16th Mediterranean Conference on Control and Automation Congress Centre, Ajaccio, France June 25-27, 2008
QFT-based design of PI+CI reset compensators: application in process control Angel Vidal and Alfonso Ba˜nos Abstract— This work approaches the problem of robust control for first-order plus time delay plants. The goal is investigate the potentials of a nonlinear/hybrid compensator previously developed by authors, referred to as the PI+CI compensator, in the control of a process with significant parametric uncertainty. To this end, the design of PI+CI has been made in two steps: first, a base robust PI compensator is designed by using QFT (Quantitative Feedback Theory) to satisfy (robust) relative stability and disturbance rejection specifications, and then a reset percentage of the integral term is chosen in order to improve the transitory response. In addition, a simple modification of the PI+CI compensator, consisting of the introduction of a reset band, is investigated regarding the improvement of the previous design.
I. INTRODUCTION This work is motivated by a problem of temperature control that is common to the thermal treatments of different products in the process industry. When a wide variety of products and different operation points (depending on the specific thermal treatment) exists, the temperature control problem is hard to solve by conventional means, for example using PID controllers with simple rules for tuning, or even autotuning. The reason is simple: the controller has to be designed for a proper treatment of the product in the presence of potentially large uncertainty with respect to the product. In addition, typically a product can be processed at different temperatures depending on the specific thermal treatment. Since the dynamic is highly non-linear and uncertain, a PID simply designed to work efficiently for a operation point and a particular product, usually degrades considerably its performance when the product and/or the operation point is changed [1]. In [1], QFT (Quantitative Feedback Theory) is used for the control of an industrial heat exchanger. QFT is a robust control technique that is especially well suited for control problems with large plant uncertainty. It has been developed since the early 60’s following the seminal works of Prof. Horowitz [9], and has been successfully applied to scalar/multivariable, LTI/nonlinear and time varying, single loop/multiloop systems. The basic idea of [1] consists of using a family of linear time invariant (LTI) plants which were obtained by local linearization of the heat exchanger nonlinear dynamic around different operation points, and This work was supported by MCYT under project DPI04-07670-C02 and Fundaci´on S´eneca under project 00507/PI/04 A. Vidal and A. Ba˜nos are with Dpt. Inform´atica y Sistemas, University of Murcia, 30071 Murcia, Spain
[email protected],
[email protected]
978-1-4244-2505-1/08/$20.00 ©2008 IEEE
for different products, following the method developed in [4]. The result is a robust PI compensator that guarantees a specified worst-case behavior for all the operation points and the different products considered. In general, the robustness and performance has to be balanced due to fundamental limitations of the resulting PI compensator, being this the main motivation of the present work, in which a type of nonlinear/hybrid control referred to as reset control is investigated with the goal of improving performance without sacrificing robustness. Reset control systems were one of the first attempts to overcome fundamental limitations of LTI control systems. Its development was initiated fifty years ago with the work of Clegg [8], that introduced a nonlinear integrator based on a reset action. Basically, the Clegg integrator (CI) consists of an integrator whose output is set to zero whenever its input is zero. Therefore, a faster system response without excessive overshooting may be expected, and limitation of its LTI counterpart is avoided. Although several other nonlinear compensators were developed, all based on describing function analysis, it is in a series works by Horowitz [10], [11], where reset control systems were impulsed by introducing the first order reset element (FORE). The implementation of reset control is very simple, it consists of resetting the state (or part of it) of a feedback LTI compensator (referred to as the base LTI compensator) at every instant in which its input is zero. Usually the design of the reset control is strongly dependent on a proper election of the base LTI control system. A common approach is to design the base LTI system to be stable and to fulfill some performance specifications, and then including reset over some compensator states to improve performance and robustness. However, this method should be carefully applied, since it is well known [5], [6] that the reset action may destabilize a base LTI control system. Thus, reset control may be used to overcome fundamental limitations of LTI control systems, but it may work worse that its base LTI system. In previous works by authors [2], [3] a reset compensator based on PI compensator has been introduced, it consists of PI plus a Clegg integrator and has been referred to as PI+CI compensator. In [2], [3] tuning rules of the PI+CI compensator for first and second order systems with time delay has been developed. In this work, the extension of this tuning techniques by using QFT will be investigated with special
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emphasis in its application to temperature control of a heat exchanger process. In Section II, the uncertain plant is defined through the parameters of nine plants; and the PI+CI controller is also studied as a type of reset controllers. In Section III, the QFT design method is applied upon the pilot plant in order to get a PI+CI controller well-tuned. Finally, in Section IV, the response of a first-order plant with time delay is improved just by modifying the controller reset condition.
of the integral term which the reset action is applied on, and it is used to set the partial reset on the integral term.
e
1 τi
kp
1 − preset
1 s
preset
CI
u
II. PRELIMINARIES AND PROBLEM SETUP Consider the feedback system given by Fig. 1, where P is the plant and C is the reset compensator.
