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DESIGN METHODS AND TUNING OF BILINEAR PROPORTIONAL-INTEGRAL-PLUS CONTROLLERS
Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain
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DESIGN METHODS AND TUNING OF BILINEAR PROPORTIONAL-INTEGRAL-PLUS CONTROLLERS S€awomirZiemian Keith J. Burnham
Control Theoy and Applications Centre, Coventry University, Coventry CVI 5FB, UK Tel, Fm:+44(O)24 76888052 email: [email protected]
Abstract: This paper describes the advantages of extending the existing linear philosophy of proportional-integral-plus (PIP) control to a bilinear case. Investigation of the quasilinearisation, design and tuning procedures together with an additional quasilinearisation parameter are presented. The role of the quasilinearisation parameter and the potential improvements in closed loop performance are demonstrated via simulation studies using a system similar to that found in high temperature furnaces.
Copyright © 2002 IFAC Keywords: bilinear systems, design, discrete systems, pole assignment
1. INTRODUCTION
(Dixon and Lees, 1994) or (Taylor et al., 2000b). However, since industrial systems exhibit nonlinear characteristics, it is often difficult to obtain an appropriate model. A wide range of systems, which exhibit operating point dependent steadystate gain and dynamic response characteristics, can be approximated by making use of bilinear equations (Bruni et al., 1994). Bilinear models hold for a range, rather than for a point as is the case for linear models. The benefits of adopting a bilinear approach for model-based techniques have been previously reported (Mohler, 1973) and successfully applied in practice - see for example (Burnham et al., 1999)and (Dunoyer et al., 1997).
In 1987 Young et al. introduced the idea of true digital control (TDC) - see (Young et al., 1987), (Young et al., 1991) and (Wang and Young, 1988). This approach assumes that a whole design procedure is carried out only in discrete time terms and is based on the definition of non-minimal state space (NMSS) (Dixon, 1996). Adoption of these two concepts in which the control is realized via state variable feedback (SVF), with only past input and current and past output values as states, creates the basis for PIP. The avoidance of a state vector observer, or estimator, hence leading to potentially greater complexity, is the main advantage of PIP. The implementation of TDCPIP is reported to be quite straightforward and can be regarded as an extension of proportionalintegral-derivative (PID) control (Dixon, 1996). In addition it has been shown that PIP can duplicate the behaviour of generalized predictive control (GPC) offering more design freedom, and further research to exploit the full potential of PIP is still ongoing (Taylor et al., 2000a). This approach has been applied to practical situations and its usefulness demonstrated - see for example
2. PROBLEM FORMULATION 2.1 Bilinear model
There are many ways of representing a bilinear system (Dunoyer et al., 1996). In this paper the bilinear single-input single-output (SISO) ARMAX structure of the following form is assumed:
373
0
A ( q - l ) y ( t ) = B(q-')u(t)
+
c na
q;y(t - i)u(t- i - k
i=l
+ 1)+
E(t)
(1)
in which the polynomials A(q-'), B(q-') are defined as follows:
+
+ u ~ Q+- .~ + anaq-nm (2) + bq-2+ . .. + bn,q-nb (3)
A(q-') = 1 a1q-I
a
a
B(q-') = b1q-I where u(t),y ( t ) , ~ ( tare ) the input, output and white noise output disturbance, respectively, k 2 1 is the system time delay expressed as an integer multiple of the sampling interval denoted T, and q-l is the backward shift operator defined as qPiy(t) y ( t - i). Note that the first k - 1 coefficients in B(q-') are equal to zero to accommodate the time delay.
0
ai = 0 for all i - full quasilinearisation with A(q-') polynomial; ai = 1 for all i - full quasilinearisation with B ( q - l ) polynomial.
Since the resulting controller gains will be dependent on numerator and denominator of the model transfer function - they are dependent on ai. As a consequence the closed loop performance will depend on the quasilinearisation.
