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Non linear Standard Regulators
Copyright © l FAC 11 th Trienn ial World Congress, Ta lli n n, [ slOn i", USSR, 1990
NON LINEAR STANDARD REGULATORS A. Balestrino*, A. Brambilla**, C. Scali** and A. Landi* *Department of Electrical Systems and Automation, University of Pisa, Via Diotisalvi, 2, 56100 Pisa, Italy **Department of Chemical Engineering, University of Pisa, Via Diotisalvi, 2, 56100 Pisa, Italy
ABSTRACT The problem of designing Non Linear Standard Regulators (NLSR) is faced for uncertain systems with time delay. Some recent rules for robust tuning of conventional controllers are recalled. The favourable characteristics of the Variable Structure Control (VSC) approach to the design of NLSR for system without delay are ghown by a comparison of performance with Standard Regulators in the presence of la!ge param~ter ~ariations. Th~ VSC ~pproach is then extended, by means of a. SUitable modification of the Smlt~ predictor control structure, to systems with time delay. A control law, able to give acceptable performance for different process parameters, is then proposed and simulation results are discussed . KEYWORDS : Variable structure control; Robust control; Nonlinear systems; Delays. INTRODUCTION Most used industrial regulators are of standard (PID) type. In a typical proctlss control instrument engineers and plant personnel have to select, install and operate hundreds of control loops. Therefore tuning rules for standard regulators are of relevant interest: the Ziegler-Nichols method (1942), is still very popular, but the subject has received new attention in recent years (Martin, Corripio and Smith, 1976; Rivera, Skogestad and Morari, 1986; Hwang and Chang, 1987). The main goal is to achieve desirable response also when process parameters are poorly known or may vary in a wide range. One of the major improvement in the technology of regulators is due to the introduction of microprocessors. Their greater flexibility has led to various alternatives for automatic tuning and for self-optimizing regulators . PID controllers can be successfully utilized in processes where time constants, rather than dead times, dominate the dynamic behaviour. When time delays are dominant, a significant improvement in the overall performance can be attained by using standard controllers in a control structure as the Smith predictor. Many different structures of PID controllers have been reported in the literature up to the point that the integral action over the error is the only common feature clearly recognizable. In practice almost all schemes are suitably modified; for example a PI controller may be coupled with an external logic to satil!fy inequality constraints in the state variables (Glattfelder, Huguenin and Schaufelberger, 1980). 323
The.PID self-tuners can be broadly classified into three groups: indirect and direct self-tuners based on pole assignment and controllers based on an explicit criterion of minimization. The implementation aspects require the introduction of practical precautions in order to take into account reset wind-up, nonlinearities or specific problems of adaptive controllers (Astrom, 1983; Wittenmark and Astrom, 1984). Often such practical precautions lead to the mtroduction of intentional nonlinearities such as saturations and bang-bang actions (Suh, Hwang and Bien, 1985). As a matter of fact the family of regulators, standard or self- tuners or nonlinear, is so large that a compilation of a complete list of references would be a too heavy burden to carry over. Henceforth only the applications related to standard regulators and their modifications are considered. Intentional nonlinearities in order to realize robust controllers for electrical drives (and robots) have been systematically introduced in the general framework of the theory of Variable Structure Systems (Utkin, 1978). The main issues to be considered are robustness of the control system and ease of tuning. The classical structure of PID standard regulators is retained, but VSC characteristics are introduced so that a typical nonlinear standard regulator is implemented. The variable structure control approach emerged in the last decade as a systematic and effective design procedure allowing a complete specification of the transient behavior by means of the sliding modes. , Strong stability characteristics, simple control laws and high speed of response are obtainable through the use of such design approach.
Moreover, typical plant nonlinearities, as saturation, are simply accounted for and suitable modifications of the control laws can be introduced in order to eliminate the phenomenon of chattering.
by adopting controller parameters given by:
Kc = 1 T+ B/2 I(p (e + 1)B
In this paper recent rules for robust tuning of standard regulators for process with time delay are recalled. The VSC approach to NLSR design is then presented for systems without delays and a comparison of performance with standard regulator is shown. The VSC approach is then extended to plants with time delay, by adopting a suitable modification of the Smith predictor control structure. Simulations results and a control law able to give acceptable performance for different process parameters, also in the presence of uncertainty in the delay, are presented. In order to take the exposition as simple as possible only the reference tracking problem will be addressed.
ROBUST TUNING OF STANDARD CONTROLLERS PID controllers are the most used in process control and in particular in chemical process control. The plants are characterized by open-loop stable and sluggish models with severe modeling problems. Robustness, reliability and steadystate performance are the most important issues, but of course the shaping of the transient response is also of interest.
