb cancel disturbances by use of disturbance estimates. For this problem, we propose a new LTR technique targetting a state feedback integral controller including a disturbance estimator. The proposed procedure recovers the target feedback properly provided a plant is minimum phase. For non-minimum phase plants, we discuss the feedback properly obtained by enforcing the LTR procedure using an explicit expression of the sensitivity matrix.

Keywords: Discrete-time systems, Non-minimum phase systems, Disturbance rejection. Integral controllers, Loop transfer recovery, Riccati equations.

1. INTRODUcrlON It is well-known that an integral controller, which achieves robust rejection of step disturbances, can be designed by the disturbance cancellation technique (e.g., Willenmark, 1989, Franklin, et al., 1990). This type of controllers bas been successfully used in practical applications. However, no systematic design method bas been proposed for the output feedback case. The LTR techniques are well-known as systematic design methods to achieve desired feedback properlies by output feedback controllers (e.g., Stein and Athans, 1987, Sabed et al., 1993). They are useful not only for tbe standard LQG design but also for problems described by an extended plant satisfying stabiIizabiIity and detectability conditions. Although Ishihara et al. (1992) and Shafai et al. (1995) have discussed discrete-time integral controller designs using the conventional LTR techniques, they do not aim at directly canceling step disturbances by disturbance estimates. It should be noted that the disturbance cancellation problems requires extended plants whicb are not stabili7..able so that the existing LTR techniques can not be applied.

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In this paper, we propose a new LTR technique for designing discrete-time integral controllers based on the disturbance cancellation. We are interested in feedback property at the input side of a plant. As a target, we choose a state feedback integral controller including a disturbance estimator. We find an LTR procedure based on the Riccati equation formalism where the stochastic model contains the fi1ter gain matrix of tbe disturbance estimator in the target. The procedure recovers the target feedback properly provided a plant is minimum phase. For nonminimum phase plants, we discuss the feedback property obtained by enforcing the LTR procedure using an explicit expression of the sensitivity matrix.

2. DISCRETE-TIME CONTROLLER BASED ON DISTURBANCE CANCELLATION Consider a discrete-time model including a disturbance term

x(k + 1) - Fx(k) + G[u(k) +d(k»), y(k + 1) - Hx(k) + D[u(k) + d(k»),

(1)

where d(k) is a step disturbance entering the plant input.

Note that the discrele-lime model have a direct feedthrough lerm even if Ihe conlinuous-time planl is slrictly proper model (e.g.• Williamson. 1991). The transfer function matrix from the control inpul u(k) 10 Ihe oulpul y(k) can

~(k + 1) - <1>~(k) + fu(k) + K[y.(k +1) - Y.(k + 1)],

where

~(I) - [x '(I) d'(t)]'.

be expressed as (2) Our primary concern is 10 design a conlroller which cancels Ihe slep disturbance d (k) . Noting thallhe step disturbance d(k) can be described by

d(k+l)-d(k),

(3)

S(k + 1) - <1>s(k) + ruCk) + "'u (k). y(k + 1) - es(k) + DU(I) + "'y(k). where

(9)

The Kalman filter gain malrix K is given by

K _ (Xe' + Qxy )(eX9' + Q y )-1.

(10)

where X is a non-negative definile solulion of the Riccati equation

X _ <1>X' -(<1>Xe' + Qxy )(eXe' + Q y )-1

we inlroduce Ihe extended planl model

(8)

(11)

(eX<1> , + Q~) + Qx' (4)

As Ihe control algorilhm canceling the disturbance by the

estimated(k). we consider

s(k) - [x'(k) d'(k)]'. <1>-[: ~], r-[:],

e-[H

(5)

D}

(6)

and '" u (k) and '" y(t) represent a slochastic disturbance and an observation noise, respectively. We assume that "'u(k) and", y(t) are zero-mean while noise processes

u(k)--LX(k)-d(k)+T(z)r(k).