2.3 Steady-state behaviour of bilinear systems
The characteristic which makes a negative bilinearity attractive is the asymptotic saturation-like phenomena. This asymptotic behaviour depends on the magnitude and sign of system parameters. For a discrete first order bilinear system the asymptotes are given by:
=
2.2 Quasilinearisation Equation (1) comprises nonlinear (or bilinear) term(s). Essentially this bilinearity, can be quasilinearised with either input or output giving a linear description at the sampling instants. Thus by analysing the model at a given sample time, it can be 'loosely' regarded as a linear model with input or output dependent coefficients (Ziemian et al., 2000). However there is no reason why this method of quasilinearisation cannot be apportioned, with the bilinear terms being combined such that both numerator and denominator are affected. In such a case the quasilinearisation parameter a is required to be introduced. Then equation (1)will take the following form (for reasons of clarity, it is assumed that ~ ( t=)0 and is omitted) :
A(q-l)y(t) =%-')u(t)
For a system in which 61 > 0, ql < 0 and 0 5 a1 < 1,the steady-state characteristic is given in Fig. 1. yss L-.......-
(4)
n,
+ C [ai+ (1- ai)]qiy(t - i)u(t- i - k + 1) i= 1
For a first order bilinear system with unit delay (i.e. n, = l,nb = 1,k = 1) the equivalent discrete transfer function takes the form:
________________________
Fig. 1. Steady-state characteristic for system with bl > 0, ql < 0 and 0 5 a1 < 1 - the dashed lines represent the asymptotes. The characteristic in Fig. 1 has to be taken into account when specifying the magnitudes of the desired input and output signals.
3. BILINEAR PIP
where
iil = a1 - (1- a1)q1u(t - 1)
(6)
The idea of TDC assumes the use of only discretetime models and discrete-time data. For a discrete SISO system the NMSS representation takes form (Dixon, 1996):
= bi + ~ q i y (-t 1) (7) In principle there are no limits on the quasilinearisation parameter a because the open loop system remains the same regardless of the value of a. However, if the upper and lower limits on control input are known - the limits on C Y ~can be set up in the controller design phase to achieve specific goals. Two cases of special interest are:
z(t)= F z ( t - 1) + gu(t - 1) + dy&) Y ( t )=H 4 t ) where the state vector is of the form:
374
(9)
s ( t ) T = [ y ( t ) y ( t - 1).. .y(t - 71,
+ 1)
3.1 Pole assignment
a(t - 1)...u(t - 126 $- 1)z ( t ) ] (10) with the integral-of-error term z ( t ) defined as:
Whilst the notion of poles is limited to linear systems theory, consideration in this paper is given to equivalent linearised systems at the sampling instants. Loosely, in the specific case of a bilinear system, the control input may need to be bounded in order to 'guarantee' stability.
z ( t ) = zz(t - 1)+ tYd(t) - y(t)l
(11) where y d ( t ) is the set point or reference. Matrices F , g , d and H are as defined in (Dixon, 1996), with elements appropriately amended to include the quasilinearised parameters.
The gain vector k is calculated according to the following algorithm (Dixon, 1996):
The PIP control law takes the familiar SVF form:
u(t)= - k z T ( t ) where the gain vector k is given by:
(12) where d d and pol are vectors of coefficients for the desired closed and open loop charaderistics polynomials, respectively, i.e.:
T
k = [fo f i . . .fna-191Q2gnb-1 - k ~ ] (13) in which the fi and g j are the coefficients of the feedback and input filter polynomials, and kl is the coefficient of the integral term. The standard implementation of the PIP servomechanism control (Taylor et a$., 1996) is shown in Fig. 2.
polT = [61 - 162 - 61 6 3 - 6 2 . . .
(16)
- an, - O...O]
. ..&,, S is the partitioned matrix I
S = [g : Fg : F2g : . .. : Fna+nb-l91
I
and
Fig. 2. Standard form of PIP. where: A = (1 - q-'), so that k l / A provides integral action, F ( 4 - l ) = fo f 1 q - l ... fn,-lq-(na-l) is the feedback filter, G(q-') = 1 g1Q-l . . . gnb-lq-(nb-l) is the input filter.