Kp
The PI and PID controller structures can be derived by using a first order Pade approximation of the time delay and then neglecting terms of higher order than those appearing in equations (3) and
(4) . Approximations, introduced by forcing the structure of the ideal controller (eq. 2) to conventional ones (eq. 3 and eq. 4), make necessary the adoption of a dimensionless tuning parameter e = >'/B. Its value must be specified in order to guarantee that the actual response will not differ more than an allowed amount from the desired one. To be noted that: - only the controller gain (Kc) depends on the parameter e, which determmes the speed of response; - the integral time (1"1) and the derivative time ~~D) depend only on the process constants T and
An example is given in Fig. 1 for a PI controller: values of the parameter e reported in Fig. 1 guarantee the achievement of a response with a limited overshoot (5% maximum), for the worst case uncertainty.
(1)
o
>.s + 1 - e- o•
PI con t roll er 2.i I - - - - - - , - O - - - - - - - " stc , . - - - - - - - - - - - . J 1
.....
2.
order + de la y mod el
I--~.,.-----------____'
~ 15 1 - - - -.......-~.. ~-------____.J •
1
(7)
Tuning correlations for robust performance are given assuming, as performance index, a small amount of overshoot in the time response of the system, in order to have a response as close as possible to an overdamped one.
H an underdamped closed loop response of first order ~ype is desired (with a time constant equal to >'), a non conventional controller structure should be used:
+1
1"D
A detuning of controllers, by increasing the value of the parameter e, is especially necessary to account for uncertainty in the model of the process. The.main causes of uncertainty are considered to derive from errors in time delay and in model order reduction.
A First Order Plus Time Delay (FOPTD) model of the process is assumed:
= 1's
(6)
T+ B/2 TB = 2T+B
,
A simple method to obtain tuning rules for standard controllers able to give a desired closed loop response is reported in (Brambilla, Chen' and Scali, 1989) . It is summarized in the following.
c
1"1 =
- Kc and 1"1 have the same expressions for PI and PID controllers, but values of the tuning parameter e are different because different approximations are introduced by the two type of controllers.
Tuning of continuous unconstrained controllers for chemical processes has been extensively studied ( Cohen and Coon, 1953; Martin, Corripio and Smith, 1976; Rivera, Skogestad and Morari, 1986; Hwang and Chang, 1987). Simple tuning relations for the controller parameters are derived for different performance indices and for different inputs."
-0. p = K pe 1+ T8
(5)
I- -
-"...
(2)
A close approximation to the desired response can be obtained by using conventional PI and PID controllers: Fig.' 1. Robust tuning correlations for PI controller and FOPTD process. (Continuous line: Nominal case; dotted line: reduction from higher order process to first order model; dashed line: 20% error in time delay)
(3) (4)
324
Analogous correlations for robust performance are given for PID controllers and for different process models, as second order plus time delay and right half plane zero systems. By extensive simulation for a wide range of 'process parameters variations, it is shown that a superior performance with respect to classical tuning rules for conventional controllers is achieved, while the loss of performance with respect to advanced controller structures is negligible. The favourable characteristics of this tuning technique is that a smooth approach to the steady state is guaranteed at the maximum speed allowed by the assumed performance specification. Of cour,se, as the controller parameters are constant, the speed of response may become slow for cases in which a large amount of uncertainty is present. A time varying parameters controller has the potential of giving better performance and also to take into account the presence of constraints on the control action. The control system studied in this paper is represented in Fig. 2: a simple FOPTD model has been assumed, but saturation limits on the plant input have been considered explicitly. This nonlinearity is present in all actuators or power amplifiers and therefore it must be considered as a very typical element of a physical plant. The saturation limits are assumed known; the plant parameters K p , T are assumed largely uncertain.
near transfer function and N is a nonlinear block. By requiring G(s)be strictly positive real and N be passive, an hyperstable system is obtained. Strong stability characteristics are assured, by using simple control laws.
:~-:
~
•
:-----,::
Fig. 3. The hyperstability structure A high speed of response is obtainable assuming the linear block with high gain and the highest admissible levels for the nonlinear part such as relay or saturation. In this way the control law usually becomes discontinuous along a number of switching hypersurfaces; on the intersections of these hypersurfaces the system dynamics is in the sliding mode and the system becomes insensitive to p.arametric variations and noise disturbances (Zinober, EI-Ghezawi and Billings, 1982). Some of the major problems are due to chattering and unlocking. If chattering occurs, then the plant input is discontinuous and this often is not physically admissible. Unlocking results in large deviations from the sliding regime and it can be due to disturbances or to a loss in the power supplied from actuators. These phenomena can be avoided by introducing some variations such as adaptive sliding (Harashima, Hashimoto and Konda, 1985) or self-tuning variable structure control systems ( Wang and Owens, 1989).
