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Simultaneous gains tuning in boiler/turbine PID-based controller clusters
using iterative feedback tuning methodology
Shu ...
controller is tuned and placed into automatic mode, the entire
system is expected to be very stable and responsive. Usuall...
increased, attention should be paid to the presence of noise at the
controller output. Derivative action tends to amplify ...
shortly after the drum level loop, because it is also difficult to
control manually.
The burner air flow rate control loop i...
The system is described by
yt
wt
!
¼ G
rt
vt
ut
0
B
@
1
C
A ð1:1Þ
where t represents the discrete time instants, G is the ...
C(k) as g and c(k), respectively, where both g and c(k) are the
corresponding time domain operators. rs, us, and ys denote...
(4) Calculate dJ=dkðk
i
Þ as
dJ
dk
ðk
i
Þ ¼ À
1
N
XN
t ¼ 1
½yn
t ŠT
Pt ~yt
 #
,
and then follow the iterative algorithm be...
Steam flow rate ¼ 80%,
Excess oxygen ¼ 3%,
Air flow rate ¼ 80%,
Drum level ¼ 0 m,
Feedwater flow rate ¼ 80%,
and all input...
Therefore, there are five PIs and only one PID, i.e. PID2, in the
cluster. Henceforth, we drop the D for simplicity unless ...
Biases are dealt with by treating all values as deviations about
a nominal value. Finally, the air/fuel flow ratio is appro...
The staircase algorithm [30] is then used to determine the
controllability and observability of the linearized system afte...
steps iterations are stopped and the controller parameters in the
eighth iteration are adopted as the optimal ones and sub...
5. Conclusions
The general area of automatic multi-loop PID tuning shows an
unexpectedly strong promise. The basic IFT con...
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Simultaneous gains tuning in boiler turbine pid-based controller clusters using iterative feedback tuning methodology

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Tuning a complex multi-loop PID based control system requires considerable experience. In today's power industry the number of available qualified tuners is dwindling and there is a great need for better tuning tools to maintain and improve the performance of complex multivariable processes. Multi-loop PID tuning is the procedure for the online tuning of a cluster of PID controllers operating in a closed loop with a multivariable process. This paper presents the first application of the simultaneous tuning technique to the multi-input–multi-output (MIMO) PID based nonlinear controller in the power plant control context, with the closed-loop system consisting of a MIMO nonlinear boiler/turbine model and a nonlinear cluster of six PID-type controllers. Although simplified, the dynamics and cross-coupling of the process and the PID cluster are similar to those used in a real power plant. The particular technique selected, iterative feedback tuning (IFT), utilizes the linearized version of the PID cluster for signal conditioning, but the data collection and tuning is carried out on the full nonlinear closed-loop system. Based on the figure of merit for the control system performance, the IFT is shown to deliver performance favorably comparable to that attained through the empirical tuning carried out by an experienced control engineer.

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Simultaneous gains tuning in boiler turbine pid-based controller clusters using iterative feedback tuning methodology

  1. 1. Simultaneous gains tuning in boiler/turbine PID-based controller clusters using iterative feedback tuning methodology Shu Zhang a , Cyrus W. Taft b , Joseph Bentsman a,n , Aaron Hussey c , Bryan Petrus a a Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206W Green Street, Urbana, IL 61801, USA b Taft Engineering, Inc., Harriman, TN, USA c EPRI, 1300 West WT Harris Boulevard, Charlotte, NC 28262, USA a r t i c l e i n f o Article history: Received 3 September 2011 Received in revised form 16 April 2012 Accepted 16 April 2012 Available online 26 May 2012 Keywords: Iterative feedback tuning PID control Multi-input–multi-output systems Boiler/turbine control a b s t r a c t Tuning a complex multi-loop PID based control system requires considerable experience. In today’s power industry the number of available qualified tuners is dwindling and there is a great need for better tuning tools to maintain and improve the performance of complex multivariable processes. Multi-loop PID tuning is the procedure for the online tuning of a cluster of PID controllers operating in a closed loop with a multivariable process. This paper presents the first application of the simultaneous tuning technique to the multi-input–multi-output (MIMO) PID based nonlinear controller in the power plant control context, with the closed-loop system consisting of a MIMO nonlinear boiler/turbine model and a nonlinear cluster of six PID-type controllers. Although simplified, the dynamics and cross- coupling of the process and the PID cluster are similar to those used in a real power plant. The particular technique selected, iterative feedback tuning (IFT), utilizes the linearized version of the PID cluster for signal conditioning, but the data collection and tuning is carried out on the full nonlinear closed-loop system. Based on the figure of merit for the control system performance, the IFT is shown to deliver performance favorably comparable to that attained through the empirical tuning carried out by an experienced control engineer. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction and background 1.1. Sequential and simultaneous tuning of individual PID controllers in a multi-input–multi-output (MIMO) PID cluster Power plant control systems have utilized proportional-inte- gral-derivative (PID) controllers as their primary control algo- rithm for over 50 years. Researchers have been developing and testing advanced power plant control laws during the past 20 years and although many of the results are positive, acceptance of the new technology has been limited. The PID control law is still the dominant algorithm in the industry. A major difference between the PID algorithm and most of the advanced algorithms is that a PID controller is a single input, single output (SISO) controller and therefore can only control a single control loop at a time. The more advanced algorithms are inherently MIMO and can control many loops in an integrated fashion. When using single loop PID controllers to regulate a process with interacting variables such as a boiler/turbine system, the process must be segregated into the SISO loops. The basic arrangement of control loops is largely done by pairing inputs and outputs that are most closely correlated. This technique is adequate for many control loops where the magnitude of the interactions is relatively small. For larger interactions, ad hoc feedforward signals are sometimes used to reduce the coupling effects. In addition to this logical pairing of inputs and outputs, the structure of the control system also utilizes many cascade control