+ +
(17)
+ + + +
M=
The gain vector k can be found by the use of any suitable control algorithm. The founders of linear PIP have shown how pole assignment and optimal control can be employed to calculate the required coefficients. The first requires the specification of the desired pole location, while in the second, the weighting matrices used in the performance index need to be chosen. For SISO systems the specification of desired poles would appear to be easier and more straightforward. The considerations in this paper are limited to the SISO case and the pole assignment strategy utilized.
0 0
0
-&n,
0 0
0 0
0 0
0 0 0 0
1 61 - 1 - 61
Properties of the controller such as robustness and disturbance rejection capability depend on its structure (Taylor et a$., 1996). In addition, the control input, in practice, is implemented in an incremental form, which can be straightforwardly realized. This allows the inclusion of both magnitude and rate constraints without additional complications, and integral wind-up is also prevented. In this paper only results regarding the standard form of PIP are presented.
62
0 0 0 0
0 1 61 - 1
...
... ...
...
... ... ...
...
...
...
0 0 0 0
..
0' 0 0 0
.
..
0
0
0
0
0
0
.
Note that when adopting a bilinear approach the gain vector is required to be recalculated at each time step. This is due to the fact that the quasilinearised model parameters are in fact input/output dependent quantities.
375
4. SIMULATION STUDIES The simulation studies presented in this paper are limited to the case of a first order bilinear system with a negative bilinearity and a unit delay. The model is given by a1 = -0.8, bl = 0.4 and r ~ 1= -0.1, with time delay k = 1. For such a system it is instructive to observe the impact of the a parameter on controller performance. For the bilinear case the gains can be represented as follows:
d l + -& + l ] (18) bi bi with 61 and 6, defined as in (6) and (7), respectively (note that the input filter G(q-') = 1). With reference to quasilinearised equation (7), it has been found in simulation studies that in the case of the 61 parameter it is better to use the most recent value of y ( t ) instead of g ( t - 1).The number of closed-loop poles to be assigned in the controlled system is n, + nb = 2, and choice of closed loop poles gives rise to the closed loop polynomial coefficients dl and dz. In this case one pole is arbitrarily assigned to zero and closed loop performance is achieved by positioning the other pole along the real axis. The model and assumed parameters are similar to that obtained from an identification exercise conducted on a high temperature furnace used in steel production. Fig. 3 illustrates how a can influence the closed loop performance, in the case where the closed loop poles are chosen to be [0,0.2]. kT =
[-(&+at)
1-
0.8 -
achieved by making use of a pseudo random binary sequence (PRBS) to excite the open-loop bilinear system. Model structure selection and estimation was carried out by the use of simplified refined instrumental variable (SRIV) (Young, 1985). The linear system is of first order with unit delay, with the following coefficients all = -0.7839 and bll = 0.3360 (the subscript 1 denotes parameters of a linearised model). Such a linearized model replicates the behaviour of the simulated bilinear system quite well, which is confirmed by the value of Young Identification Criteria (YIC) of well below -10 and other indices provided by the Captain Toolbox. 4.1 Choice of a pammeter This section investigates the factors affecting choice of the quasilinearisation parameter a. It is found that one of the main factors on which the choice of quasilinearisation parameter depends is the sign of coefficients and the magnitude of control input for which a system is to be operated. To obtain a suitable value of a the measure of performance used here is the integral of absolute errors (IAE) between reference signal (output of a prefilter with added delay) and output of the system. It is interesting to note that when 0 5 a 5 1, the value of a which gives rise to a minimum IAE depends on the desired closed loop pole locations. In particular, when the desired pole is close to the 'natural pole' of the linear part of the system (e.g. 0.8) the value of a = 1 is found to be best, implying full quasilinearisation with B(q-'). The 'optimum' values of a for various pole locations in the range 0 - 1,when the reference input is set to 1.0, are given in Fig. 4.
e
3 0.6rn
2
0.4 0.2 -
O
;
;
4
No.5of samples 6
;
8
9
Fig. 3. Impact of a parameter on the output. The desired response is given by the dash-dot line, the solid (thick) line represents the response for linear PIP, and the solid (feint), dashed and dotted lines represent the responses of the resulting bilinear PIP schemes. It may be observed that the performance of bilinear controller is better for all values the quasilinearisation parameter investigated; however in general this may not always be the case. Note that to realize the linear PIP, a linearized model is required to be identified. This was
376
Fig. 4. Best value of a vs desired pole location. Extending the study to include the magnitude of the reference input in the range 0 - 1.5 leads to the results shown in Fig. 5 It is interesting to note that for lower values of reference, the range for which a = 0 (i.e. full quasilinearisation with A ( q - l ) ) is increased.
