© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004
A multivariable generalised minimum-variance stochastic self-tuning controller with pole-zero placement Ali S. Zayed, Amir Hussain & L.S. Smith
Department of Computing Science and Mathematics, University of Stirling, FK9 4LA, Scotland Tel: +44 1786-467437, Fax: +44 1786-464551
Corresponding Author's E-mail:
[email protected]
Abstract The paper presents the derivation of a new robust multivariable adaptive controller, which minimises a cost function, incorporating system input, system output and set point. It provides an adaptive mechanism which ensures that both the closed-loop poles and zeros are placed at their pre-specified positions. The proposed design overcomes the shortcomings of other pole placement designs by combining the robustness of classical control strategy of pole-zero placement with the flexibility of self-tuning generalised minimum variance control. It tracks set-point changes with the desired speed of response, penalises excessive control action, and can be applied to non-minimum phase systems. Additionally, at steady state, the controller has the ability to regulate the constant load disturbance to zero. Example simulation results using both simulated and real plant models demonstrate the effectiveness of the proposed controller.
Keywords Multivariable self-tuning design, zero-pole placement, minimum-variance control, multivariable water bath system and non-minimum phase system.
1. Introduction The generalised minimum variance controller [1] was developed by Clarke and Gawthrop and provided several advantages over the original minimum variance strategy of Astrom and Wittenmark [2]. By introducing a cost
1
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 function combining the system input, system output and set point, the facility for control weighting and set point following is provided in addition to the capability of dealing with non-minimum phase systems. The generalised minimum variance self-tuning control has been extensively applied in literature and has gained considerable theoretical development. For instance, it was extended to include MIMO systems by Koivo [3], extended to have a PID structure by Cameron and Seborg [4] and Yusof et al. [5], and it was developed to achieve pole placement by Allidina and Hughes [6] and Zayed et al. [7]. It was also extended to achieve pole restriction by Hang et al. [8] and more recently combined with neural networks by Zhu et al. [9], Bittanti and Piroddi [10] and Hussain et al. [11]. On the other hand, during the past three decades, a great deal of attention has also been paid to the problem of designing pole-placement controllers and self-tuning regulators. Various self-tuning controllers based on classical poleplacement idea have been developed and employed in real applications, e.g. Wellstead et al. [12], Sirisena and Teng [13], Mansour and Linkens[14], and Davis and Zarrop [15]. The popularity of pole-placement techniques is due to the following main reasons[13]: 1) In the regulator case they provide mechanisms to over-come the restriction to minimum-phase plants of the original minimum variance self-tuner of [2]. 2) In the servo case, they give the ability to directly introduce bandwidth and damping ratio as tuning parameters. Comparatively, only little attention is given to zeros since they are considered to be less crucial than poles. Most of the previous discussions on zeros are centred around the choice of the sampling time so that the resulting system is invertible. However, it is important to note that zeros may be used to achieve better set point tracking, Puthenpura and MacGregor [16], and they may also help reduce the magnitude of the control action Sirisena and Teng [13]. The multivariable generalised minimum variance controller was extended to achieve pole-placement by Zayed et al. [7] and Zhu et al. [9]. However, the zeros are not considered in these designs. The multivariable zero-pole placement has been considered by Sirisena and Teng [13]. However, their work is restricted to stable systems and no stochastic disturbances are considered in deriving the controller. In this paper a new multivariable generalised minimum variance stochastic adaptive controller with pole-zero placement is presented. It builds on the previous works of Yusof et al. [5] and Zayed et al. [17]. The desired zeros can also be introduced in order to achieve better set point tracking and to help reduce the excessive control input. In addition the design ensures zero offset at steady state under constant load disturbances. The paper is organised as follows: the derivation of the control law is discussed in section 2. In section 3, various simulation case studies are carried out in order to demonstrate the effectiveness of the proposed controller in the performance of the closed loop system. Finally, some concluding remarks are presented in section 4.