(12)

where r(t) is a reference inpul. the matrix L is an appropriate slate feedback gain matrix and 1'(z) is an appropriate precompensator. The matrix L can be determined by the standard LQ method or the pole assigument procedure; The precompensalor T(z) can be designed by a model matching melhod.

with the covariance matrices define as

UI,(k)] Q-Cov [ UI y (k)

[Q

The following result give Ihe controller Iransfer function x

(7)

'" '.xy

Lemma 2.2: The transfer function malrix from the inpul

NOle thallhe correlalion between "'u(k) and Uly(t) is indispensable 10 discuss an LTR technique. To guaranlee Ihe eslimalion of Ihe dislurbance d(k), assume the following conditions:

matrix.

we

u(t) 10 the oulpul yet) can be expressed as Cyu(z) - -z{l + 'l'(zl -

<1> + K6)-1(f - KD)}-l

(13)

'l'(zl-+Ke)-lK. where 'l'-[L I].

Cl:(F,G,H,D) is a minimal realizalion.

For the sensitivity matrices at the input of the plant, we can

C2: The matrix D is non.singular.

obtain Ihe following expression after some simple matrix calculalions.

C3: The plant Iransfer function matrix (2) has no zero at z=1.

Under the above conditions, we can easily prove the following fact for Ihe extended system (4) by use of PBH lesl for the observability. Lemma 2.1: If tile condilions Cl,C2 and C3 hold, then (e. <1» is observable. By the above lemma. we can eslimale the state and the disturbance by the following Kalman filler:

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Proposition 2.1: Define Ihe partilion of Ihe Kalman filler

gain matrix as K-[K~

Kd]',

(14)

which is compalible wilh the definition of the extended slale (5). Lel :1:(z) denole Ihe sensilivily matrix al the plant input side of Ihe conlrol syslem consisting ofthe planl (1) and Ihe controller (12). Then :1:(z) can be expressed as

l:(z) - (z -1)[1 + L(zl - F)G]-I[1 + L(zl - F

+ KxH)-1 (G - KxD)]R-I(Z) ,

matrix (I - KdG) are less than unity. (15) From (17) and (18), we can easily obtain the following result.

where

R(z) -zl-I +KdD +KdH(zl- F +K.H)-l(G-KxD).

Lellllllll 3.1: For the controller given by (17) and (18), the (16)

controller transfer function matrix from x(t) to u(t) is given by C",,(z) - -(z -1)-I[(zl-1 + KdG)L + ~(zl - F)]. (19)

3. LOOP TRANSFER RECOVERY

To design the controller consisting (8) and (12), it is necessary to detennine the state feedback gain matrix L, the Kalman filter gain matrix K and the precompensator 7'(z). In this section, we propose a new LTR technique which provides an efficient determination of the matrices Land K to achieve desired feedback property. For the conventional discrete-time LTR techniques, see Maciejowski (1985), Zhang and Freudenherg (1993), Ishihara et 01.(1993) and It should he noted that the Saheri et al.(1993). conventional LTR techniques can not he applied for our problem since the extended plant (8) is not stabilizable. The precompensator 7'(z), which has apparently no influence on the feedback property, can he designed by varions methods after the feedback property is fixed by K and L. We first give a target appropriate for our present problem and then provide a recovery procedure based on the Riccati equation formalism.

Using the above result, we can easily obtain the following result for the target feedback property. Proposition 3_1: The sensitivity matrix at the input side of the plant in the control system shown can he expressed as

3.2. LTR Procedure based on the IUccali equation Formalism By the analogy with the Riccati equation formalism for the standard continuous-time LTR (e.g.• Stein and Athans, 1987, Saheri et al., 1993), we consider the stochastic tenns in (4) are given by ",u(!) - fW(k), "'y(k) - Dw(k) +.jO v(k),

3.1. Target feedback property As a target of the LTR, we choose a state feedback integral controller including a disturbance estimator. The algorithm of the controller is given by

u(k) - -lx(k) -d (k) + T(z)r(k) ,

ii(k +1) - (I-KdG)ii(k)

+ Kd [x(k + 1) - Fx(k) - Gu(k)] ,

where w(k) and v(k) are mutually independent zeromean white noise processes with the identity covariance

matrices,

0

is a positive scalar and

(17)

where d (I) is an estimate generated by the disturbance estimator given by (18)

(22) Note that f includes the estimator gain matrix Kd' For the stochastic terms (21), the covariance matrices in (7) are given by

Dx-ff',

where Kd is the estimator gain matrix. The gain matrix

Kd can he detennined by different methods, e.g., the Kalman filtering technique or the pole assignment procedure. For our purpose, it is sufficient to assume the following condition regardless of the method used. C4: The absolute value of all of the eigenvalue of the

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(21)

Qxy-fD',

Dy-o/+DD'.