loops. Tuning a complex PID based control system requires considerable expertise and experi- ence. While single loop PID tuning is well understood and supported by several commercial tuning software packages, it is not clear how to simultaneously optimize several interacting PID control loops. Thus, there is a need in the industry for better tuning tools to provide a high performance control of complex multivariable processes. The most common method of tuning boiler/turbine control systems today is the empirical method, also called the trial and error method. It applies to both SISO and MIMO systems. The general approach is to start with all the loops in the manual mode or significantly detuned and begin tuning controllers at the bottom of the hierarchy, working upwards. After each controller is tuned it is placed in automatic mode. Where multiple con- trollers exist at the same level, the controller with the least interactions is generally tuned first. There are some exceptions to this rule, but this is the general approach. When the last Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.04.003 n Corresponding author. Tel.: þ1 217 244 1076; fax: þ1 217 244 6534. E-mail address: jbentsma@illinois.edu (J. Bentsman). ISA Transactions 51 (2012) 609–621
  2. 2. controller is tuned and placed into automatic mode, the entire system is expected to be very stable and responsive. Usually it is necessary to go back and fine tune some of the controllers once the whole system is in automatic mode. When tuning a series of controllers sequentially, it is impor- tant to provide some additional stability margin on the first loops tuned, because as other loops are later tuned and placed in automatic mode, the whole system can become unstable or very oscillatory. This is where the experience of the tuner is most beneficial. It requires judgment to provide the right amount of stability margin without sacrificing good performance. For multi- loop systems the method is carried out in a prescribed sequence, tuning one controller at a time, and therefore belongs to the class of sequential tuning methods. Simultaneous tuning of individual controllers in the MIMO PID controller clusters in the multi-loop MIMO configurations can potentially capitalize on the interactions among process variables and loops to attain better overall performance. Unlike the sequential empirical loop tuning or sequential loop closure [1], simultaneous tuning can incorporate, in addition to classical SISO time domain performance objectives, the true MIMO control objectives, such as robust performance specified in terms of the so-called sensitivity and complementary sensitivity functions in linear case, or gap metric measures in nonlinear case. MIMO PID cluster tuning is better developed in the linear case; however, even then it is still highly non-trivial. For example, the fully linear problem of simultaneous tuning of PID clusters of arbitrary structure in a MIMO system to attain closed-loop stability belongs to a class of the so-called computationally NP-hard (nondetermi- nistic polynomial time) problems characterized by the explosive computational demand and is in general nonconvex, i.e. not amenable to simple gradient-type solutions for attainment of global optimal solution. Nevertheless, a significant progress has been made in MIMO PID cluster tuning that can parlay specific application knowledge into successful tuning procedures [1–3]. Simultaneous tuning of individual PID controllers in a MIMO PID-based closed-loop subsystem can be accomplished in the closed-loop or the open-loop manner. Closed loop is often pre- ferable; however, it has limitations and might present difficulty in re-identifying subsystem dynamics for controller tuning, or more generally, redesign. As indicated in [1], the simultaneous MIMO PID tuning techniques can be broadly classified into the model- free and the model-based methods, with the latter being non- parametric, partially parametric, and parametric. The model-free technique is represented by the iterative feedback tuning (IFT) [4–10] and the controller parameter cycling [11]. The nonpara- metric, partially parametric, and parametric model-based meth- ods are represented, respectively, by the relay autotuning technique [12], the subspace identification technique [13–15], and the optimal restricted structure design and related methods [16–28]. Iterative feedback tuning is a model-free direct multi-loop PID tuning method whose performance is usually superior to sequen- tial tuning methods. Ref. [4] extends IFT to the case where both the plant and the controller can be nonlinear. It shows that an estimate of the gradient can be constructed using only the signal based information. In [7], an overview of IFT is presented for both SISO and MIMO linear systems. Stability and robustness aspects are covered. A survey of existing extensions, applications, and related methods is also provided. In [8], IFT addresses tuning of the PID parameters in the applications where the objective is to achieve a fast response to the set point changes. The performance of these IFT-tuned PID controllers is compared with the perfor- mance achieved by four classical PID tuning schemes widely used in industry. Simulations show that IFT always achieves a perfor- mance that is at least as good as that of the classical PID tuning schemes, and often dramatically better: faster settling time and less overshoot. Nakamoto [9] provides an application of IFT that adjusts parameters of multiple PID controllers in a MIMO power plant process. In the IFT applications for the SISO systems, depending on the controller structure, only two or three closed- loop experiments are required. However for the MIMO systems the number of experiments increases proportionally to the dimension of the controller, making the tuning process time- consuming. In [10] several methods are proposed to reduce the experimental time by approximating the gradient of the cost function. 