2
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004
2 Derivation of control law In deriving the multivariable self-tuning control law we assume that the process is described by the following matrix polynomial (CARMA) model:
A( z −1 )y (t ) = B( z −1 )u(t − k ) + C( z −1 )ξ (t )
(1)
where y (t ) is the measured output vector with dimension ( n × 1 ), u(t ) is the measured control input vector ( n × 1 ),
ξ (t ) is an uncorrelated sequence of random variables with zero mean, k is the time delay in the integer sampling interval and (t ) denotes the sampling instant, t = 1, 2, 3,....., . The polynomial matrices A ( z −1 ) , B( z −1 ) and C( z −1 ) are expressed in terms of the backwards shift operator,
z −1 {i.e. z −1 x(t ) = x(t − 1) }, and are given as:
A(z −1) = I + A1z −1 + A2 z −2 + ............ +An z−na
(2)
a
B( z −1 ) = B 0 + B 1 z −1 + ...... +B n z
− nb
b
C(z−1) = I + C1z−1 + C2 z −2 + ..........+Cn z
, B(0) ≠ 0
(3)
−nc
(4)
c
where n a , n b , and n c are the degrees of the polynomials. The coefficients of the above polynomials are ( n × n ) matrices and I is the ( n × n ) identity matrix. In order to realise the algorithm, the following assumptions are made (Yusof et al. [5]):
a) The zeroes of the det C( z −1 ) lie inside the unit disc of the z-plane. b) The polynomial matrices A( z −1 ) and B( z −1 ) are coprime. c) nc ≤ na . The control law minimises the variance of an auxiliary output φ (t ) :
φ (t ) = P( z −1 )y (t ) + Q ′( z −1 )u(t − k ) − R ( z −1 )w (t − k )
(5)
Here w (t ) is the ( n × 1 ) set point vector and P( z −1 ) , Q ′( z −1 ) and R ( z −1 ) are the user-defined transfer functions in the backward shift operator z −1 . P( z −1 ) is rational matrix which can be expressed as: 3
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 P ( z −1 ) = Pn ( z −1 )Pd−1 ( z −1 )
(6)
here Pn ( z −1 ) and Pd ( z −1 ) are respectively monic ( n × n ) numerator and denominator matrices with degrees n p
n
and n p . The performance of closed loop system is determined by the selection of the polynomial matrices d
P( z −1 ) , Q ′( z −1 ) and R ( z −1 ) which are important design decisions. Let A ′( z −1 ) = A( z −1 )Pd ( z −1 )
(7)
and [ A ′( z −1 )] −1 C( z −1 ) = C ( z −1 )[ A ′( z −1 )] −1
(8)
It is assumed that: det C ( z −1 ) = det C( z −1 ) and A ′(0) = C (0) = I Now we can introduce the following identity: Pn ( z −1 ) C ( z −1 ) = E ′( z −1 ) A ′( z −1 ) + z − k F ′( z −1 )
(9)
where E ′( z −1 ) and F ′( z −1 ) denote ( n × n ) polynomial matrices with the following form: E ′( z −1 ) = I + E1′ z −1 + ....... + E ′k −1 z − ( k −1) F ′( z −1 ) = F0′ + F1′ z −1 + ........ + Fn′ z
(10)
−n f ′
(11)
f′
where n f ′ = n p + n a − 1
(12)
d
In the above equations, k, na and n p
d
are the time delay in the integer sample interval, the degree of polynomial
A( z −1 ) and the degree of polynomial Pd ( z −1 ) , respectively. ~ ~ ~ Since matrices are non-commutative, the polynomials C′( z −1 ) , E ′( z −1 ) and F ′( z −1 ) are introduced such that the following relationships are satisfied [5]: ~ ~ E ′( z −1 )C −1 ( z −1 ) = C −1 ( z −1 )E ′( z −1 )
(13)
~ ~ F ′( z −1 ) C −1 ( z −1 ) = C −1 ( z −1 )F ′( z −1 )
(14)
~ ~ where det C( z −1 ) = det C( z −1 ) and C(0) = I Combining (8), (9), (13) and (14) gives: ~ ~ ~ C( z −1 )Pn ( z −1 ) = E ′( z −1 ) A ′( z −1 ) + z − k F ′( z −1 )
(15)
Post multiplying (15) by Pd −1 ( z −1 ) and using (1) and (13) yields:
4
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 ~
~