illustrated in Fig. 7. The x, y and z axes show the perturbations on the a l , bl and 171 parameters, respectively. The values on the grid show the best values of a obtained. In this case the input reference is 1.0 and the poles are arbitrarily chosen as [0;0.2]. Again the ‘best’ values of a is that which results in a minimum IAE with respect to the desired responce in Fig. 3. -0.09
.92
r- -0.1
Fig. 5. Best a parameter vs pole location and magnitude of reference.
.94 “.,a
This behaviour is also visible in Fig. 6. It can be concluded that if the linear dynamics of the system are fast and desired pole slow - the value of a shall be 1. Quite often the task to be performed by a controller is to force a system with slow dynamics to respond faster. If this is the case then the qusilinearisation parameter should be chosen more carefully. Its ‘optimal’ value could lie within the range 0 < a < 1. Fig. 6 shows the ‘best’ values of a when the reference level is again fixed at 1.0 and the closed loop pole location and natural pole of the linear part of the system are varied over the range 0 - 1.0. Similar observations can be made when varying other model parameters. For example if bl is increased, the range over which a = 0 is increased. Increase in the bilinear coefficient 171 also affects the value of a.
Fig. 7. Best a parameters in perturbed models.
5. CONCLUSIONS In this paper a novel approach of apportioning the bilinear coefficient to achieve an effective quasilinearisation has been investigated. This allows greater flexibility and more freedom in designing the bilinear controller using a proportionalintegral-plus technique. Preliminary results indicating the potential improvement together with quasilinearisation parameter dependency on other model parameters of the system together with the operating set-point have been shown. Further work is currently on-going in order t o evaluate the advantages of the resulting bilinear controllers based on the proposed quasilinearisation.
1
0.8
4
0.6
v)
0.4 0.2 0 0
6. ACKNOWLEDGEMENTS 0
a,
Desired pole location
Fig. 6. Best a parameter versus desired pole location.
a1
The authors wish t o acknowledge the use of the Captain Toolbox, courtesy of its creators from the Centre for Research on Environmental Systems and Statistics (CRES), Lancaster University, Lancaster, United Kingdom.
value and
7. REFERENCES
In reality the model of a system is only an approximation of the system itself, with errors in modelling the actual behaviour being inevitable. With this in mind, it is useful and instructive to explore the effect on a of variations in the model parameters. The result of such a study is
Bruni, C., G. Dipillo and G. Koch (1994). Bilinear systems: An appealing class of “nearly linear”systems in theory and applications. IEEE Trans. on Automatic and Control 19, 334348.
377
Burnham, K.J., A. Dunoyer and S. Marcroft (1999). Bilinear controller with pid structure. IEE Computing €9 Control Engineering Journal 10, 63-69. Dixon, R. (1996). Design and Application of Proportional-Integral-Plus (PIP) Control Systems. PhD thesis. Lancaster University. Lancaster, UK. Dixon, R. and M.J. Lees (1994). Implementation of multivariable proportional-integralplus (PIP) control design for a coupled drive system. Proc. loth International Conference on Systems Engineering, Coventry University 1, 268-293. Dunoyer, A., K.J. Burnham and T.S. McAlpine (1997). Self-tuning control of an industrial pilot-scale reheating furnace: Design principles and application of a bilinear approach. IEE Control Theory Appl. 144, 25-31. Dunoyer, A., L. Balmer, K.J. Burnham and D.J.G. James (1996). On the characteristics of practical bilinear model structures. Systems Science 22, 43-58. Mohler, R.R. (1973). Bilinear Control Processes With Applications to Engineering, Ecology, and Medicine. Academic Press. New York and London. Taylor, C.J., A. Chotai and P.C. Young (2000a). State space control system design based on non-minimal state-variable feedback: further generalization and unification results. Int. J. Control 73, 1329-1345. Taylor, C.J., P.C. Young, A. Chotai, A.R. McLeod and A.R. Glasock (2000b). Modelling and proportional-integral-plus control design for free air carbon dioxide enrichment systems. Journal of Agricultural Engineering Research 75, 375-382. Taylor, C.J., P.C. Young, A. Chotai, W. Tych and M.J. Lees (1996). The importance of structure in pip control design. UKACC’96 427, 1196-1201. Wang, C.L. and P.C. Young (1988). Direct digital control by input-output, state variable feedback: theoretical background. Int. J. Control 47,97-109. Young, P., M.A. Behzadi, C.L. Wang and A. Chotai (1987). Direct digital and adaptive control by input - output state variable feedback pole assignment. Int. J. Control 46, 1867-1881. Young, P.C. (1985). The instrumental variable method: a practical aproach to identification and system parameter estimation. In: Identification and System Parameter Estimation (H.A. Barker and P.C. Young, Eds.). Vol. 1. Pergamon Press. Oxford. Young, P.C., A. Chotai and W. Tych (1991). True digital control: A unified design procedure for linear sampled data control systems. In:
378
Lectuw Notes in Control and Infomation Sciences (M. Thoma and A. Wyner, Eds.). Vol. 158. pp. 71-109. Springer-Verlag. Ziemian, S., K.J. Burnham and R. Dixon (2000). An extended bilinear proportional-integralplus controller. Pmc. 14th Int. Conf. on Systems Engineering, ICSE’2000, Coventy University, UK 2, 626-631.
www.elsevier.com/locate/ifac
DESIGN METHODS AND TUNING OF BILINEAR PROPORTIONAL-INTEGRAL-PLUS CONTROLLERS S€awomirZiemian Keith J. Burnham
Control Theoy and Applications Centre, Coventry University, Coventry CVI 5FB, UK Tel, Fm:+44(O)24 76888052 email: [email protected]
Abstract: This paper describes the advantages of extending the existing linear philosophy of proportional-integral-plus (PIP) control to a bilinear case. Investigation of the quasilinearisation, design and tuning procedures together with an additional quasilinearisation parameter are presented. The role of the quasilinearisation parameter and the potential improvements in closed loop performance are demonstrated via simulation studies using a system similar to that found in high temperature furnaces.
Copyright © 2002 IFAC Keywords: bilinear systems, design, discrete systems, pole assignment
1. INTRODUCTION
(Dixon and Lees, 1994) or (Taylor et al., 2000b). However, since industrial systems exhibit nonlinear characteristics, it is often difficult to obtain an appropriate model. A wide range of systems, which exhibit operating point dependent steadystate gain and dynamic response characteristics, can be approximated by making use of bilinear equations (Bruni et al., 1994). Bilinear models hold for a range, rather than for a point as is the case for linear models. The benefits of adopting a bilinear approach for model-based techniques have been previously reported (Mohler, 1973) and successfully applied in practice - see for example (Burnham et al., 1999)and (Dunoyer et al., 1997).
In 1987 Young et al. introduced the idea of true digital control (TDC) - see (Young et al., 1987), (Young et al., 1991) and (Wang and Young, 1988). This approach assumes that a whole design procedure is carried out only in discrete time terms and is based on the definition of non-minimal state space (NMSS) (Dixon, 1996). Adoption of these two concepts in which the control is realized via state variable feedback (SVF), with only past input and current and past output values as states, creates the basis for PIP. The avoidance of a state vector observer, or estimator, hence leading to potentially greater complexity, is the main advantage of PIP. The implementation of TDCPIP is reported to be quite straightforward and can be regarded as an extension of proportionalintegral-derivative (PID) control (Dixon, 1996). In addition it has been shown that PIP can duplicate the behaviour of generalized predictive control (GPC) offering more design freedom, and further research to exploit the full potential of PIP is still ongoing (Taylor et al., 2000a). This approach has been applied to practical situations and its usefulness demonstrated - see for example
2. PROBLEM FORMULATION 2.1 Bilinear model
There are many ways of representing a bilinear system (Dunoyer et al., 1996). In this paper the bilinear single-input single-output (SISO) ARMAX structure of the following form is assumed:
373
0
A ( q - l ) y ( t ) = B(q-')u(t)
+
c na
q;y(t - i)u(t- i - k
i=l
+ 1)+
E(t)
(1)
in which the polynomials A(q-'), B(q-') are defined as follows:
+
+ u ~ Q+- .~ + anaq-nm (2) + bq-2+ . .. + bn,q-nb (3)
A(q-') = 1 a1q-I
a
a
B(q-') = b1q-I where u(t),y ( t ) , ~ ( tare ) the input, output and white noise output disturbance, respectively, k 2 1 is the system time delay expressed as an integer multiple of the sampling interval denoted T, and q-l is the backward shift operator defined as qPiy(t) y ( t - i). Note that the first k - 1 coefficients in B(q-') are equal to zero to accommodate the time delay.