The presence of the nonlinearity can be easily accounted for by using the variable structure control approach; moreover in this way no parameter identification is generally needed allowing a faster transient response. Large uncertainty in the gain Kp and in the plant parameter T can be easily dealt with, while uncertainty in the time delay () requires some special precautions.
The mathematical models often give a rough description of our plants and devices; therefore in order to cope with this uncertainty we can use at the beginning the hyperstability approach and then remove the implementation difficulties related to a VSC by a suitable modification of the control law (Balestrino, Gallanti and Kalas, 1988a). A PI with the addition of a suitable bangbang action gives a very satisfactory performance (Balestrino, Gallanti and Kalas, 1988b), so that a simple NLSR can be designed by taking into account some modifications in order to avoid problems due to chattering and unlocking.
Fig. 2. First Order Plus Time Delay (FOPTD) process with input saturation
VSC APPROACH TO NLSR DESIGN APPLICATION OF NLSR
It :s well known that a large class of nonlinear plants can be controlled by using adaptive model following control with nonlinear high gain subsystems (Balestrino, De Maria and Zinober, 1984). A complete specification of the transient behavior is possible by means of sliding modes, the structure of which can be established by using Lyapunov theory or hyperstability theory (Popov, 1973), usually assuming discontinuous laws (Balestrino and Innocenti, 1989).
On the basis of this previous experience, a modified structure of adaptive regulator, able to avoid the above mentioned problems, is proposed. Advantages of the NLSR over a conventional PI controller for a first order system (no delay) are evident from Fig. 4 and Fig. 5, where responses to a step set-point changes are shown in the ~re sence of a saturation on the control action. The NLSR controller is still chosen with Proportional and Integral action, but with time variant Pafameters.
The design via hyperstability leads to a typical scheme as in Fig. 3, where G(s) denotes a li-
325
Plont
When process parameters are considered perfectly known (nominal case), results for the two controllers are identical. This is shown in Fig. 4, where the integral time constant 1"[ of the linear PI controller has been assumed equal to the time constant of the process T.
Fig. 6. The proposed control system: NLSR in a Smith predictor control structure 0.8
Obviously, if a perfect matching between model and plant is assumed, the behaviour of the system is the same as in the case without delay. The basic ideas to select a suitable control law can be summarized as follows: i) the integral action is essentially needed when the error is small, while the proportional action is essentially needed when the error signal and the saturation level of the actuator are of the same order of magnitude; ii) the signals generated by the controller must be always compatible with the saturation constraints; iii) for large mismatch between plant and model parameters, the choice of the controller parameters has to be more conservative.
°0~--~0.~'--~----~1.,----~--~--~
Fig. 4. Overlapping time responses of NLSR. and PI Standard for a process without delay in the nominal case (no uncertainty)
In the presence of uncertainty in the process parameters, maintaining the same controller constants, performance of the linear PI controller deteriorate, while the NLSR is not affected by parametric variations. This is shown in Fig. 5, where the process time constant T changes from 1 to 10.
Among the many choices of possible control laws, the one proposed here is easy to implement on a microprocessor. The constants of of the proportional and integral component of the control action are given respectively by: "IP =
--
1.'r---~--------~--------------' /-
/
/
-.
~---------------=-=-==-==4
________~"I~r~o______~ (1 + "IP8 Ib. 1)(1 + "IPele!)
(8)
(9)
0.0
where: is the absolute value of the error; - (b. = () - 8) is the uncertainty on the time delay; - "fpo and "110 are values depending on the process gain and on the saturation level of the control action; - "floand "IPO take into account the effect of mismatch on the time delay; they must be related also to the ratio 8IT in order to reduce gains furtherly, for plants in which time delay is dominant respect to the time constant; - "IPe and "lIe introduce the desired adaptive characteristics in the controller parameters; - 6 is a very small value constant to be selected in order to give very large values of the integral ·gain when lel = 1b.1 = o.
- lel
oL-____________________~----~
o
t.~
Fig. 5. Time responses of NLSR (solid line) and PI Standard (dashed line) for a process w~thout delay in the presence of parameter uncertamty
When in addition to input saturation, a time delay is present in the process, we propose to apply a NLSR to a Smith predictor control structur~, which makes use of an estimated model of the lInear part of the plant (Fig. 6).