~
P( z −1 )y (t + k ) = C−1 ( z −1 )[F′( z −1 )Pd −1y (t ) + E′( z −1 )B( z −1)u(t )] + E′( z −1 )ξ (t + k )
(16)
Using (16) and (5) yields: ~
~
~
φ (t + k ) = C −1 ( z −1 )[F ′( z −1 )Pd −1 ( z −1 )y (t ) + (E ′( z −1 )B( z −1 ) + Q)u(t )] − Rw (t ) + E ′( z −1 )ξ (t + k )
(17)
We define φ * (t + k / t ) as the optimum prediction of φ (t + k ) based on the measurement up to time (t ) and ~
φ (t + k ) as the prediction error. Therefore, we have ~
~
~
φ * (t + k / t ) = C −1 ( z −1 )[F ′( z −1 )Pd −1 ( z −1 ) y (t ) + (G ′( z −1 ) − Q( z −1 ))u(t )] − R ( z −1 )w (t ) ~
(18)
φ (t + k ) = φ (t + k ) − φ * (t + k / t ) = E ′ξ (t + k )
(19)
~ where Q( z −1 ) = C( z −1 )Q ′( z −1 )
(20)
~ ~ G ′ = E ′( z −1 )B( z −1 )
(21)
~ where the coefficients of G ′( z −1 ) are ( n × n ) matrices. Also n g~ = n ~e + n b = k − 1 + n b
(22)
~ where n g~ is the degree of G ′( z −1 ) , nb is the degree of polynomial B( z −1 ) and k is the time delay in integer interval. In the rest of the derivation the argument z −1 is omitted to make the notation easier. The variance of φ (t + k ) is minimised by selecting the control u(t ) such that φ * (t + k / t ) in (18) is set to zero. We therefore obtain: ~ ~ ~ (G ′ + Q)u(t ) = [CRw (t ) − F ′Pd−1 y (t )]
(23)
which is the generalised minimum variance control law.
2.1 Multivariable Adaptive Pole-Zero placement The next target is to incorporate the flexibility of generalised minimum variance control discussed in previous section with robustness of pole-zero placement. ~ Let F ′Pd−1 = Pd−1 F ′ ,
(24)
5
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 ~ ~ where det F ′ = det F ′ , F ′(0) = F ′(0) , det Pd = det Pd and Pd (0) = Pd (0) = I . Pre-multiplying (23) by P ( z −1 ) and setting the user transfer function R ( z −1 ) such that ~ R = C −1 Pd−1 H
(25)
Make use of (23), (24) and (25) we obtain: ~ ( Pd G ′ + Pd Q)u(t ) = [Hw (t ) − F ′y (t )]
(26)
The control law in above equation can also be expressed as: qu (t ) = [Hw (t ) − F ′y (t )]
(27)
~ here q = ( Pd G ′ + Pd Q)
(28)
where q and H are still the user transfer functions matrices since they depend on Q and R , respectively. We further assume that q can also be expressed as: q = ∆ q and H = F (1) H
(29)
in this case,
q ( z −1 ) = I + q1 z − 1 + ....... + q n z
−nq
(30)
q
∆ = (1 − z −1 )I , and q are the ( n × n ) polynomial matrices, and H is a user-defined polynomial matrix. We assume that: [ q ] −1 F ′ = F * [q * ] −1 ,
(31)
At steady state ( z = 1 ) the following equation is satisfied: [ q ] −1 F ′(1) = F * (1)[q * ] −1
(32)
Using (27), (29), (31) and (32), the control law becomes: ∆u(t ) = F * (1)[q * ] −1 Hw (t ) − F * [q * ] −1 y (t )
(33)
The closed loop system is obtained by using (33) and (1): y (t ) = q * (∆Aq * + z − k BF * ) −1 [ z − k BF * (1)(q * ) −1 Hw (t ) + ∆Cξ (t )] where (∆Aq * + z − k BF * ) is the characteristic equation. 6
(34)
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 Now we can introduce the identity: ~ Aq * + z −k BF * = T
(35)
~ where A = ∆A
(36)
and T is the desired closed loop polynomial matrix In this case T( z −1 ) = I + T1 z −1 + ...... + Tn T z − n T
(37)
For (37) to have a unique solution, the order of the polynomials F * , q * and T have to be {Wellstead and Zarrop. [18], and Prager and Wellstead [19]}: n
f*
n
q*
= n a~ − 1 = n a = nb + k − 1
(38)
n T ≤ n a~ + nb + k − 1 here n a~ , n
f*
n
q*
~ nT and k are the orders of the polynomials A , F * , q * , T and the time delay respectively.