Fig. 2.
It has been shown in [2], [3] that this partial reset of the integral term results in an improvement on the transitory closed loop response, reducing the overshoot percentage and settling time corresponding to the design without reset.
d h e6
Fig. 1.
v- ? h u-
C
P
y -
In the state-space, a PI+CI controller can be expressed by using two states, one corresponding to the I term, an the other corresponding to the CI term: xr = (xi , xci )⊤ .
Reset compensator C applied to a LTI plant P
Specifically, in our work the plant will be represented by a set of first-order with time delay transfer functions, given by P (s) =
k e−hs τs + 1
Gain k 0.11 0.06 0.10 0.04 0.10 0.10 0.03 0.08 0.04
Time constant τ (s) 55.56 43.48 58.82 30.30 45.46 52.63 33.33 43.48 47.62
x˙ r x+ C : r v
(1)
where the parameters have been experimentally obtained [7], as given in Table I. Plant Pi 1 2 3 4 5 6 7 8 9
PI+CI Controller Structure
(2)
+ + where x+ r or xr (t ) is the value xr (t + ǫ) with ǫ → 0 . The matrices Br , Aρ , Cr and Dr are given by
Br =
Delay h (s) 31.7 15 15 20 15 41.5 49 53.2 16.8
= Br e, e 6= 0 = Aρ xr , e=0 = Cr xr + Dr e
Cr =
kp τi
1 1
,
1 − preset
Aρ =
preset
1 0 ,
0 0
Dr = kp
As it was remarked in [12], reset systems, such as (2), are prone to the presence of Zeno solutions. To avoid this kind of solutions, the notion of temporal regularization is used here in the same way it was already used in [12]. Therefore, any implementation of the reset controller (2) would require the time regularization to avoid Zeno solutions.
TABLE I P ILOT P LANT U NCERTAINTY
A. PI+CI compensation On the other hand, the reset PI+CI controller will be used. It can be interpreted as a PI controller with reset action on its integral term [2], [3]. Specifically, a PI+CI controller is defined simply by adding a Clegg integrator (CI) [8] to a PI controller. In this perspective, the PI+CI controller will have three terms as shown in Fig. 2. In this compensator, kp and τi are the proportional gain and the integral time constant of its counterpart PI compensator. The reset percentage preset represents the part
In the frequency domain, the describing function of the PI+CI compensator can be obtained by adding the describing function of a Clegg integrator [8] to a PI controller as shown in Fig. 2. In that way, the describing function of a PI+CI controller is simply given by
1 − preset 1 preset 1.62 −j38.1◦ DP I+CI (jω) = kp 1+ + e τi jω τi ω (3) As it can be seen [3], the difference between a PI+CI controller and its counterpart PI is that the reset action will give extra phase lead of more than 50◦ over a (LTI)
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integrator [8]. This has been one the main reasons to use PI+CI compensation in order to overcome fundamental limitations due to time delay.
of a proper reset percentage. First of all, plant templates are compute taking into account parametric uncertanty as given in (1)-Table I, being the nominal plant P8 (s) =
B. Quantitative Feedback Theory QFT is a robust control technique that is especially well suited for control problems with large plant uncertainty. It has been developed since the early 60’s by Prof. Horowitz and his collaborators [9], and it has been applied to scalar/multivariable, LTI/nonlinear and time varying, single loop/multiloop systems. QFT works in the frequency domain, thus plant models can be derived from transfer functions (usually with parametric uncertainty) or directly by sets of frequency responses. QFT basically consists of several design steps:
Secondly, boundaries are computed for each frequency from design specifications by using templates. Design specifications will be a (robust) phase margin of 30◦ , and a minimum disturbance rejection at the plant input seen in Table II. Frequency, (rad/s) 0.0025 0.005 0.014
Maximun Disturbance Rejection (dB) -24 -21 -15 TABLE II
MAXIMUM DISTURBANCE REJECTION AT THE PLANT INPUT
Note that a relatively small phase margin has been specified for the base LTI control system. This is due to the fact that the reset PI+CI compensator will be designed to improve this minimum phase margin for all the plants set. The next step consists of shaping the nominal open loop. Here a PI controller is tuned by making the open-loop gain fits both the stability and disturbances rejection boundaries. A shaping is shown in Fig. 3
20
0.0025 rad/s 0.001 rad/s
15 10
Magnitude (dB)
3. Nominal open loop shaping. Once the boundaries are computed, the next design step is to compute (shape) the open loop gain that fits them in some optimal way. In general, this is a hard computational problem that usually has been solved heuristically. Once the open loop gain is obtained, the feedback compensator is directly computed. In addition, a precompensator must be added to the feedback structure if tracking specifications are to be satisfied.