Adopting this algorithm, the integrator windup is an extremely rare event; however an anti-windup device can be added if necessary. Also, a parameter identification procedure can be avoided without a heavy worsening of the global performance, by introducing some simple automatic gain adjustment. For instance, assume that
326
As it can be seen by comparing the two cases, performance of the NLSR is affected by an increase in the ratio 0IT, as the problem of controlling delay dominant processes becomes more difficult. An analogous role is played by an increase in delay uncertainty. A control law more complex than the one adopted here (eq. 8 and eq. 9) can give better results in this case.
the parameters defining the proportional action have been designed with reference to the lowest gain of the plant: in this case 1po can be reduced by introducmg in eq. 8 a factor which depends on the derivative of the error, evaluated by the microprocessor on the basis of the difference between errors at two successive sampling times. These considerations are supported by simulation results. For the system reported in Fig. 6, assuming a time delay 0 = 1 and an uncertainty I~I equal to 10% on the delay, different ratios oC-OfT have been investigated.
Delay dominant processes with large uncertainty will be the subject of further investigations. Also, future research will be devoted to: - obtaining correlations between controller and process parameters, in order to make possible an easier tuning; - two-degree of freedom structures, in order to allow independent controllers adjustment both for reference tracking and disturbance suppression.
Values of controller parameters for 0IT varying in the range: 0.1 - 1, are: 1po = 0.01 OfT; 1 P 9 = 7o.00fT; 1Pe = 2.0; 110 = 20.0; 119 = 70.00/1'; 1Ie = 2000.; 6 = 0.01
CONCLUSIONS
Results of the simulation are shown in Fig. 7 and in F,ig. 8, as response to step set-point changes for the cases 0IT = 0.1 and 0IT = 1.
The Variable Structure Control approach to the design of Non Linear Standard Regulator has been presented as an effective way of accounting for problems arising from nonlinearities in the system and uncertainty in the process parameters.
The two curves in each figure refer respectively to an underestimation (0 < 0) and to an overestimation (0) 0) of the plant delay 0; better performance (smaller overshoot) is obtained for the case of overestimation of the delay: this confirms a previous result presented by Hocken, Saleh and Marshall (1983).
By adopting a simple control law, with Proportional Integral action, advantages over standard PI controllers have been shown in the case of large parameter uncertainty, for systems without delays. By using a modified Smith predictor control structure and accounting for the delay uncertainty in controller tuning, the same control law gives acceptable performance also in the presence of uncertain delay in the process. Further work will be devoted to delay dominant systems with large uncertainty, to problems of simultaneous reference tracking and disturbance suppression in two-degree of freedom structure and to correlate controller and process parameters.
Time
Fig. 7. Time response of the proposed NLSR for a process with time delay: 9fT = 0.1, I~I = 10% O. (Solid line: < 0; dashed line: 0)
e
e>
NOMENCLATURE
"
c e
./
Kc Kp T :
D.'
1P : DD~--~--~----7---~----~--~
Fig. 8. Time response of the proposed NLSR for a process with time delay: OfT = 1.0, I~I = 10% O. (Solid line: 0 < 0; dashed line: 0 > 0)
11
.:
~
).
: :
"D "I
: :
o
:
o :
327
Tuning parameter Error Standard controller proportional gain Process gain Process time constant NLSR proportional gain NLSR integral time Uncertainty in process delay Time constant of desired response Standard controller derivative time Standard controller integral time Process time delay Model time delay
REFERENCES Astrom, K. J. (1983). Theory and applications of adaptive control: a survey. Automatica, EO, 471-483. Balestrino, A., G. De Maria and A.S.1. Zinober. (1984). Multivariable variable structure adaptive model following control systems. Automatica, 5, 559-568. Balestrino, A., G. Gallanti and D. Kalas. (1988a). High performance dc drive using variable structure control theory. Proc. ICEM, 9539-543. Balestrino, A., G. Gallanti and D. Kalas. (1988b). On the implementation of variable structure control systems. IMACS Symp. 267-272. Balestrino, A. and M. Innocenti. (1989). The hyperstabilitr approach to VSCS design. In A.S.1. Zinober (ed.). Variable Structure Control Systems.Peter Peregrinus Publ., London. Brambilla A., S. Chen and C. Scali. (1989). Tunhtg of conventional controllers for robust performance . IFAC Symp. Low Cost Automation.(to appear). Cohen, G. D. and G. A. Coon. (1953). Theoretical consideration for retarded controllers. 1rans. ASME, 75, 827-833. Glattfelder, A. 11., F. Huguenin, and W. Schaufelberger. (1984). Microcomputer based self-tuning and selt-selecting controller. Automatica, 16, 1-8. Harashima, F., H. Hashimoto and S. Kondo. (1985). MOSFET converter-fed position servo systems with sliding mode control. IEEE, Trans. on lE, IE-9£, 238-244. Hocken~ R. D., S. V. Saleh and J. E. Marshall. (1983 . Time-delay mismatch and the"performance 0 predictor control schemes. lnt. J. Control, 98, 433-437. Hwang, S. and H. Chang. (1987). A theoretical examination of closed loop properties and tuning methods of single loop PI controllers. Chem. Eng. Sci., 4£, 2395-2399. Martin, J.Jr., A. B. Corripio and C. L. Smith. (1976). How to select modes and tuning parameters from simple process models. ISA 1rans., 15,314-319 . .Popov, V. M. (1973). Hyperstability of control systems. Springer Verlag, Berlin. Rivera, D. E., S. Skogestad and M. Morari. (1986). Internal model control. 