Using (34), (35) and (36) yields: y (t ) = q * T −1 [ z − k BF * (1)(q * ) −1 Hw (t ) + ∆Cξ (t )]
(39)
It is obvious that the closed loop poles are in their desired locations. The additional closed loop zeros can be introduced by setting the user-defined polynomial H as following: H = h[h(1)] −1
(40)
where h is the desired closed loop zeros polynomial matrix and can be expressed as: h( z −1 ) = I + h1 z −1 + ...... + h n z
− nh
(41)
h
In practice the orders of T and h are most of the time selected as 1 or 2 [14]. It is clear that the polynomials F * and q * change continuously on line in order to satisfy (35) and (38) [20]. The polynomial F * is automatically tuned by continuously adjusting the pre-filter polynomials Pd and Pn in order to achieve the pole-placement, whereas the polynomial q * is automatically tuned by adjusting on line the user transfer function Q . However, it is not necessary to explicitly compute the polynomial P , Q and R every sampling instant [6,17,18].
7
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 In the SISO case, the idea of using the user defined transfer functions P and Q in order to achieve pole-placement and using the polynomial R to ensure zero steady state error was first developed by Allidina and Hughes [6] (see also Wellstad and Zarrop, [18]). In this paper the polynomial R has a double effect it is used to ensure zero steady state error and to achieve zero placement. In addition it is clear from equation (29) that the integral action to reject the constant load disturbances is also provided by automatically adjusting the user-defined polynomial Q .
2.2 Implementation of control law After obtaining the polynomials q * and F * from (35) the control law proposed in section 2.1 may be implemented in the following three ways: 1) In the first way the control law is directly implemented by using (33): det(q * )∆u(t ) = F * (1){adj(q * )}Hw (t ) − F * {adj(q * )}y (t )
(42)
where [q * ] −1 = [det(q * )] −1{adj(q * )}
(43)
2) The second way is to compute q and F ′ from (31) and then use (29) and (27), respectively. 3) In the third way the control law may also be directly implemented using (33) through the use of the instrumental ~ (t ) and ~ variables w y (t ) given by [21]: ~ (t ) = Hw (t ) q*w
(44)
which can be written as:
~ (t ) = w
n * q
~ (t − i)] +Hw (t ) [ −q *i w
(45)
i =1
q* ~ y (t ) = y (t )
(46)
which can also be expressed as:
~ y (t ) =
n * q
[−q *i ~ y (t − i)] +y (t )
(47)
i =1
~ (t ) and ~ After computing w y (t ) from (45) and (47) we can use the control law in (33) as follows: ~ (t ) − F * ~ ∆u(t ) = F * (1)w y (t )
(48)
8
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 The second method of implementing the control law may be preferred since it utilises routines already required in order to compute the pole placement law parameters q * and F * . However, in the situation where the number of inputs and outputs is large the third method may be preferred because the computation time is reduced and microcomputer implementation is made easier since the computation of matrix inversion is avoided. The algorithm for the pole-zero placement can then be summarised as follows: Step 1. Select the desired closed-loop system poles and zeros polynomials T and h . Step 2. Read the new values of y (t ) and w (t ). ˆ using least squares algorithm. After that compute qˆ * and ˆ , Bˆ , and C Step 3. Estimate the process parameters A Fˆ * using the identity (35).
Step 4. Apply the control law using one way from those were discussed in section 2.2. Step 5. Steps 1 to 4 are to be repeated for every sampling interval.