(4)
In this case, the templates are obtained for a wide set of working frequencies, from 0.0001 up to 0.025 rad/s.
1. Computation of templates. A template represents, at a given frequency, the uncertainty of the plant. It is a region of the Nichols Plane, being each point given by the phase and magnitude of a plants set element. For a set of working frequencies, the first design step consists of computing the templates set. 2. Computation of boundaries. Given (robust) stability and performance specifications such as disturbance rejection, each template generates a boundary. If the nominal open loop gain avoids the boundaries, one boundary at every working frequency, then closed loop specifications are satisfied for all the plants considered in the template.
0.08 e−53.2s 43.48s + 1
III. PI+CI DESIGN
0.0001 rad/s
5 0 −5
0.005 rad/s
Since the main goal of this work is to design a robust control system that guarantees both a stable and optimal operation for the set of plants considered in Table I, QFT (Quantitative Feedback Theory) will be used to design the base LTI control system. Design specifications are closed-loop stability and disturbances rejection at the plant input. In a second stage, this base compensator will be used to design the final reset control system.
−10
0.025 rad/s
0.014 rad/s
−15 −20 −25 −350
−300
−250
−200
−150
−100
−50
0
Phase (degrees)
Fig. 3.
Open Loop Design
A. Base PI tuning
corresponding to a proportional gain of kp = 0.80 and an integral time constant of τi = 6.42 s.
In the design of the base LTI compensator, a PI structure will be used to shape the loop gain in basis to design specifications. After that, the PI parameters will be used to design the PI+CI compensator, being a key issue the election
Finally, once the base PI compensator is tuned, a closedloop system analysis is done to ensure that the system fits the given stability and disturbance rejection specifications. This
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is usually referred as to compensator validation in QFT. This analysis has to be done for a wider set of frequencies than the one used in the templates computation, and in this case, this set goes from 0.00001 up to 100 rad/s with a much bigger gridding. In Fig. 4, the stability analysis is shown, whereas the disturbances rejection analysis is done in Fig. 5.
10 0
Stability Specification Closed−Loop System
Magnitude (dB)
−10 −20 −30 −40
of this parameter it is important to note that for high values of preset a worse closed-loop response can be expected [2], [3]. Here, the tuning rules derived in the above work are used to tuning this parameter for the nominal loop. The result is an increment of the phase margin and thus a better transitory response for the nominal loop, and also for rest of loops due to the extra phase lead given by reset compensation for all the plant set. A value preset = 0.2(20%) is obtained. For comparison of results, phase margins are computed for both compensator: the base PI compensator (kp = 0.80, τi = 6.42), and the PI+CI compensator (kp = 0.80, τi = 6.42 and preset = 0.2). Bode plots of the nominal loops with both compensators are shown in Fig. 6, where it can be seen the extra phase lead given by the PI+CI compensator without significative change in the magnitude, in relation to the base PI design.
−50
BODE PLOT Magnitude (dB)
−60 −70
10
−4
−3
10
10
Frequency (rad/sec)
Fig. 4.
40
−2
−1
10
0
10
1
10
Stability Analysis
Phase (degrees)
−80 −5 10
20
Magnitude (dB)
20 0 −20 −4 10
−3
10
−2
Frequency (rad/s)
−1
10
10
PI Controller PI+CI Controller
−80 −100 −120 −140 10
Disturbances Rejection Specification Closed−Loop System
PI Controller PI+CI Controller
40
−4
−3
10
Fig. 6.
−2
Frequency (rad/s)
10
−1
10
Bode Plot of both PI and PI+CI Controllers
0
Note that, although reset compensation by PI+CI can considerably improve the phase margin of a PI controller, these improvements need to be carefully interpreted since in this case the closed-loop system is nonlinear, and the results have been obtained by using the describing function of the PI+CI system (3).
−20
−40
−60
To analyze the robustness of the phase margin with respect to the parametric uncertainty, the phase margin of all the loops with both compensators, PI and PI+CI are computed. The result is shown in Table III.