4: PID controller design. lnd. Eng. Chem. Process Des . De"., E5,252-265. Suh, I. S., S.H.Hwang and Z. Bien. (1985). A design and experiment of speed controller with PI plus bang-bang action for a DC servomotor with transistorized PWM drives. IEEE 1rans. on lE, IE- 91, 338-345. Utkin, V. I. (1978). Sliding Modes and their Application in Variable Structure Systems. (1st. English Version). MlR, Moscow. Wang, L. and D. H.Owens (1989). Self tuning/adaptive control using approximate models. Proc. ICCON, TP 6-6 . " Wittenmark, B. and K.J. Astrom. (1984). Practical issues in the implementation of self tuning control. Automatica, £0, 595-605. Ziegler, J. G. and N. B. Nichols. (1942). Optimum settings for automatic controller. Trans. ASME, 75, 827-833.
Zinober, A. S. I., O. M. E. EI-Ghezawi and S. A. Billings. (1982). Multivariable variable structure adaptive model following control systems. lEE Proc. D, 1£9,6-12.
328
NON LINEAR STANDARD REGULATORS A. Balestrino*, A. Brambilla**, C. Scali** and A. Landi* *Department of Electrical Systems and Automation, University of Pisa, Via Diotisalvi, 2, 56100 Pisa, Italy **Department of Chemical Engineering, University of Pisa, Via Diotisalvi, 2, 56100 Pisa, Italy
ABSTRACT The problem of designing Non Linear Standard Regulators (NLSR) is faced for uncertain systems with time delay. Some recent rules for robust tuning of conventional controllers are recalled. The favourable characteristics of the Variable Structure Control (VSC) approach to the design of NLSR for system without delay are ghown by a comparison of performance with Standard Regulators in the presence of la!ge param~ter ~ariations. Th~ VSC ~pproach is then extended, by means of a. SUitable modification of the Smlt~ predictor control structure, to systems with time delay. A control law, able to give acceptable performance for different process parameters, is then proposed and simulation results are discussed . KEYWORDS : Variable structure control; Robust control; Nonlinear systems; Delays. INTRODUCTION Most used industrial regulators are of standard (PID) type. In a typical proctlss control instrument engineers and plant personnel have to select, install and operate hundreds of control loops. Therefore tuning rules for standard regulators are of relevant interest: the Ziegler-Nichols method (1942), is still very popular, but the subject has received new attention in recent years (Martin, Corripio and Smith, 1976; Rivera, Skogestad and Morari, 1986; Hwang and Chang, 1987). The main goal is to achieve desirable response also when process parameters are poorly known or may vary in a wide range. One of the major improvement in the technology of regulators is due to the introduction of microprocessors. Their greater flexibility has led to various alternatives for automatic tuning and for self-optimizing regulators . PID controllers can be successfully utilized in processes where time constants, rather than dead times, dominate the dynamic behaviour. When time delays are dominant, a significant improvement in the overall performance can be attained by using standard controllers in a control structure as the Smith predictor. Many different structures of PID controllers have been reported in the literature up to the point that the integral action over the error is the only common feature clearly recognizable. In practice almost all schemes are suitably modified; for example a PI controller may be coupled with an external logic to satil!fy inequality constraints in the state variables (Glattfelder, Huguenin and Schaufelberger, 1980). 323
The.PID self-tuners can be broadly classified into three groups: indirect and direct self-tuners based on pole assignment and controllers based on an explicit criterion of minimization. The implementation aspects require the introduction of practical precautions in order to take into account reset wind-up, nonlinearities or specific problems of adaptive controllers (Astrom, 1983; Wittenmark and Astrom, 1984). Often such practical precautions lead to the mtroduction of intentional nonlinearities such as saturations and bang-bang actions (Suh, Hwang and Bien, 1985). As a matter of fact the family of regulators, standard or self- tuners or nonlinear, is so large that a compilation of a complete list of references would be a too heavy burden to carry over. Henceforth only the applications related to standard regulators and their modifications are considered. Intentional nonlinearities in order to realize robust controllers for electrical drives (and robots) have been systematically introduced in the general framework of the theory of Variable Structure Systems (Utkin, 1978). The main issues to be considered are robustness of the control system and ease of tuning. The classical structure of PID standard regulators is retained, but VSC characteristics are introduced so that a typical nonlinear standard regulator is implemented. The variable structure control approach emerged in the last decade as a systematic and effective design procedure allowing a complete specification of the transient behavior by means of the sliding modes. , Strong stability characteristics, simple control laws and high speed of response are obtainable through the use of such design approach.