1.2. Sequential tuning—a simple empirical tuning example To help clarify the general empirical tuning procedure described in the previous section, the tuning process will be demonstrated on a simple boiler control system structure shown in Fig. 1. This control system is typical of many drum boiler control system architectures; although a real control system has many additional details that have not been included in this example. This example assumes that no prior loop tuning has been done, so that the system is completely detuned. All tuning should be done with the unit at steady load with no operational disturbances, such as sootblowing, in progress. It is preferable for the unit load to be at or near full load, but if that is not possible, the load should be set as high as possible. The control system at hand is seen to have a definite hierarchy and to include several cascade control loops. On any drum boiler, the first loop to be tuned should be the feedwater one followed by the drum level control loop. This is not due to its position in the hierarchy, but rather because the level is non-self-regulating and therefore troublesome to operate in manual mode. For this reason, it is important to get this loop in automatic mode as soon as possible. The empirical requirements for the controller perfor- mance are specified in Table 1. Prior to tuning the feedwater flow controller, the drum level controller in set to manual or greatly detuned. The feedwater flow rate dynamics has a fast rise time, usually around 5 s, so it is relatively quick and easy to tune empirically. Tuning starts with the integral gain and the derivative gain set to zero and the proportional gain set well lower than its expected final value. A step change to the feedwater flow rate setpoint is introduced and the response is observed. The change in the latter setpoint can be produced by changing the drum level setpoint. The dynamics of the drum level controller is much slower than that of the feed- water system, so it will not interfere with the feedwater loop tuning. The proportional gain is then increased until a small amount of overshoot is seen in the response and then it is reduced to eliminate overshoot. The integral gain is then increased until the flow rate error returns to zero quickly, but with little or no overshoot. It may be necessary to reduce the proportional gain slightly as the integral gain is increased to maintain a quick return to setpoint with no overshoot. Derivative action is rarely, if ever, used on the feedwater flow rate control due to noise in the process measurement and no real need to decrease the response time of the loop beyond what proportional and integral actions alone can provide. Once the feedwater flow rate control is tuned, the drum level controller should be tuned. The method is similar to the feed- water flow rate tuning, except that the final loop tuning must be done by introducing a setpoint change into the loop, usually through the feedwater flow rate. Also, because the process has an integrating type response, very little integral action is needed on the controller. Derivative action may be used on drum level control, but often is not, and when used, its tuning should be done prior to that of the integral action. As derivative gain is S. Zhang et al. / ISA Transactions 51 (2012) 609–621610
  3. 3. increased, attention should be paid to the presence of noise at the controller output. Derivative action tends to amplify any noise that exists in the process measurement and the noise ultimately limits the amount of the derivative gain that can be used. As derivative gain is increased, it may be possible to increase the proportional gain without introducing any overshoot due to the damping effect of the derivative gain. Drum level response may exhibit a nonminimum phase shrink and swell effect in response to drum pressure and steam flow changes. This effect causes the level to move in the opposite direction initially before moving in the expected direction. If this type of response is observed, less control gain will be possible and the level response will be worse. After tuning the drum level, the steam flow rate feedforward should be scaled so that at full load the steam flow rate and feedwater flow rate readings agree in the feedwater controller. It should be checked again at mid-load, and if the two readings differ by more than 1%, consideration should be given to replacing the linear scaling with a function generator to allow a better match between feedwater and steam flow rates over the load range. The typical control structure shown in Fig. 1 does not include a furnace pressure control loop. If it did, that loop should be tuned PID PIDPIDPID Air Flow Demand Fuel Flow DemandFeedwater Flow Demand Superheat Spray Valve Demand Heat Distribution Demand (Burner Tilts) PID Turbine Valve Demand X PID PID ATTTTT MW PTLT PID PID Unit Load Reheat Steam Temperature Superheat Steam Temperature Drum Level Excess Oxygen Throttle Pressure FT Steam Flow Typical Control System Structure for Drum Boiler 3 5 6 41 9 8 2 10 Typical Tuning Order K A f(x) f(x) KK f(x) K A K A K A K A K K FT Air Flow K FT Feeder Speed K FT 0.7 to 1.3 (+/- 30%) These K’s convert EUs to % FW Flow PID 7 K TT Desuper- heater Outlet Fig. 1. Boiler/turbine control structure in SAMA format. Table 1 Control performance requirements. Controlled variable Primary objective Tuning test(s) Unit load Setpoint following Closed loop load ramp at 5%/min Throttle pressure Disturbance rejection Closed loop load ramp at 5%/min Drum level Disturbance rejection Closed loop load ramp at 5%/min Excess oxygen Disturbance rejection Closed loop load ramp at 5%/min S. Zhang et al. / ISA Transactions 51 (2012) 609–621 611
  4. 4. shortly after the drum level loop, because it is also difficult to control manually. The burner air flow rate control loop is tuned next. The air flow represents total air flow including primary and secondary air flows that go into the furnace. This task is generally straightfor- ward, with a time constant of around 20–30 s, depending on the control means. Fan inlet vanes, fan blade pitch, and fan speed are all used to control air flow rate. Again, the goal should be to tune for a small amount of overshoot, and then eliminate it by slightly reducing the gain. The fuel flow rate control is tuned next. The loop is often that of a feeder speed (or flow rate) control, with the bulk of the control done within the feeder control system. If a heat release type of fuel measurement is used, the tuning is more difficult and that system is beyond the scope of this study. At this point all the low level flow control loops are tuned and it is time to begin tuning the upper level controllers. The first upper level controller to be tuned should be the throttle pressure controller. Again, the tuning procedure is similar to that for the feedwater flow rate controller described above. Initial tuning can be done by making setpoint changes at constant load. Because the pulverizer response is very slow, it is generally necessary to use derivative action in the throttle pressure controller. During the initial tuning little or no derivative action should be used. This should wait until the second round of tuning. The objective at this point is still stability without using excessive over- or under- firing. Once the loop is tuned for setpoint response, small load changes or fuel disturbances should be introduced to check the response under those conditions. Excess oxygen loop should be tuned next. The initial tuning can be done for tracking in response to setpoint changes, but the final tuning must be carried out for disturbance rejection through actually generating air or fuel flow rate disturbances. Noise is usually a factor in excess oxygen measurement, so it may be difficult to use much derivative action in this loop. At this point, it would probably be best to do an initial tuning on the superheat temperature control and the burner tilt control. The superheat temperature control is another cascade loop with the