(23)

As in the continuous-time case, the formal LTR procedure is to let a tend to zero. The LTR design consists of two phases:

1.

Determine the feedback gain matrix L and the estimator gain matrix Kd such that the target possesses the desired feedback property.

2.

Construct the output feedback controller with covariance matrices (23). To determine the filter gain matrix K. decrease the parameter 0 until the output feedback controller provides sufficiently close feedback property to the target.

al., 1995) is a discrete-time version of the continuous-time decomposition found by Enns (1984). For simplicity, we assume that (24) has a single unstable zero. Since the result gives a method for extracting an unstable zero, it is an easy exercise to extend the result to multiple unstable zeros.

The above LTR procedure differs from the standard procedure in the point that the flfst step includes the determination of the estimator gain matrix Kd' This

Lemma 3.5:

estimator gain matrix is not directly used in the actual controller but influences to the actual filter gain matrix K through the covarianee matrix (23). In the following, we discuss feedback property achieved by this procedure to clarify the effectiveness of the second step.

which has a single real unstable zero q

For the above stochastic model, we can easily obtain the following result.

Define

Q(z) - 9(zl - <1»-1

and 1 q

the matrices rq

where

!;

r + D,

(25)

(Iql > 1) .

as

r. -r+-;})!;t]',

(26)

l. -1-(q+1)tjt]',

(27)

and

t]

are vectors satisfying

Lemma 3.2: Consider the stochastic model (4) with the stochastic terms given by (21). Then the transfer function matrix from w(k) to y(k) is given by

r

Q(z) - z-l[ 9(zl - <1»-1 +D]- z-1[H (z/ - F)-IG +D][I +(z-l)-I~G].

Defme

(28) In addition, define

(24)

fm-f.l q , D,.-Dlq .

(29)

Then the transfer function matrix Q(z) defined in (25) can Noting that the matrix Kd G is non-singular under the

be decomposed as

assnmption C4, we can easily check the following fact by use of PBH test.

where

Lemma 3.3: Assnme that the conditions Cl and C4 are satisfied. Then (<1>, r) is controllable. The general result on the Riccati equation for not necessarily strictly proper plants found by Kueera (1991) does nOl provide an explicit condition to guarantee the existence of a stabilizing filter gain matrix for the covariance matrices (23). However, for 0> 0, we can easily obtain the following result using a similar technique as for Riccati equations for strictly proper plants. Lemma 3.4: Assume that the conditions Cl, C2, C3 and C4 are satisfied. Then, the Riccati equation (11) for the covariance matrix (23) where a> 0 has a unique nonnegative definite solution X that makes the matrix <1> - K9 , where K is defined by (14), is stable. To discuss the solution for 0 _ 0, we need a minimumphase/all-pass decomposition of the transfer function defined in (24). The result of Ou et al. (1989) can be used for this purpose but it requires the solution of a singular Riccate equation. The following simpler result (e.g., Ouoet

1278

(30) (31)

is minimum phase and Qa (z) - (I

z-q -t]t]') - --t]t]' qz -1

(32)

is an-pass, i.e., (33)

Noting the expression (24), we can easily check that, under the condition C4, the unstable zero q of Q(z) is also contained in the plant transfer function matrix (2). For the case 0 _ 0, we have the following result for the solution of the Riccati equation and the optimal filter gain

matrix. Lemma 3.6: Assume that the conditions Cl, C2, C3 and C4 are satisfied. In addition, assume that the plant transfer function matrix (2) has a single unstable zero q. Consider

the Riccati equation (11) with the covariance matrices

(42) (34) Prom (26) and (39), we can write

i.e.,

x - X' - (X8' + I'D')(<3X8' + DD')-l (8X' + Dr') + rr' .

_

_

(35)

J;;l _I~',

(37)

f .. ,

-(1- q ;1'l'l}

Using (42) and (44) in (43), we have the expression (40). The expressions in (41) follow from (29) and (37). Using the above result in Proposition 2.1, we have the following expression for tbe sensitivity matrix achieved by using the LTR procedure.