Moreover, typical plant nonlinearities, as saturation, are simply accounted for and suitable modifications of the control laws can be introduced in order to eliminate the phenomenon of chattering.
by adopting controller parameters given by:
Kc = 1 T+ B/2 I(p (e + 1)B
In this paper recent rules for robust tuning of standard regulators for process with time delay are recalled. The VSC approach to NLSR design is then presented for systems without delays and a comparison of performance with standard regulator is shown. The VSC approach is then extended to plants with time delay, by adopting a suitable modification of the Smith predictor control structure. Simulations results and a control law able to give acceptable performance for different process parameters, also in the presence of uncertainty in the delay, are presented. In order to take the exposition as simple as possible only the reference tracking problem will be addressed.
ROBUST TUNING OF STANDARD CONTROLLERS PID controllers are the most used in process control and in particular in chemical process control. The plants are characterized by open-loop stable and sluggish models with severe modeling problems. Robustness, reliability and steadystate performance are the most important issues, but of course the shaping of the transient response is also of interest.
Kp
The PI and PID controller structures can be derived by using a first order Pade approximation of the time delay and then neglecting terms of higher order than those appearing in equations (3) and
(4) . Approximations, introduced by forcing the structure of the ideal controller (eq. 2) to conventional ones (eq. 3 and eq. 4), make necessary the adoption of a dimensionless tuning parameter e = >'/B. Its value must be specified in order to guarantee that the actual response will not differ more than an allowed amount from the desired one. To be noted that: - only the controller gain (Kc) depends on the parameter e, which determmes the speed of response; - the integral time (1"1) and the derivative time ~~D) depend only on the process constants T and
An example is given in Fig. 1 for a PI controller: values of the parameter e reported in Fig. 1 guarantee the achievement of a response with a limited overshoot (5% maximum), for the worst case uncertainty.
(1)
o
>.s + 1 - e- o•
PI con t roll er 2.i I - - - - - - , - O - - - - - - - " stc , . - - - - - - - - - - - . J 1
.....
2.
order + de la y mod el
I--~.,.-----------____'
~ 15 1 - - - -.......-~.. ~-------____.J •
1
(7)
Tuning correlations for robust performance are given assuming, as performance index, a small amount of overshoot in the time response of the system, in order to have a response as close as possible to an overdamped one.
H an underdamped closed loop response of first order ~ype is desired (with a time constant equal to >'), a non conventional controller structure should be used:
+1
1"D
A detuning of controllers, by increasing the value of the parameter e, is especially necessary to account for uncertainty in the model of the process. The.main causes of uncertainty are considered to derive from errors in time delay and in model order reduction.
A First Order Plus Time Delay (FOPTD) model of the process is assumed:
= 1's
(6)
T+ B/2 TB = 2T+B
,
A simple method to obtain tuning rules for standard controllers able to give a desired closed loop response is reported in (Brambilla, Chen' and Scali, 1989) . It is summarized in the following.
c
1"1 =
- Kc and 1"1 have the same expressions for PI and PID controllers, but values of the tuning parameter e are different because different approximations are introduced by the two type of controllers.
Tuning of continuous unconstrained controllers for chemical processes has been extensively studied ( Cohen and Coon, 1953; Martin, Corripio and Smith, 1976; Rivera, Skogestad and Morari, 1986; Hwang and Chang, 1987). Simple tuning relations for the controller parameters are derived for different performance indices and for different inputs."
-0. p = K pe 1+ T8
(5)
I- -
-"...