inner loop controlled variable being the temperature imme- diately downstream of the spray injection point, referred to as the desuperheater outlet temperature. This measurement has a quick response to changes in spray flow, typically about a 25–30 s time constant. As a result, it is easy to tune and requires no derivative action. The outer loop, with the final steam temperature as the controlled variable, is tuned next. This loop is often the most difficult to tune due to its slow response time and frequent disturbances. Initial tuning can be done under setpoint changes, but final tuning should be done under temperature disturbances. At this point, it is not necessary to tune the feedforward signals for the superheat control. The burner tilt control can be difficult to tune because the burner tilt response is often inconsistent throughout its operating range. The response times are also slow. At this point it is not necessary to tune the feedforward signals for the burner tilt control. The final loop to be tuned on the first pass through the control system is the megawatt control loop. The latter loop may be tuned for a very quick or a more sluggish response depending on the operating goals of the unit. If tuned for fast response, the throttle pressure control will not be as precise as it would if a more sluggish response were the goal. For this example, the goal for megawatt control will be a fast tight response. This can be done by making fairly quick, but small, load ramps in the MW setpoint. These small ramps approximate step changes. A response with no overshoot should be the goal for this initial tuning. This concludes the initial tuning of the system. At this time all loops should be in automatic mode and stable under steady load conditions. Slow load ramps (o0.5%/min) should be possible without major oscillations in key variables. The final tuning is primarily focused on improving the unit’s load ramp response. The most common control objective for the complete unit is to ramp load at a given rate, while maintaining key parameters within certain limits. The key loops for maximiz- ing unit load rates are the pressure control and the steam temperature control ones. These most often limit a unit response time, calling for more attention during the second phase of the tuning. The other loops are certainly important as well; however, being inherently faster, they do not constitute the limiting factor. In the final phase of the tuning, the unit load, or MW (mega- watt), setpoint is ramped up and down, while the response of individual loops is monitored. In addition to the tuning of the PID controllers for each loop, the calibration of feedforward signals for the pressure, air flow rate, and steam temperature loops is important. If a PID controller is not responding well during a load ramp, its tuning should be re-examined, with the objective shifting somewhat towards responsiveness over stability. On drum boilers it is important to overfire and underfire during load changes to compensate for changes in energy storage in the boiler and to reduce the response time of the pulverizers. Dynamic ‘‘kicker’’ functions are added to the firing rate demand signals to accom- plish this. The setting of these kickers is always a balance between improving the response and limiting the overfiring to maintain good combustion and furnace characteristics. With tight tuning on the MW control, the pressure control needs good feedforward action to prevent oscillations in its response. The initial load ramps should be done at ramp rates around 1.0%/min. As the unit is fine-tuned and the response improves, faster rates can be used. The load change during the ramps should be large enough to ensure that the system has time to arrest the initial drop or increase in throttle pressure before the ramp ends. The feedforward signals for pressure and air flow are generally just static load index type functions. To calibrate these signals, the output of the feedforward function generator should agree with the control loop demand over the load range. Steady state data must be collected at several load points and used to set the feedforward signals. If the latter are set correctly, the pressure controller and the excess oxygen controller outputs under steady state conditions will be very close to 0.0 throughout the load range. The steam temperature feedforward signals are usually more complex than those shown in this example and often include more than one signal. Tuning these signals involves determining how each signal affects the steam temperature and devising a function to counteract that effect. Open loop step tests or normal load changes can be used to quantify the effects of each signal. While empirical methods can often find suitable feedfor- ward functions, this is one area where analytical tools can be used to good advantage. 1.3. Simultaneous tuning—basic setting for MIMO iterative feedback tuning Iterative feedback tuning [4–10] is a model-free direct tuning method using the closed-loop experimental data. The method is based on numerical optimization, with the use of an unbiased gradient estimate in each iteration step. The unbiased gradient estimates provide convergence of the tuning sequence to a stationary point of the control criterion provided the closed loop signals remain bounded throughout the iterations. The block diagram in Fig. 2 illustrates the feedback loop for a discrete time linear time-invariant (LTI) system as a suitable candidate for IFT. S. Zhang et al. / ISA Transactions 51 (2012) 609–621612
  5. 5. The system is described by yt wt ! ¼ G rt vt ut 0 B @ 1 C A ð1:1Þ where t represents the discrete time instants, G is the (general- ized) plant consisting of the true plant and possibly some frequency weighting filters and a reference model and is repre- sented by a transfer function matrix, rt represents external user controlled signals such as set-points, vt represents unmeasurable signals such as (process) disturbances and noise, wt represents the measured output errors (deviations from the setpoints) that will be utilized in control laws, and ut represents the control signals that control the plant. Furthermore, yt represents the variables of interest to be included in the controller tuning criterion (e.g. selected measured outputs from wt and control signals selected from ut if some of them need to be penalized). Relating these variables to Fig. 1, wt includes the seven incre- mental outputs (respective deviations from the setpoints): mega- watt output, throttle pressure, steam flow rate, excess oxygen, air flow rate, drum level, and feedwater flow rate, whereas four of them, megawatt output, throttle pressure, excess oxygen, and drum level, form vector yt in the tuning criterion. None of the control variables are penalized in the present work, and hence none are included into yt. The external signal z is used for gradient estimation. The controller for this system is given by ut ¼ cðk,wtÞ ð1:2Þ where c(k,wt) is a linear time domain