0
ai = 0 for all i - full quasilinearisation with A(q-') polynomial; ai = 1 for all i - full quasilinearisation with B ( q - l ) polynomial.
Since the resulting controller gains will be dependent on numerator and denominator of the model transfer function - they are dependent on ai. As a consequence the closed loop performance will depend on the quasilinearisation.
2.3 Steady-state behaviour of bilinear systems
The characteristic which makes a negative bilinearity attractive is the asymptotic saturation-like phenomena. This asymptotic behaviour depends on the magnitude and sign of system parameters. For a discrete first order bilinear system the asymptotes are given by:
=
2.2 Quasilinearisation Equation (1) comprises nonlinear (or bilinear) term(s). Essentially this bilinearity, can be quasilinearised with either input or output giving a linear description at the sampling instants. Thus by analysing the model at a given sample time, it can be 'loosely' regarded as a linear model with input or output dependent coefficients (Ziemian et al., 2000). However there is no reason why this method of quasilinearisation cannot be apportioned, with the bilinear terms being combined such that both numerator and denominator are affected. In such a case the quasilinearisation parameter a is required to be introduced. Then equation (1)will take the following form (for reasons of clarity, it is assumed that ~ ( t=)0 and is omitted) :
A(q-l)y(t) =%-')u(t)
For a system in which 61 > 0, ql < 0 and 0 5 a1 < 1,the steady-state characteristic is given in Fig. 1. yss L-.......-
(4)
n,
+ C [ai+ (1- ai)]qiy(t - i)u(t- i - k + 1) i= 1
For a first order bilinear system with unit delay (i.e. n, = l,nb = 1,k = 1) the equivalent discrete transfer function takes the form:
________________________
Fig. 1. Steady-state characteristic for system with bl > 0, ql < 0 and 0 5 a1 < 1 - the dashed lines represent the asymptotes. The characteristic in Fig. 1 has to be taken into account when specifying the magnitudes of the desired input and output signals.
3. BILINEAR PIP
where
iil = a1 - (1- a1)q1u(t - 1)
(6)
The idea of TDC assumes the use of only discretetime models and discrete-time data. For a discrete SISO system the NMSS representation takes form (Dixon, 1996):
= bi + ~ q i y (-t 1) (7) In principle there are no limits on the quasilinearisation parameter a because the open loop system remains the same regardless of the value of a. However, if the upper and lower limits on control input are known - the limits on C Y ~can be set up in the controller design phase to achieve specific goals. Two cases of special interest are:
z(t)= F z ( t - 1) + gu(t - 1) + dy&) Y ( t )=H 4 t ) where the state vector is of the form:
374
(9)
s ( t ) T = [ y ( t ) y ( t - 1).. .y(t - 71,
+ 1)
3.1 Pole assignment
a(t - 1)...u(t - 126 $- 1)z ( t ) ] (10) with the integral-of-error term z ( t ) defined as:
Whilst the notion of poles is limited to linear systems theory, consideration in this paper is given to equivalent linearised systems at the sampling instants. Loosely, in the specific case of a bilinear system, the control input may need to be bounded in order to 'guarantee' stability.
z ( t ) = zz(t - 1)+ tYd(t) - y(t)l
(11) where y d ( t ) is the set point or reference. Matrices F , g , d and H are as defined in (Dixon, 1996), with elements appropriately amended to include the quasilinearised parameters.