−80 10
−5
10
−4
Fig. 5.
−3
10
10
−2
Frequency (rad/sec)
−1
10
0
10
1
10
Disturbances Rejection Analysis
As it can be seen, the designed closed-loop system satisfies both stability and disturbances rejection specifications, for every frequency, thus the base PI compensator design is finished. B. PI+CI tuning Once the base PI compensator has been design, the obtained PI parameters are chosen as part of the PI+CI parameters [3]. But, in addition, an additional parameter, the reset percentage (preset ), must be tuned. For a proper election
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Plant, Pi 1 2 3 4 5 6 7 8 9
Phase Margin, PI 40.5◦ 69.2◦ 52.8◦ 77.7◦ 57.6◦ 38.5◦ 73.9◦ 43.2◦ 74.1◦
Phase Margin, PI+CI 52.8◦ 82.5◦ 65.6◦ 91.3◦ 70.4◦ 50.6◦ 87.4◦ 55.3◦ 87.6◦
TABLE III P HASE M ARGINS
As it can be seen, the phase margin is always increased just by adding a reset percentage of 20%. On the other hand, in the Fig. 7 it is seen the improvement of the reset for disturbance rejection. In this figure the nominal plant (4) is considered. It shows that both systems (for PI and PI+CI controllers) are equal until the reset action is experimented by the system. After reset, the PI+CI system reaches the steady-state faster than the PI one and less undershoot.
2
Product Temperature Variation (ºC)
PI Controller PI+CI Controller 1.5
1
0.5
0
−0.5
−1 0
100
Fig. 7.
200
300
400
500
Time (s)
600
700
800
900
1000
System Responses of both PI and PI+CI Controllers
In Fig. 8, the output of different plants controlled by the designed PI+CI compensator, in response to a step disturbance at the plant input, are given. As a result, the PI+CI compensator is shown to be robust against parametric uncertainty with respect the disturbance rejection.
Product Temperature Variation (ºC)
2.5 Plant 1 Plant 3 Plant 6 Plant 8
2
1 0.5 0 −0.5
Fig. 8.
100
200
300
400
Time (s)
500
600
700
IV. PI+CI COMPENSATOR WITH RESET BAND The reset action is a good choice to improve the transitory response of a closed-loop system. But there are systems whose response improvement is not as good as it can be expected. This is due to the presence of a dominant time delay in the plant. The problem in these dominant delay systems is that the reset action is done at reset times (τk ), but the system suffers the reset action at another time, specifically at τk + h, where h is the time delay of the plant. This lack of coordination between the reset times (when error is equal to zero) and the times when the system undergoes the reset action can be overcome by modifying the reset condition. Until now, the reset condition in the PI+CI controller has been given by the fact that the error signal had to be equal to zero. In this Section, the reset condition is modified so as the PI+CI controller to do reset before the error signal is zero. The objective of this modification consists of doing reset some time before the crossing between the error signal and zero. As a consequence of this modification, now the new reset condition consists of doing reset when the error signal is decreasing and when the system output reaches some specified value. This value will be fixed by the parameter reset band, δ. The value of this parameter must be fixed in such a way that all plants undergo the reset action around the reset times. When the plant has uncertainty the fixing of δ can be more complicated. In this case, with time delay uncertainty, the δ value should be fixed in such a way that the plant with higher time delay undergoes the reset action after reset times. In that way, we will ensure that most of possible plants undergo the reset action around the reset times. Consider the systems responses given in Fig. 8, that is, a step disturbance will act at the input of the considered plants (1, 3, 6 and 8). The control will be done by the PI+CI controller tuned in Section III, where its parameters were kp = 0.80, τi = 6.42 and preset = 0.2. Reference input to the closed-loop system is also considered equal to zero. In addition, an initial value of the reset band is chosen, in this case, δ = 0.7 for Fig. 9.
1.5
−1 0
significatively improve control system performance.
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Representative Plants Responses with a PI+CI Controller
Although the responses given in Fig. 8 fulfill design specifications, it can be seen that the effect of reset takes place after the crossing of the output with zero, due to the existence of delays in the plants. Even although these responses are better that those corresponding to PI compensation. In the next it will be investigated how a simple modification of the reset PI+CI compensator can
Comparing closed-loop responses in Fig. 8 and Fig. 9, it can be said that closed-loop responses are considerably improved for all plants by adding the reset band. This is due to the fact that most of closed-loop systems undergo the reset action around the reset times. In the same way as this value of δ was chosen, other values could have been chosen too. Finally, consider two different values of the reset band for the same system at the same conditions. First, a higher value of δ is fixed, δ = 0.9. Fig. 10 shows that with this δ,
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all plants undergo the reset action before the reset times. In Fig. 10, all closed-loop responses are also improved in comparison with Fig. 8 when δ = 0.9. Next, a lower value of the reset band will be fixed. In Fig. 11, a reset band equal to 0.5 is chosen.