operator mapping from wt to ut parameterized by some parameter vector k. A differentiable control tuning criterion is chosen as JðkÞ ¼ 1 2N E XN t ¼ 1 ½ytðkފT PtytðkÞ " # ð1:3Þ where E[ Á ] denotes expectation with respect to the disturbance v and Pt is the weighting matrix. The objective is to minimize this criterion by the Newton–Raphson type iterative algorithm to find PID coefficients that best make system track reference trajectories: k iþ 1 ¼ k i ÀgiRÀ1 i dJ dk ðk i Þ, i ¼ 0,1,2. . ., ð1:4Þ dJ dk ðk i Þ ¼ À 1 N E XN t ¼ 1 @yt @k ðk i Þ !T Ptytðk i Þ " # ð1:5Þ where i’s are the iteration numbers and Ri is some appropriate positive-definite matrix, typically a Gauss–Newton approxima- tion of the Hessian of J(k), while gi, the step size, is a positive real scalar which is adjustable to ensure convergence. Since the exact calculation of expectation of [(@yt/@k)(ki )]T Ptyt(ki ) is impossible for complex systems, it is replaced by finding the unbiased estimates of the product [(@yt/@k)(ki )]T Ptyt(ki ) that are obtained by performing experiments on the closed-loop system by using a suitable signal z located as shown in Fig. 2. Further details of the algorithm are given in Section 2. In [6], it is shown how the unbiased estimates at each IFT iteration step (1.5) can be obtained for multivariable LTI systems. Particular attention is given to the issue of keeping the experi- ment time to a minimum and several efficient algorithms are presented. It is shown that, for tuning of an arbitrary LTI MIMO controller with nw inputs and nu outputs, 1þnu  nw experi- ments are sufficient in each iteration step of the algorithm. For the disturbance rejection, an alternative algorithm is proposed which requires nuþnw experiments. As an illustration, the method is applied to a simulation model of a gas turbine engine. 1.4. Advantages/disadvantages The iterative feedback tuning (IFT) method [7] is a model-free approach, a rather attractive feature that potentially eliminates or reduces the modeling effort. It uses the closed loop data collection which is superior to the open loop one. It can handle different cost functions that can be designed to address the robustness issues. The method has been extended to be applicable to some common classes of nonlinear plants. It can be used to tune a known control structure with no need for making changes to it. The fundamental drawback of IFT is that it is based on the gradient search, i.e. is an essentially convex method, whereas the MIMO PID tuning is known to be a nonconvex problem in general. Thus, for a MIMO PID cluster tuning, IFT can get stuck in a local minimum that could be located arbitrarily far from the optimal tuning point. In this regard, IFT should be supplied with a good starting parameter value point, possibly obtainable through experience or other techniques, such as parametric HN restricted structure PID rede- sign [22]. The operational drawback of IFT is that the number of experiments required for gradient computation could be large and quite comparable to the effort needed to perform system identification. The production of the unbiased Hessian matrix – the square matrix of second-order partial derivatives of a function – appears to involve quite a heavy computational load and requires additional experiments and in some cases two additional identifica- tion steps along with the one required for gradient estimation. 2. MIMO iterative feedback tuning—techniques and application 2.1. Iterative feedback tuning of MIMO PID clusters—tuning procedure Consider an unknown true closed loop LTI MIMO continuous time plant/controller system described as follows: ysðkÞ ¼ GusðkÞ, usðkÞ ¼ CðkÞðrsÀysðkÞÞ ð2:1Þ where GG(s) is the p  m plant transfer function matrix whose elements are Laplace (transform) transfer functions, C(k)C(s,k) is the m  p controller transfer function matrix whose elements are Laplace transfer functions parameterized by some parameter vector kARf , m and p are the number of inputs and outputs, respectively. Denote the time domain representations of G and r v y u z w C G Fig. 2. Closed-loop configuration for the iterative feedback tuning. S. Zhang et al. / ISA Transactions 51 (2012) 609–621 613
  6. 6. C(k) as g and c(k), respectively, where both g and c(k) are the corresponding time domain operators. rs, us, and ys denote the Laplace transforms of the corresponding continuous time signals rtARp , utARm , and ytARp representing, respectively, the external user controlled signals such as the set-points or more general reference signals, the control signals, and the plant output signals. The argument k in these signals indicates their dependence on the changes in the parameter vector k in the controller. The corre- sponding block diagram is shown in Fig. 3. The external signal zsARm , Laplace transform of ztARm , in Fig. 3 will be used to obtain the gradient estimate in each iteration step. We note that for the applicability of the IFT procedure there is no restriction on the number of plant or controller inputs with respect to the number of outputs as long as G can be associated with controllable and observable (or, more generally, stabilizable and detectable) state space realization [30] and the controller is stabilizing, i.e. closed loop remains asymptotically stable. In the application considered, the p  m plant is ‘‘non-square’’ – it has four inputs (m¼4) and seven outputs (p¼7), while controller, respectively, has seven inputs and four outputs, having control signal u as a 4-vector, as indicated above. However, only four of the seven output components are subjected to tracking setpoints according to the system tracking performance objectives, while the remaining three components are used by the controller to generate control signals making themselves internal signals, posing only the regulation problem in terms of their stabiliz- ability. Therefore the plant is essentially a 4  4 system. The criterion chosen for the present work is the quadratic one: JðkÞ ¼ 1 2N E XN t ¼ 1 ½~ytðkފT Pt ~ytðkÞ # ð2:2Þ where ~ytðkÞ ¼ rtÀytðkÞ is the difference between the setpoints rt and the achieved output yt. Since yt is the continuous time function, the criterion J(k) is seen to be based on the samples of yt at t¼1, 2, y, N s, i.e. the sampling interval of 1 s. This sampling interval is commensurable with the response of the plant model used further in the MIMO PID cluster tuning. Pt is the weighting matrix. The optimal controller parameter kn is defined by k n ¼ arg min k JðkÞ: ð2:3Þ Since the IFT typically performs a local gradient search in the vicinity of an operating point of a closed loop with the controller and the plant being both nonlinear, as is the case in the present work, the problem of minimizing J(k) in the actual IFT application is in general not convex. To obtain a stationary point of J(k), we would like to find a solution to the equation: 0 ¼ dJ dk ðkÞ ¼ À 1 N E XN t ¼ 1 @yt @k ðkÞ !T Pt ~ytðkÞ # : ð2:4Þ If the gradient dJ/dk could be computed, then the solution of the above equation