3 Simulation results The objective of this section is to study the ability of the proposed multivariable pole-zero placement controller in controlling a process under set point changes. Two simulation examples will be carried out in order to observe the ability of the proposed algorithm to locate the closed loop poles and zeros at the pre-specified locations. The simulation study also includes an investigation of the influence of the constant load disturbances and stochastic disturbances on the systems. Both simulation examples were performed over 600 samples with set point change every 100 sampling instants. In order to see clearly the effect of the zeros in the performance of the closed loop system, the controller is arranged to work as either a pole placement or zero-pole placement as follows: a)
From 0th to 250th sampling instant only the multivariable pole-placement controller is on-line.
b) The multivariable pole-zero placement controller is switched on from 251th to 600th sampling instant.
3.1 Example (1) The algorithm is tested on a minimum phase multivariable water bath system treated previously by Yusof et al. [22] and Zayed et al. [7,11,20] and described by the following transfer function: (I + A 1 z −1 )y (t ) = B 0 u(t − 1) + ξ (t ) , where A 1 =
− 0.411 − 0.634 0.492 0.085 , B0 = and the sample time = 30 sec. − 0.103 − 0.885 0.041 0.237 9
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 The simulations were performed over 600 samples (300 minutes) under set point w (t ) =
w 1 (t ) w 2 (t )
changes every 100
sampling instants as follows: 1) w 1 (t ) changes from 60 0 C to 80 0 C and from 80 0 C to 60 0 C . 2) w 2 (t ) changes from 35 0 C to 55 0 C and from 55 0 C to 35 0 C . The closed loop poles and zeros are respectively chosen as: T = I +
− 0.5 0 0.4 0 −1 z −1 and h = I + z . 0 − 0.5 0 0.4
ˆ and Bˆ are estimated using the least squares estimator and In each sampling instant the parameter estimations A 1 0 the second method of control law implementation discussed in section 2.2 was used. The outputs and the control inputs are respectively shown in the Figures (1a) and (1b). 90
80
y1
80
60
70 40 60
y2
50 40
u1
20 0
30
u2
-20
20 -40
10 0
0
100
200
300
400
500
-60
600
0
Figure(1a): the outputs
100
200
300
400
500
600
Figure(1b): the control inputs
We can see clearly from above figures that the excessive control action which produced from set-point changes is tuned after using pole-zero placement from the sampling interval 250th . In order to see the influence of the poles and the zeros on the performance of the closed loop system the desired poles and zeros polynomials are changed as follows: T=I+
− 0.35 0 −1 − 0.3 0 − 2 − 0.8 0 z + z z −1 h = I + − 0.6 0 0.4 0 0 0
The outputs and the control inputs are respectively shown in the Figures (2a) and (2b). It is obvious from these figures that the performance of the closed loop system can be controlled by both the polynomial T and h . We can also see that the output response y 1 tracks the set point more closely when the polezero placement is switched on from 251th until 600th sampling intervals.
10
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 80
60
y1
70
40
u1
60 20
y2
50 40
0
30
u2
-20
20 -40
10 0
0
100
200
300
400
500
-60
600
0
100
Figure(2a): the outputs
200
300
400
500
600
Figure(2b): the control inputs
3.2 Example (2) The algorithm was also applied to the following MIMO plant, originally introduced by Prager and Wellstead [19]: (I + A 1 z −1 + A 2 z −2 )y (t ) = z −1 (B 0 + B 1 z −1 )u (t ) + (I + C1 z −1 )ξ (t ) where A 1 =
− 1.4 − 0.2 0.48 0.1 1 0 1.5 1 − 0.5 0 , A2 = , B0 = , B1 = , C1 = − 0.1 − 0.9 0 0.2 0 0 0 1 0.1 − 0.3
and the ξ (t ) is a white-noise vector sequence with zero mean and variance R ′ =
0.1 0 . 0 0.1
Notice that the plant is non-minimum phase system and also has different time delays in the two channels. The set point w (t ) changes every 100 as follows: 1) w 1 (t ) changes from 5 to 10 and from 10 to 5 . 2) w 2 (t ) changes from 15 to 20 and from 20 to 15 . − 0.5 0 0.4 0 −1 z −1 and h = I + z . − 0.7 0 0 0.3
The closed loop poles and zeros are respectively chosen as: T = I +
The outputs and the control inputs are respectively shown in the Figures (3a) and (3b). 25
6
y2
20
15
u2
5 4 3
y1
10
2 1
5
0
0
-5
u1
-1
0
100
200
300
400
500
-2
600
Figure(3a): the outputs
0
100
200
300
400
500
600
Figure(3b): the control inputs
We can see clearly that the excessive control inputs which are produced from set-point changes in the first 250 sampling intervals are tuned after the sampling interval 251th when the pole-zero placement controller is used 11
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 The last task is to see the effect of the load disturbances on the closed loop system when pole-zero placement 1 1
controller is used. The polynomials T and h were kept fixed as before and constant load disturbances of value
were added to the outputs
y1 y2
from the 350th sampling instant to 600th sampling time instant.