(2)
A close approximation to the desired response can be obtained by using conventional PI and PID controllers: Fig.' 1. Robust tuning correlations for PI controller and FOPTD process. (Continuous line: Nominal case; dotted line: reduction from higher order process to first order model; dashed line: 20% error in time delay)
(3) (4)
324
Analogous correlations for robust performance are given for PID controllers and for different process models, as second order plus time delay and right half plane zero systems. By extensive simulation for a wide range of 'process parameters variations, it is shown that a superior performance with respect to classical tuning rules for conventional controllers is achieved, while the loss of performance with respect to advanced controller structures is negligible. The favourable characteristics of this tuning technique is that a smooth approach to the steady state is guaranteed at the maximum speed allowed by the assumed performance specification. Of cour,se, as the controller parameters are constant, the speed of response may become slow for cases in which a large amount of uncertainty is present. A time varying parameters controller has the potential of giving better performance and also to take into account the presence of constraints on the control action. The control system studied in this paper is represented in Fig. 2: a simple FOPTD model has been assumed, but saturation limits on the plant input have been considered explicitly. This nonlinearity is present in all actuators or power amplifiers and therefore it must be considered as a very typical element of a physical plant. The saturation limits are assumed known; the plant parameters K p , T are assumed largely uncertain.
near transfer function and N is a nonlinear block. By requiring G(s)be strictly positive real and N be passive, an hyperstable system is obtained. Strong stability characteristics are assured, by using simple control laws.
:~-:
~
•
:-----,::
Fig. 3. The hyperstability structure A high speed of response is obtainable assuming the linear block with high gain and the highest admissible levels for the nonlinear part such as relay or saturation. In this way the control law usually becomes discontinuous along a number of switching hypersurfaces; on the intersections of these hypersurfaces the system dynamics is in the sliding mode and the system becomes insensitive to p.arametric variations and noise disturbances (Zinober, EI-Ghezawi and Billings, 1982). Some of the major problems are due to chattering and unlocking. If chattering occurs, then the plant input is discontinuous and this often is not physically admissible. Unlocking results in large deviations from the sliding regime and it can be due to disturbances or to a loss in the power supplied from actuators. These phenomena can be avoided by introducing some variations such as adaptive sliding (Harashima, Hashimoto and Konda, 1985) or self-tuning variable structure control systems ( Wang and Owens, 1989).
The presence of the nonlinearity can be easily accounted for by using the variable structure control approach; moreover in this way no parameter identification is generally needed allowing a faster transient response. Large uncertainty in the gain Kp and in the plant parameter T can be easily dealt with, while uncertainty in the time delay () requires some special precautions.
The mathematical models often give a rough description of our plants and devices; therefore in order to cope with this uncertainty we can use at the beginning the hyperstability approach and then remove the implementation difficulties related to a VSC by a suitable modification of the control law (Balestrino, Gallanti and Kalas, 1988a). A PI with the addition of a suitable bangbang action gives a very satisfactory performance (Balestrino, Gallanti and Kalas, 1988b), so that a simple NLSR can be designed by taking into account some modifications in order to avoid problems due to chattering and unlocking.
Fig. 2. First Order Plus Time Delay (FOPTD) process with input saturation
VSC APPROACH TO NLSR DESIGN APPLICATION OF NLSR
It :s well known that a large class of nonlinear plants can be controlled by using adaptive model following control with nonlinear high gain subsystems (Balestrino, De Maria and Zinober, 1984). A complete specification of the transient behavior is possible by means of sliding modes, the structure of which can be established by using Lyapunov theory or hyperstability theory (Popov, 1973), usually assuming discontinuous laws (Balestrino and Innocenti, 1989).
On the basis of this previous experience, a modified structure of adaptive regulator, able to avoid the above mentioned problems, is proposed. Advantages of the NLSR over a conventional PI controller for a first order system (no delay) are evident from Fig. 4 and Fig. 5, where responses to a step set-point changes are shown in the ~re sence of a saturation on the control action. The NLSR controller is still chosen with Proportional and Integral action, but with time variant Pafameters.
The design via hyperstability leads to a typical scheme as in Fig. 3, where G(s) denotes a li-
325
Plont
When process parameters are considered perfectly known (nominal case), results for the two controllers are identical. This is shown in Fig. 4, where the integral time constant 1"[ of the linear PI controller has been assumed equal to the time constant of the process T.
Fig. 6. The proposed control system: NLSR in a Smith predictor control structure 0.8
Obviously, if a perfect matching between model and plant is assumed, the behaviour of the system is the same as in the case without delay. The basic ideas to select a suitable control law can be summarized as follows: i) the integral action is essentially needed when the error is small, while the proportional action is essentially needed when the error signal and the saturation level of the actuator are of the same order of magnitude; ii) the signals generated by the controller must be always compatible with the saturation constraints; iii) for large mismatch between plant and model parameters, the choice of the controller parameters has to be more conservative.
°0~--~0.~'--~----~1.,----~--~--~
Fig. 4. Overlapping time responses of NLSR. and PI Standard for a process without delay in the nominal case (no uncertainty)
In the presence of uncertainty in the process parameters, maintaining the same controller constants, performance of the linear PI controller deteriorate, while the NLSR is not affected by parametric variations. This is shown in Fig. 5, where the process time constant T changes from 1 to 10.