The gain vector k is calculated according to the following algorithm (Dixon, 1996):
The PIP control law takes the familiar SVF form:
u(t)= - k z T ( t ) where the gain vector k is given by:
(12) where d d and pol are vectors of coefficients for the desired closed and open loop charaderistics polynomials, respectively, i.e.:
T
k = [fo f i . . .fna-191Q2gnb-1 - k ~ ] (13) in which the fi and g j are the coefficients of the feedback and input filter polynomials, and kl is the coefficient of the integral term. The standard implementation of the PIP servomechanism control (Taylor et a$., 1996) is shown in Fig. 2.
polT = [61 - 162 - 61 6 3 - 6 2 . . .
(16)
- an, - O...O]
. ..&,, S is the partitioned matrix I
S = [g : Fg : F2g : . .. : Fna+nb-l91
I
and
Fig. 2. Standard form of PIP. where: A = (1 - q-'), so that k l / A provides integral action, F ( 4 - l ) = fo f 1 q - l ... fn,-lq-(na-l) is the feedback filter, G(q-') = 1 g1Q-l . . . gnb-lq-(nb-l) is the input filter.
+ +
(17)
+ + + +
M=
The gain vector k can be found by the use of any suitable control algorithm. The founders of linear PIP have shown how pole assignment and optimal control can be employed to calculate the required coefficients. The first requires the specification of the desired pole location, while in the second, the weighting matrices used in the performance index need to be chosen. For SISO systems the specification of desired poles would appear to be easier and more straightforward. The considerations in this paper are limited to the SISO case and the pole assignment strategy utilized.
0 0
0
-&n,
0 0
0 0
0 0
0 0 0 0
1 61 - 1 - 61
Properties of the controller such as robustness and disturbance rejection capability depend on its structure (Taylor et a$., 1996). In addition, the control input, in practice, is implemented in an incremental form, which can be straightforwardly realized. This allows the inclusion of both magnitude and rate constraints without additional complications, and integral wind-up is also prevented. In this paper only results regarding the standard form of PIP are presented.
62
0 0 0 0
0 1 61 - 1
...
... ...
...
... ... ...
...
...
...
0 0 0 0
..
0' 0 0 0
.
..
0
0
0
0
0
0
.
Note that when adopting a bilinear approach the gain vector is required to be recalculated at each time step. This is due to the fact that the quasilinearised model parameters are in fact input/output dependent quantities.
375
4. SIMULATION STUDIES The simulation studies presented in this paper are limited to the case of a first order bilinear system with a negative bilinearity and a unit delay. The model is given by a1 = -0.8, bl = 0.4 and r ~ 1= -0.1, with time delay k = 1. For such a system it is instructive to observe the impact of the a parameter on controller performance. For the bilinear case the gains can be represented as follows:
d l + -& + l ] (18) bi bi with 61 and 6, defined as in (6) and (7), respectively (note that the input filter G(q-') = 1). With reference to quasilinearised equation (7), it has been found in simulation studies that in the case of the 61 parameter it is better to use the most recent value of y ( t ) instead of g ( t - 1).The number of closed-loop poles to be assigned in the controlled system is n, + nb = 2, and choice of closed loop poles gives rise to the closed loop polynomial coefficients dl and dz. In this case one pole is arbitrarily assigned to zero and closed loop performance is achieved by positioning the other pole along the real axis. The model and assumed parameters are similar to that obtained from an identification exercise conducted on a high temperature furnace used in steel production. Fig. 3 illustrates how a can influence the closed loop performance, in the case where the closed loop poles are chosen to be [0,0.2]. kT =
[-(&+at)
1-
0.8 -
achieved by making use of a pseudo random binary sequence (PRBS) to excite the open-loop bilinear system. Model structure selection and estimation was carried out by the use of simplified refined instrumental variable (SRIV) (Young, 1985). The linear system is of first order with unit delay, with the following coefficients all = -0.7839 and bll = 0.3360 (the subscript 1 denotes parameters of a linearised model). Such a linearized model replicates the behaviour of the simulated bilinear system quite well, which is confirmed by the value of Young Identification Criteria (YIC) of well below -10 and other indices provided by the Captain Toolbox. 4.1 Choice of a pammeter This section investigates the factors affecting choice of the quasilinearisation parameter a. It is found that one of the main factors on which the choice of quasilinearisation parameter depends is the sign of coefficients and the magnitude of control input for which a system is to be operated. To obtain a suitable value of a the measure of performance used here is the integral of absolute errors (IAE) between reference signal (output of a prefilter with added delay) and output of the system. It is interesting to note that when 0 5 a 5 1, the value of a which gives rise to a minimum IAE depends on the desired closed loop pole locations. In particular, when the desired pole is close to the 'natural pole' of the linear part of the system (e.g. 0.8) the value of a = 1 is found to be best, implying full quasilinearisation with B(q-'). The 'optimum' values of a for various pole locations in the range 0 - 1,when the reference input is set to 1.0, are given in Fig. 4.