The outputs and control inputs are shown in figures (4a) and (4b). It is clear from both figures (4a) and (4b), that at steady state the controller has the ability to regulate constant load disturbances to zero. 25
6 5
20
y2
15
u2
4 3
y1
10
2 1
5
u1
0 0
-5
-1
0
100
200
300
400
500
-2
600
0
100
200
300
400
500
600
Figure(4b): the control inputs
Figure(4a): the outputs
4 Conclusions In this paper, an algorithm to incorporate the robustness of classical pole-zero placement into the generalised minimum variance stochastic self-tuning controller for multivariable systems has been proposed. The resulting self-tuning controller provides an adaptive mechanism, which ensures that the closed loop poles and zeros are located at their pre-specified positions. The design was successfully tested on both simulated and real plant models. The results presented here indicate that the controller tracks set point changes with the desired speed of response, penalises the excessive control action, and can deal with non-minimum phase systems. It was shown that the zeros have a considerable influence in the performance of the closed loop system. In addition, the controller has the ability to ensure zero steady state error if the system is subjected to constant load disturbances.
References [1] D.W. Clarke & P.J. Gawthrop, Self-tuning control, Proc. Inst. Electr. Engineering, Part D, 126, 1979, 633-640. [2] K.J. Astrom & B. Wittenmark, On self-tuning regulators,. Automatica, 9, 1973, 185-199. [3] H.N., Koivo, A multivariable self-tuning controller, Automatica, 16, 1980, 351-366. [4] F. Cameron & D. E. Seborg, A self-tuning controller with a PID structure, Int. J. Control, 1983, 38, 401-417. 12
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 [5] R. Yusof, S. Omatu & M. Khalid, Self-tuning PID control: a multivariable derivation and application, Automatica, 30, 1994, 1975-1981. [6] A.Y. Allidina & F.M. Hughes, Generalised self-tuning controller with pole assignment, Proc. Inst. Electr. Engineering, Part D, 127(1), 1980, 13-18. [7] A.S. Zayed, L. Petropoulakis
& M.R. Katebi, An explicit multivariable self-tuning pole-placement PID
controller, 12th International Conference on systems Engineering (ICSE’97), Coventry, UK., Sept. 9-11, 1997, 778-785. [8] C. Hang, K. Lim & W. Ho, Generalised minimum-variance stochastic self-tuning with pole restriction, Proc. Inst. Electr. Engineering, Part D, 138(1), 1991, 25-32. [9] Q. Zhu, Z. Ma & K. Warwick, Neural network enhanced generalised minimum variance self-tuning controller for nonlinear discrete-time systems, IEE Proc. Contol theory Appl., 146, 1999, 319-326. [10] S. Bittanti, & L. Piroddi, Neural implementation of GMV control schemes based on affine input/output models, IEE Proc. Contol theory Appl., 144(6), 1997, 521-530. [11] A Hussain, A. Zayed & L. Smith, “A New Neural Network and pole-placement Based Adaptive composite self-tuning controller”, proceedings 5th IEEE international multi-topic Conference (INMIC’2001), Lahore, 28-31 Dec 2001, 267-271 [12] P.E. Wellstead, J.M. Edmunds, D Pragerand & P. Zanker, Self-tuning pole/zero assignment regulators, Int. J. Control, 30(1), 1979, 1-26. [13] H. R. Sirisena & F. C. Teng, Multivariable pole-zero placement self-tuning controller, Int. J. Systems Sci., 17(2), 1986, 345-352. [14] N.E. Mansour & D.A. Linkens, Self-tuning pole-placement multivariable control of blood pressure for post-operative patients: a model based study, Proc. Inst. Electr. Engineering, Part D, 137, 1990, 13-29. [15] R. Davies & M. Zarrop, On reduced variance overparametrized