would be obtained by the following iterative algorithm: k iþ 1 ¼ k i ÀgiRÀ1 i dJ dk ðk i Þ, i ¼ 0,1,2,. . . ð2:5Þ where Ri is some appropriate positive-definite matrix, typically a Gauss–Newton approximation of the Hessian of J(k), while gi, the step size, is a positive real scalar which is adjustable to ensure convergence and whose choice is case-dependent. Large gi usually gives fast convergence rate but also could lead to instability. Small gi has slow convergence rate but will ensure stability. In order to solve this problem, one thus needs to generate the unbiased estimate of the product @yt @k ðkÞ !T Pt ~ytðkÞ: ð2:6Þ Denoting by y0 (k) the gradient of y(k) w.r.t. an arbitrary element of parameter vector k, Fig. 4 illustrates how to obtain the unbiased estimate of y0 ðkÞT Pt ~ytðkÞ from the closed-loop experiments. The case when both controller and plant are linear will be henceforth referred to as the nominal condition. The standard procedure of IFT is given in this case by the following steps where the subscript t denotes the time domain signals and the subscript s denote their corresponding Laplace transform signals. (0) Initialization: choose some initial set of controller parameters k0 – these could be arbitrary numbers or the values provided by an experienced control engineer, both could yield good results – and specify setpoints rt as some fixed values. (1) At each iteration i, perform a closed-loop experiment of time length N using specified rt as the input and with the controller parameters set to the values ki for which the derivative dJ/dk(ki ) is to be computed. Collect measurements ~yt, t ¼ 1,. . .,N s. during this experiment, further referred to as the normal experiment. (2) Use the output ~yt collected in step (1) as input and calculate the corresponding output ~yout,t. Here ~yout,s ¼ C0 ðk i Þ~ys, where C0 (ki )¼@C(ki )/@k and C(k) is the Laplace transfer function matrix of the linear controller and C0 (ki ) is the partial derivative matrix of C(k) with respect to the controller parameters k at ki . (3) Set the setpoints rt to zero and perform another closed loop experiment referred to as gradient experiment while adding the signal ~yout,t obtained in step (2) at the position ~yout,t as shown in Fig. 4. Collect measurements yn t , t ¼ 1,. . .,N s. It has been shown in [6] that ½yn t ŠT Pt ~ytðk i Þ is an unbiased estimate of ½ð@yt=@kÞðk i ފT Pt ~ytðk i Þ. However, if we do not introduce noise into our MATLAB/SIMULINK model for simplicity, they are exactly equal to the ½ð@yt=@kÞðk i ފT Pt ~ytðk i Þ, given that the system and the controller are both LTI [10]. If controller parameter vector k has dimension nk, that is k i ¼ ½k i 1,k i 2,. . .,k i nk ŠT , then for each k i l, l ¼ 1,2,. . .,nk, we need to repeat (2) and (3) once: For each k i l, l ¼ 1,2,. . .,nk, we need to compute C0 ðk i lÞ ¼ @ Cðk i lÞ=@kl and then compute ½ð@yt=@k i lÞðk i lފT . So, for each iteration, we need to perform nk gradient experiments. C k G r z u y Fig. 3. Laplace transform representation of the feedback system. G 0r y ( )C k ( )C k y yout Fig. 4. Set-up for the exact gradient experiment. S. Zhang et al. / ISA Transactions 51 (2012) 609–621614
  7. 7. (4) Calculate dJ=dkðk i Þ as dJ dk ðk i Þ ¼ À 1 N XN t ¼ 1 ½yn t ŠT Pt ~yt # , and then follow the iterative algorithm below to calculate the next optimal controller parameters for i¼iþ1: k iþ 1 ¼ k i ÀgiRÀ1 i dJ dk ðk i Þ: Then repeat (1)–(4) with ki þ 1 , for i¼0,1,2,y until certain termination criterion regarding J(k) is satisfied. The flow chart of the IFT algorithm for nominal condition is shown in Fig. 5. The case when both controller and plant are nonlinear will be henceforth referred to as the actual condition. Let the nonlinear controller representation be denoted as f(k,yt,rt), where f(k,yt,rt) is the time-domain operator mapping output and setpoints vectors into control signal vector, parameterized by some parameter vector kARnk . For simplicity, we omit yt, rt and use f(k) to represent f(k,yt,rt) for illustration. The procedure of IFT for actual condition is similar to the nominal condition above, while involving the linearization of the nonlinear controller around operation condition. The procedure of IFT for the actual nonlinear system is as follows: (0) Initialization: choose some initial set of controller parameters k0 and specify setpoints rt as some fixed values, same as nominal condition. (1) Normal experiment: At each iteration i, perform a closed-loop experiment using specified rt as the input and collect corre- sponding output response ~yt. The closed-loop system used here consists of the nonlinear plant and the nonlinear con- troller f(k) with k set to the value ki . (2) Controller linearization: Linearize the nonlinear controller f(k) around operating condition to obtain the linearized controller fL(k) and its Laplace transform transfer function matrix FL(k)FL(s,k). Similarly to the nominal condition above, use the output ~yt measured in step (1) as input and calculate the corresponding output according to ~yout,t. Here ~yout,s ¼ F0 Lðk i Þ~ys and F0 Lðk i Þ ¼ @ FLðk i Þ=@k, where F0 Lðk i Þ is the partial derivative transfer function matrix of FL(k) with respect to k at ki . (3) Gradient experiments: Follow the same procedure as the step (3) of the nominal condition except that the ~yout,t in step (2) is generated by the partial derivative of the linearized controller F0 Lðk i Þ and all the nk closed loop experiments in this step are performed in terms of the original nonlinear controller f(k) and the nonlinear plant. (4) Controller parameter update: Use the identical update routine in step (4) of nominal condition to calculate the next optimal controller parameter set kiþ 1 . Then repeat (1)–(4) again with kiþ 1 , for i¼0,1,2,y until the same termination criteria regarding J(k) are satisfied. The IFT procedure for actual condition follows the same flow chart in Fig. 5 as well. Under the actual condition the plant is nonlinear and infinite- dimensional due to existence of time delays. However, using the corresponding procedure above for the actual condition, this poses no problem for the applicability of the IFT technique. This is due to the nonlinearity around the operating point being smooth (differentiable) and relatively weak, making the plant well linearizable. Also, the delay can be accurately finite-dimen- sionalized, with the resulting linear plant approximation being controllable and observable. This results in the closed-loop operation with both the linearization and the original nonlinear infinite-dimensional plant being asymptotically stable and almost identical in terms of time domain responses to practically mean- ingful changes in setpoints. 