e
3 0.6rn
2
0.4 0.2 -
O
;
;
4
No.5of samples 6
;
8
9
Fig. 3. Impact of a parameter on the output. The desired response is given by the dash-dot line, the solid (thick) line represents the response for linear PIP, and the solid (feint), dashed and dotted lines represent the responses of the resulting bilinear PIP schemes. It may be observed that the performance of bilinear controller is better for all values the quasilinearisation parameter investigated; however in general this may not always be the case. Note that to realize the linear PIP, a linearized model is required to be identified. This was
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Fig. 4. Best value of a vs desired pole location. Extending the study to include the magnitude of the reference input in the range 0 - 1.5 leads to the results shown in Fig. 5 It is interesting to note that for lower values of reference, the range for which a = 0 (i.e. full quasilinearisation with A ( q - l ) ) is increased.
illustrated in Fig. 7. The x, y and z axes show the perturbations on the a l , bl and 171 parameters, respectively. The values on the grid show the best values of a obtained. In this case the input reference is 1.0 and the poles are arbitrarily chosen as [0;0.2]. Again the ‘best’ values of a is that which results in a minimum IAE with respect to the desired responce in Fig. 3. -0.09
.92
r- -0.1
Fig. 5. Best a parameter vs pole location and magnitude of reference.
.94 “.,a
This behaviour is also visible in Fig. 6. It can be concluded that if the linear dynamics of the system are fast and desired pole slow - the value of a shall be 1. Quite often the task to be performed by a controller is to force a system with slow dynamics to respond faster. If this is the case then the qusilinearisation parameter should be chosen more carefully. Its ‘optimal’ value could lie within the range 0 < a < 1. Fig. 6 shows the ‘best’ values of a when the reference level is again fixed at 1.0 and the closed loop pole location and natural pole of the linear part of the system are varied over the range 0 - 1.0. Similar observations can be made when varying other model parameters. For example if bl is increased, the range over which a = 0 is increased. Increase in the bilinear coefficient 171 also affects the value of a.
Fig. 7. Best a parameters in perturbed models.
5. CONCLUSIONS In this paper a novel approach of apportioning the bilinear coefficient to achieve an effective quasilinearisation has been investigated. This allows greater flexibility and more freedom in designing the bilinear controller using a proportionalintegral-plus technique. Preliminary results indicating the potential improvement together with quasilinearisation parameter dependency on other model parameters of the system together with the operating set-point have been shown. Further work is currently on-going in order t o evaluate the advantages of the resulting bilinear controllers based on the proposed quasilinearisation.
1
0.8
4
0.6
v)
0.4 0.2 0 0
6. ACKNOWLEDGEMENTS 0
a,
Desired pole location
Fig. 6. Best a parameter versus desired pole location.
a1
The authors wish t o acknowledge the use of the Captain Toolbox, courtesy of its creators from the Centre for Research on Environmental Systems and Statistics (CRES), Lancaster University, Lancaster, United Kingdom.
value and
7. REFERENCES
In reality the model of a system is only an approximation of the system itself, with errors in modelling the actual behaviour being inevitable. With this in mind, it is useful and instructive to explore the effect on a of variations in the model parameters. The result of such a study is
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