pole-assignment control, Int. J. Control, 69(1), 1998, 131-144. [16] S. Puthenpura & J. MacGregor, Pole-zero placement controllers and self-tuning regulator with better set-point tracking, Proc. Inst. Electr. Engineering, Part D, 134, 1987, 26-30. [17] A. Zayed, A. Hussain & L. Smith, “A modified minimum-variance stochastic self-tuning controller with polezero placement”, proceedings 5th IEEE international multi-topic Conference (INMIC’2001), Lahore, 28-31 Dec 2001, 252-256. [18] P.E. Wellstead & M. Zarrop, Self-tuning systems: Control and signal processing (John Wiley & sons, 1991).
13
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 [19] D.L. Prager & P.E., wellstead, Multivariable pole-placement Self-tuning regulators, Proc. Inst. Electr. Engineering, Part D, 128, 1980, 9-18. [20] A.S. Zayed, Minimum Variance Based Adaptive PID Control Design, M.Phil Thesis, Industrial Control Centre, University of Strathclyde, Glasgow, U.K, 1997. [21] B., Gong, correspondence on: multivariable pole-placement self-tuning regulators, Proc. Inst. Electr. Engineering, Part D, 130, 1983, 200. [22] R. Yusof, & S. Omatu, A multivariable PID controller, Int. J. Control, 57(6), 1993, 1387-1403.
Authors’ Biographies: 1) Ali Zayed obtained a BSc in Electronic Engineering from the Higher Institute of Electronics (Libya), in 1989. He joined Sirte Oil Company as an electronic engineer in (1989-1998) and was awarded a scholarship to pursue an MPhil program in Electrical and Electronic Engineering at the University of Strathclyde, UK. (1996-1997). In 1998 he was promoted to a senior engineer and in 1999, he joined the Department of Electrical Engineering at the University of 7th April as a Lecturer. He was awarded a scholarship by the 7th April University to pursue his PhD program at the University of Stirling (2001- present). 2) Amir Hussain graduated with a BEng in Electronic & Electrical Engineering in 1992, from the University of Strathclyde in Glasgow, UK. From 1992-1995, he worked as a University of Strathclyde sponsored Doctoral Researcher and Teaching Assistant and was awarded his PhD in Electronic & Electrical Engineering in 1996. From 1996-1998, he worked as a post-doctoral Research Fellow at the Department of Electronic Engineering of the University of Paisley, Scotland, UK. From 1998-2000, he was employed as an academic staff member of the Department of Applied Computing Science at the University of Dundee, Scotland, UK. In summer 1999, he was an invited Visiting Faculty staff member at GIK Institute of Engineering Science and Technology, Pakistan. Since 2000, he is working as an academic staff member in the Department of Computing Science at the University of Stirling, Scotland, UK. Dr. Hussain has one international patent and over 50 publications to-date in various journals, books and refereed international conferences. His research interests include computational intelligence, adaptive non-linear speech signal-processing and control, digital and mobile (tele)communications technologies and applications. He is a member of the IEEE and Guest Editor of the Journal of Control & Intelligent Systems Special Issue on Non-linear Speech Processing (2002).
14
© International Journal of Control and Intelligent Systems, Vol.32, No.1, pp.35-44, 2004 3) Leslie S. Smith (B.Sc 1973 Ph.D 1981) has worked on neural networks, neuromorphic systems and early auditory processing. He is currently Head of Department at the Department of Computing Science and Mathematics of Stirling. Professor Smith is a member of the IEEE and the ASA.
15