2.2. Application of IFT to a specific PID cluster The IFT procedure described in the previous section has been applied to a typical, but simplified, PID based boiler/turbine control system cluster shown in Fig. 6. A real control system has many additional details which have not been included in this structure. The cluster is seen to consist of four control loops, namely, the ones for power output, throttle pressure, drum level, and excess oxygen. The drum level control and excess oxygen control ones are cascade loops with feedwater flow rate and air flow rate as the inner controlled variables, respectively. A non- linear ratio controller is included for the excess oxygen control. A total of six PID-type controllers are used, with five of them being tuned as PI controllers and one as a PID controller. This provides a total of 13 tuning parameters to be adjusted. The model is an incremental one describing the dynamics of all the deviation variables with respect to the nominal operating condition and designed to represent a 250 MW plant dynamics around 80% of the operating point. The model dynamics were selected based on [29] and the authors’ experience with similar plants. Additional details about the process can be found in [29]. The nominal operating point values are specified as follows: Megawatt output ¼ 200 MW, Throttle pressure ¼ 12:5  106 Pa,Fig. 5. Flowchart of the standard IFT procedure. S. Zhang et al. / ISA Transactions 51 (2012) 609–621 615
  8. 8. Steam flow rate ¼ 80%, Excess oxygen ¼ 3%, Air flow rate ¼ 80%, Drum level ¼ 0 m, Feedwater flow rate ¼ 80%, and all inputs are 80%: The process outputs in this model are: Dy1—MW, unit load (megawatts), Dy2—TP, throttle pressure (Pa), Dy3—SF, steam flow rate (%), Dy4—O2, excess oxygen (%), Dy5—AF, air flow rate (%), Dy6—DL, drum level (m), Dy7—FW, feedwater flow rate (%). The controller inputs to the process are: Du1—TV, turbine valve demand (%), Du2—FR, firing rate demand (%), Du3—FD, FD fan damper demand (%), Du4—FWV, feedwater valve position demand (%), and k—controller parameter vector. This control system model structure provides a simple but non-trivial testbed for the multi-loop tuning method. We define the nonlinear controller as containing a lookup table, bias and a multiplication operator as shown in Fig. 6. Thus, the controller in Fig. 6 is given by the six-PID cluster that includes one lookup table and one multiplication operator and two biases, making the cluster nonlinear. In order to apply IFT, a linearized prefilter F0 L kð Þ defined in step (2) of the IFT procedure for the actual condition needs to be constructed that would preprocess the data for the application of the IFT to the actual closed loop system. In the present work this prefilter is constructed on the basis of the linearized PID cluster model fL(k). For this purpose, we linearized the nonlinear PID cluster f(k) around the system operating point to obtain fL(k). We indexed the 6 different PIDs from top to bottom as 1–6. Therefore PID1 is the MW controller, PID2 is the throttle pressure controller, PID3 is the air flow rate controller, PID4 is the excess oxygen controller, PID5 is the drum level controller, and PID6 is the feedwater flow rate controller. Then the Laplace transfer function matrix of the linearized controller cluster FL(k) is shown in Fig. 7. In Fig. 7, we have the following definitions: us ¼ Du1 Du2 Du3 Du4 2 6 6 6 6 4 3 7 7 7 7 5 , es ¼ DrsÀDys ¼ Dr1ÀDy1 Dr2ÀDy2 Dr3ÀDy3 Dr4ÀDy4 Dr5ÀDy5 Dr6ÀDy6 Dr7ÀDy7 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 , ð2:7Þ FLðkÞ ¼ PID1 0 0 0 0 0 0 0 PID2 À1 0 0 0 0 0 PID2 Â PID3 ÀPID3 PID4 Â PID3 PID3 0 0 0 0 ÀPID6 0 0 PID5 Â PID6 PID6 2 6 6 6 6 4 3 7 7 7 7 5 : ð2:8Þ where FL(k) is the Laplace transfer function matrix of the linear- ized controller fL(k) and us and es are the Laplace transforms of time domain control signals ut and error signals et, respectively. We denote the three gains in each PID as ki j, i¼1, y, 6, j¼1,2,3, where i represents the PID number and j represents the corre- sponding proportional, integral and differential gains. In Laplace notation each PID has continuous system representation as follows: PIDi ¼ ki1 þ ki2 s þki3s: The initial controller parameters were chosen empirically and were deliberately suboptimal. They are given as k11 ¼ 1, k12 ¼ 0:1, k21 ¼ 0:05, k22 ¼ 0:00001, k23 ¼ 2, k31 ¼ 4, k32 ¼ 0:1,k41 ¼ 4, k42 ¼ 0:4, k51 ¼ 1, k52 ¼ 0:0001, k61 ¼ 20, k62 ¼ 0:6 ð2:9Þ Fig. 6. SIMULINK representation of the nonlinear PID cluster to be tuned. C k e r y u Fig. 7. Laplace transform representation of the linearized controller. S. Zhang et al. / ISA Transactions 51 (2012) 609–621616
  9. 9. Therefore, there are five PIs and only one PID, i.e. PID2, in the cluster. Henceforth, we drop the D for simplicity unless confusion occurs. The controller tuning criterion has been selected as J ¼ 1 2 Â 2001 X2000 t ¼ 0 eT 1P1e1 þeT 2P2e2 þeT 4P4e4 þeT 6P6e6 Â Ã ð2:10Þ where the error signals are ei ¼ri Àyi, i¼1,2,4,6, and weighting matrices are defined as P1 ¼ 10I, P2 ¼ I, P4 ¼ 10I, P6 ¼ 100I ð2:11Þ where I is the identity matrix. The optimal controller parameter vector is then obtained by k nþ 1 ij ¼ k n ijÀgn ijðRn ijÞÀ1 dJ dkij ðk n ijÞ, i ¼ 1,. . .,6, j ¼ 1,2,3 ð2:12Þ where n¼0, 1, y is the iteration number and k 0 ij is given in (2.9). Then, the estimated gradient of J(k) for each particular kij is obtained by dJ dkij ðk n ijÞ ¼ À 1 2001 X2000 t ¼ 0 @y1 @kij ðk n ijÞ h iT P1e1ðk n ijÞþ @y2 @kij ðk n ijÞ h iT P2e2ðk n ijÞ þ @y4 @kij ðk n ijÞ h iT P4e4ðk n ijÞþ @y6 @kij ðk n ijÞ h iT P6e6ðk n ijÞ 0 B B @ 1 C C A: ð2:13Þ The Rn ij is an approximation of the Hessian of J(k), which is chosen as in [9]: Rn ij ¼ 1 2001 X2000 t ¼ 0 @y1 @kij ðk n ijÞ T P1 @y1 @k ðk n ijÞþ @y2 @kij ðk n ijÞ T P2 @y2 @kij ðk n ijÞ þ @y4 @kij ðk n ijÞ T P4 @y4 @kij ðk n ijÞþ @y6 @k ðk n ijÞ T P6 @y6 @kij ðk n ijÞ 2 6 6 4 3 7 7 5: ð2:14Þ Nine iterations were performed until J(k) stops to decrease. Constant step size was used for each controller parameter, i.e. gn ij are the same for all iterations n¼0,1,y. The step size for each controller parameter is defined as gn 11 ¼ 1, gn 12 ¼ 0:01, gn 21 ¼ 1, gn 22 ¼ 1, gn 23 ¼ 1, gn 31 ¼ 1, gn 32 ¼ 0:1, gn 41 ¼ 1, gn 42 ¼ 1, gn 51 ¼ 1, gn 51 ¼ 0:001, gn 61 ¼ 1, gn 62 ¼ 0:1: for n ¼ 0,1,2,. . . In each iteration step, as many gradient experiments are needed to be carried out as there are parameters to tune, which is 13 in our case, namely one normal experiment and 13 gradient experiments. The normal experiment setup is shown in Fig. 8. In the normal experiment, the setpoints are set as follows: r1 ¼ 10 MW, r2 ¼ 6895 Pa, r4 ¼ 3%, r6 ¼ 0:0254 m, r3 ¼ r5 ¼ r7 ¼ 0: Here, the external signal z is not used. Then, the error signals et(k) are obtained from this closed-loop experiment. The setup for gradient experiments is shown in Fig. 9. In each gradient experiment, the error signals e generated from the normal experiment are filtered through a linear filter whose Laplace transfer function matrix is F0 LðkijÞ which is the derivative of the linearized controller transfer function matrix FL(k) in Eq. (2.8) with respect to a particular controller parameter kij where kij ¼ k n ij. These filtered signals, working as the external signal z, are then used as input to the original closed-loop system, while setting the setpoint r to zero, that is r1 ¼ r2 ¼r3 ¼r4 ¼r5 ¼ r6 ¼r7 ¼0. According to IFT, the corresponding output signal y that is obtained in each gradient experiment is the estimate of the ð@yt=@kijÞðk n ijÞ. After ð@yt=@kijÞðk n ijÞ is obtained, the gradient estimate dJðk n ijÞ=dkij is calculated according to Eq. (2.13). In Figs. 8 and 9 the closed loop represents the dynamics of the actual controller and plant, whereas the linearized filter in Fig. 9 is part of the IFT tuning software. The simulation results of the above procedure are presented in Section 4. 