Product Temperature Variation (ºC)
2,5 Plant 1 Plant 3 Plant 6 Plant 8
2
0,7 0,5 0
100
200
300
400
Time (s)
500
600
700
800
System Responses of PI, PI+CI and band-PI+CI Controllers
2,5
Product Temperature Variation (ºC)
Many common control problems in the process industry, such as for example temperature control, are represented by first order plus time delay plants. This work has investigated the potentials of a nonlinear/hybrid compensator (PI+CI) in the robust control of these types of processes. A two-step design technique has been developed, based on the use of QFT to shape a nominal base PI compensator. It has been proved how two simple ideas, that is the partial reset of the integral term and the use of a reset band, can significatively improve closed loop performance, in spite of parametric uncertainty both in the time constant and the time delay. All the results have been checked by means of simulations over experimentally obtained plant models. Note that although the work is tailored to a specific temperature control problem of a heat exchanger, that serves to describe the design technique, thus the method can be applied to systems of high order plus time delay from which a frequency response model set is available.
1
Fig. 9.
Plant 1 Plant 3 Plant 6 Plant 8
2 1,5
0,9 0,5
R EFERENCES
0 −0,5 −0.9 0
100
Fig. 10.
System Responses of PI, PI+CI and band-PI+CI Controllers
200
300
400
Time (s)
500
600
700
2.5
Product Temperature Variation (ºC)
V. CONCLUSIONS
1,5
−0,5 −0.7 0
closed-loop responses are improved too. Therefore, it can be concluded that the reset band can have more than one value so as to improve the PI+CI responses. In this case, the reset band can take values from 0.5 up to 0.9. Of course, the purporse of this analysis has been to investigate the benefits of using a reset band, a more detailed analysis including some more formal tuning of the new parameter δ deserves more research effort.
800
Plant 1 Plant 3 Plant 6 Plant 8
2 1.5 1 0.5 0 −0.5 0
Fig. 11.
100
200
300
400
Time (s)
500
600
700
800
System Responses of PI, PI+CI and band-PI+CI Controllers
In this case, Fig. 11 shows that even though most of systems undergo the reset action after the reset times, the
[1] Ba˜nos, A., P. Garcia and L. Checa (2006). Robust Control of thermal treatments in can industry. Proceedings of the 4th IFAC/CIGR Workshop - Control Appliations in Post - Harvest and Processing Technology (CAPPT 2006). Potsdam, Germany. [2] Ba˜nos, A., A. Vidal, (2007). Design of PI+CI Reset Compensators for second order plants. 2007 IEEE International Symposium on Industrial Electronics, Vigo, Spain, June 4-7, 2007 [3] Ba˜nos, A., A. Vidal, (2007). Definition and tuning of a PI+CI reset controller. Proceedings of the European Control Conference 2007, Kos, Greece, July 2-5, 2007. [4] Ba˜nos, A., O. Yaniv., and F. J. Montoya (2003). Nonlinear QFT synthesis by local linearization. Int. Journal of Control, 76, 5, pp. 429-436. [5] Beker, O., Analysis of reset control systems, Ph. D. Thesis, University of Massachusetts Amherst, 2001. [6] Beker, O, C.V. Hollot, Y. Chait and H. Han, (2004). Fundamental properties of reset control systems. Automatica, 40, pp.905-915. [7] Cervera, J., Ajuste autom´atico de controladores en QFT mediante estructuras fraccionales, Ph. D. Thesis, University of Murcia, Spain, 2006. [8] Clegg, J. C. (1958), A nonlinear integrator for servomechnisms, Transactions A.I.E.E.m, Part II, 77, pp. 41-42. [9] Horowitz, I.M. (1992). Quantitative Feedback Theory. QFT Press. [10] Horowitz, I. M., and Rosenbaum (1975), Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty, International Journal of Control, 24, 6, pp. 977-1001. [11] Krishman, K.R, and Horowitz, I. M. (1974), Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances, International Journal of Control, 19, 4, pp. 689-706. [12] Zaccarian, L., D. Neˇsi´c, A.R. Teel (2005). First order reset elements and the Clegg integrator revisited. In American Control Conference, pages 563-568, Portland (OR), USA, June 2005.
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