D.. are given by (29).

Proof: See Ishihara et al. (1993). Por the minimum phase case, we immediately have the following result. Lemma 3.7: Assume that the conditions Cl, C2, C3 and C4 are satisfied. In addition, assume that the plant transfer function matrix (2) is minimum phase. Then the Riccati equation (35) has is a unique solution X _ 0 that makes - K<3 stable and the corresponding optimal filter gain matrix is given by K* _ f D-l.

(38)

Proposition 3.1: Assume the conditions the conditions Cl, C2, C3 and C4 are satisfied. In addition, assume that the plant transfer function matrix (2) has a single unstable zero q. Consider the covariance matrices given by (23). Then, as the variance 0" tends to zero, the sensitivity matrix l:(z) at the plant input side approaches

l:*(z) - (z -1)[1 + L(zJ - F)Grl{I + L(zI _ F)-l [G - GmP,;l(z)p(z)]}[zI - I + Kd GP,;l (z)p(z)rl , (45) where

G.. -GqJq ,

For the non-minimum phase case, we need more explicit expressions for the ftl1er gain matrix, which is given as follows. Lemma 3.8: Assume tbatthe conditions in Lemma 3.6 are satisfied. Write

(39)

(46)

P(z) - H(zI -F)-lG +D, Pm(z)-H(zI

-F) -1 G.J.

+DJ. .

which are defined in (26) and (37), respectively.

(47) (48)

Proof: By the matrix inversion lemma. we have

(zI -F +K;H)-l - (zI _F)-l{I -K; [I +H(zI -F)-lK;)-lH(zI _F)-I}.

(49)

It follows from the first expression in (41), (46). (47), (48) and (49) that

(zJ -F +K;H)-l(G -K;D)

(50)

_ (zI -
Then (40)

and (41)

Proof: Define 1; - [ 1; ~ and (28) that

(44)

(36)

where 1; is defined by (28), is a unique non-negative definite solution that makes - K8 stable and the corresponding optimal filter gain matrix is given by

where the matrices

(43)

Note that

Then the matrix

X. _(q2

q2_1

(Kd G ). - Kd G - --1;d'l' . q

!; d ]' . It follows from (6), (22)

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Using the second expression in (41) and (50), we can write (16) as (51) Substituting (50) and (51) into (15). we have the expression (45).

Note that the matrices G. and J, in (46) are obtained for the decomposition of Q(z) but not for P(z). In general, we can not guarantee that p.(z) is minimum pbase. Roughly speaking, by romparing (45) with the target sensitivity matrix (20), we can say that the feedback property achieved for the nOD-minimum phase case is sufficiently closed to the target in the frequency region where the difference between pet) and p.(:) is insignificant. For the scalar case, p.(z) is a minimum phase part of the all-pasS/minimum phase decomposition of P(:). Consequently, we have the following result. Corollary 3.1: [n addition to the assumptions in Proposition 3.1, assume that the plant is single-inputsingle-output. Then

1;" (z) - (z -1)(1 + L(zJ - F)Gr l If + L(zJ - Frl [G - G.. P" (z)]}[zJ - 1+ KdGP" (z)r l ,

(52)

where p" (z) is an all-pass part of the decomposition of PC:) and coincides with that of Q(z). For the minimum case, we can easily show that the perfect recovery is possible. Corollary 3.2: In addition to the assumptions in Proposition 3.1, assume that the plant is minimum phase. Then 1;"(z) - S(z) , where S(z) is the sensitivity matrix of the target defined in (20).

4. CONCLUDlNG REMARKS For discrete-time models with direct feedtbrough terms, we have proposed a new LTR design of integral controllers based on the disturbance cancellation technique. Unlike the standard LTR techniques, the first step of the proposed LTR proeedure requires the determination of a fictitious By this trick, we can estimator gain matrix Kd. successfully apply the LTR technique for the problem which does not satisfy the stabilizability condition. The proposed design can be performed by using tbe standard matrix computation without numerical difficulty. Although tbe feedback property achieved by enforcing the LTR proeedure has been discussed, we can show that this feedback property can also be obtained by the partial LTR technique (Moore and Xia, 1987) as is recently discussed by Guo et al. (1995) for a discrete-time integral controller design.

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