Among the many choices of possible control laws, the one proposed here is easy to implement on a microprocessor. The constants of of the proportional and integral component of the control action are given respectively by: "IP =
--
1.'r---~--------~--------------' /-
/
/
-.
~---------------=-=-==-==4
________~"I~r~o______~ (1 + "IP8 Ib. 1)(1 + "IPele!)
(8)
(9)
0.0
where: is the absolute value of the error; - (b. = () - 8) is the uncertainty on the time delay; - "fpo and "110 are values depending on the process gain and on the saturation level of the control action; - "floand "IPO take into account the effect of mismatch on the time delay; they must be related also to the ratio 8IT in order to reduce gains furtherly, for plants in which time delay is dominant respect to the time constant; - "IPe and "lIe introduce the desired adaptive characteristics in the controller parameters; - 6 is a very small value constant to be selected in order to give very large values of the integral ·gain when lel = 1b.1 = o.
- lel
oL-____________________~----~
o
t.~
Fig. 5. Time responses of NLSR (solid line) and PI Standard (dashed line) for a process w~thout delay in the presence of parameter uncertamty
When in addition to input saturation, a time delay is present in the process, we propose to apply a NLSR to a Smith predictor control structur~, which makes use of an estimated model of the lInear part of the plant (Fig. 6).
Adopting this algorithm, the integrator windup is an extremely rare event; however an anti-windup device can be added if necessary. Also, a parameter identification procedure can be avoided without a heavy worsening of the global performance, by introducing some simple automatic gain adjustment. For instance, assume that
326
As it can be seen by comparing the two cases, performance of the NLSR is affected by an increase in the ratio 0IT, as the problem of controlling delay dominant processes becomes more difficult. An analogous role is played by an increase in delay uncertainty. A control law more complex than the one adopted here (eq. 8 and eq. 9) can give better results in this case.
the parameters defining the proportional action have been designed with reference to the lowest gain of the plant: in this case 1po can be reduced by introducmg in eq. 8 a factor which depends on the derivative of the error, evaluated by the microprocessor on the basis of the difference between errors at two successive sampling times. These considerations are supported by simulation results. For the system reported in Fig. 6, assuming a time delay 0 = 1 and an uncertainty I~I equal to 10% on the delay, different ratios oC-OfT have been investigated.
Delay dominant processes with large uncertainty will be the subject of further investigations. Also, future research will be devoted to: - obtaining correlations between controller and process parameters, in order to make possible an easier tuning; - two-degree of freedom structures, in order to allow independent controllers adjustment both for reference tracking and disturbance suppression.
Values of controller parameters for 0IT varying in the range: 0.1 - 1, are: 1po = 0.01 OfT; 1 P 9 = 7o.00fT; 1Pe = 2.0; 110 = 20.0; 119 = 70.00/1'; 1Ie = 2000.; 6 = 0.01
CONCLUSIONS
Results of the simulation are shown in Fig. 7 and in F,ig. 8, as response to step set-point changes for the cases 0IT = 0.1 and 0IT = 1.
The Variable Structure Control approach to the design of Non Linear Standard Regulator has been presented as an effective way of accounting for problems arising from nonlinearities in the system and uncertainty in the process parameters.
The two curves in each figure refer respectively to an underestimation (0 < 0) and to an overestimation (0) 0) of the plant delay 0; better performance (smaller overshoot) is obtained for the case of overestimation of the delay: this confirms a previous result presented by Hocken, Saleh and Marshall (1983).
By adopting a simple control law, with Proportional Integral action, advantages over standard PI controllers have been shown in the case of large parameter uncertainty, for systems without delays. By using a modified Smith predictor control structure and accounting for the delay uncertainty in controller tuning, the same control law gives acceptable performance also in the presence of uncertain delay in the process. Further work will be devoted to delay dominant systems with large uncertainty, to problems of simultaneous reference tracking and disturbance suppression in two-degree of freedom structure and to correlate controller and process parameters.
Time
Fig. 7. Time response of the proposed NLSR for a process with time delay: 9fT = 0.1, I~I = 10% O. (Solid line: < 0; dashed line: 0)
e
e>
NOMENCLATURE
"
c e
./
Kc Kp T :
D.'
1P : DD~--~--~----7---~----~--~
Fig. 8. Time response of the proposed NLSR for a process with time delay: OfT = 1.0, I~I = 10% O. (Solid line: 0 < 0; dashed line: 0 > 0)
11
.:
~
).
: :
"D "I
: :
o
:
o :
327
Tuning parameter Error Standard controller proportional gain Process gain Process time constant NLSR proportional gain NLSR integral time Uncertainty in process delay Time constant of desired response Standard controller derivative time Standard controller integral time Process time delay Model time delay
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