3. Simulation testbed: process and controller models Although the process model used in this work is a simplified one based on transfer functions, it provides a realistic platform for the comparison between the sequential heuristic and the simul- taneous optimization-based multi-loop tuning methods. The process model is shown in Fig. 10 and is seen to compute the generator power output, the turbine throttle pressure, the drum level, the excess oxygen, and the fuel, air, and feedwater flow rates. The inputs to the process model are the turbine valve position, firing rate demand, forced draft fan damper demand, and feedwater valve position demand. The model is an incremental model and is designed to represent a 250 MW plant. The model is nonlinear as shown in Fig. 10. Deadtimes are included in the model, i.e. blocks ‘‘TV to MW3’’, ‘‘FR to PT2’’ and ‘‘FR to FF2’’ to represent the time delays inherent in the processes, such as coal pulverizer dynamics. There are cross couplings in the model between several inputs and outputs. The turbine valve position affects both the power output and the throttle pressure as does the firing rate demand. The latter also affects the excess oxygen. The power output (steam flow rate) also affects the drum level. The control system model structure used in the closed-loop simulation is given in Fig. 6. To carry out controller tuning, step change inputs are provided on all setpoints, and the MW load setpoint also includes a ramp input. Although the IFT algorithm is developed for LTI systems, it has been found [7] that IFT can still generate the true gradient up to the first order if the nonlinear system can be well approximated by its linearized model around its operating point corresponding to a specific reference input. Such linearization of the nonlinear model around its operating point is carried out as follows: In our case, there are four main nonlinear components: a nonlinear gain, delays, biases, and calculation of the air-flow/fuel-flow ratio. The nonlinear gain, described by a lookup table in the model, can be approximated as linear gain of 0.5 around operating point. Delays are approximated by the fourth-order Pade approximation, i.e. eÀ2s % s4 À10s3 þ45s2 À105sþ105 s4 þ10s3 þ45s2 þ105sþ105 , eÀ30s % s4 À0:6667s3 þ0:2s2 À0:03111sþ0:002074 s4 þ0:6667s3 þ0:2s2 þ0:03111sþ0:002074 : Fig. 8. Setup for the normal experiment. 0r e y Nonlinear PID Cluster Nonlinear Plant Linearized Filter - generated from normal experiment in Fig. 8 Setpoint Fig. 9. Setup for the gradient experiments. S. Zhang et al. / ISA Transactions 51 (2012) 609–621 617
  10. 10. Biases are dealt with by treating all values as deviations about a nominal value. Finally, the air/fuel flow ratio is approximated as a summation for small deviations. The air flow and fuel flow units are both percentage values of full-load nominal value. Since our model is an incremental one, these units are both deviations from operating point which is set at 80% load. Therefore, the model does this ratio calculation as A=F ¼ AF þ80 FF þ80 À1 where AF and FF are the air flow and fuel flow rate deviations, respectively. However, when those deviations are small A=F % 1 80ðAFÀFFÞ: Thus the linearized system is obtained in the form of a 7 Â 4 transfer function matrix given by MW TP SF O2 AF DL FW 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 ¼ H11ðsÞ H12ðsÞ 0 0 H21ðsÞ H22ðsÞ 0 0 H31ðsÞ H32ðsÞ 0 0 0 H42ðsÞ H43ðsÞ 0 0 0 H53ðsÞ 0 H61ðsÞ H62ðsÞ 0 H64 sð Þ 0 0 0 H74 sð Þ 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 TV FR FD FWV 2 6 6 6 4 3 7 7 7 5 ð3:1Þ where Fig. 10. Schematic diagram of the simplified process model in Simulink. H11ðsÞ ¼ 900s5 À9000s4 þ4:05 Â 104 s3 À9:45 Â 104 s2 þ9:45 Â 104 s 2760s6 þ2:784 Â 104 s5 þ1:266 Â 105 s4 þ3:007 Â 105 s3 þ3:153 Â 105 s2 þ2:551 Â 104 sþ105 , H12ðsÞ ¼ 0:0001454s4 À9:691 Â 10À5 s3 þ2:907 Â 10À5 s2 À4:523 Â 10À6 sþ3:015 Â 10À7 s6 þ0:6747s5 þ0:2054s4 þ0:03273s3 þ0:00233s2 þ1:762 Â 10À5 sþ6:844 Â 10À8 , H21ðsÞ ¼ À3:677s2 À0:0352sÀ0:001043 5s3 þ1:042s2 þ0:008645sþ4:9 Â 10À5 , H22ðsÞ ¼ 0:000603s4 À0:000402s3 þ0:0001206s2 À1:876 Â 10À5 sþ1:251 Â 10À6 s6 þ0:6747s5 þ0:2054s4 þ0:03273s3 þ0:00233s2 þ1:762 Â 10À5 sþ6:844 Â 10À8 , H31ðsÞ ¼ 180s5 À1800s4 þ8100s3 À1:89 Â 104 s2 þ1:89 Â 104 s 2760s6 þ2:784 Â 104 s5 þ1:266 Â 105 s4 þ3:007 Â 105 s3 þ3:153 Â 105 s2 þ2:551 Â 104 sþ105 , H32ðsÞ ¼ 2:907 Â 10À5 s4 À1:938 Â 10À5 s3 þ5:815 Â 10À6 s2 À9:045 Â 10À7 sþ6:03 Â 10À8 s6 þ0:6747s5 þ0:2054s4 þ0:03273s3 þ0:00233s2 þ1:762 Â 10À5 sþ6:844 Â 10À8 , H42ðsÞ ¼ À0:008151s4 þ0:005434s3 À0:00163s2 þ0:0002536sÀ1:691 Â 10À5 25s8 þ226:5s7 þ226:7s6 þ105:6s5 þ28:51s4 þ4:664s3 þ0:4349s2 þ0:01827sþ6:762 Â 10À5 , H43ðsÞ ¼ 0:25 3600s4 þ2040s3 þ409s2 þ34sþ1 , S. Zhang et al. / ISA Transactions 51 (2012) 609–621618
  11. 11. The staircase algorithm [30] is then used to determine the controllability and observability of the linearized system after transforming the transfer function into a state-space representation characterized by 52 state variables. From staircase algorithm, it has been shown that there are also 52 controllable states and 52 observable states. Therefore, by the definitions of the controllability and observability [31], the linearized system is both controllable and observable, although it is essentially a 4 Â 4 system since only four out of seven outputs need to track the setpoint changes, whereas the rest three can be viewed as internal variables only. In most practical conditions, the variables of the nonlinear process in Fig. 10 will stay in the vicinity of its operating point where the linearized model above is a good approximation of its nonlinear counterpart. It is, therefore, likely that IFT would per- form adequately, showing a degree of robustness to nonlinearities. This is indeed demonstrated by the simulation results in Section 4. 4. Tuning results and comparisons The results of the IFT method and their comparison with the results of the empirical tuning procedure are presented next. Comparisons are provided for both the control system figure of merit and the time response plots of the controlled variables. The IFT procedure was applied to the closed loop system consisting of the nonlinear plant of Fig. 10 and the nonlinear PID cluster of Fig. 6, with a linear prefilter outside of the loop, as shown in Fig. 8 (normal experiment) and 9 (gradient experiment) in Section 2.2. Table 2 shows the design criteria and controller parameter values during each iteration step. The closed-loop system was tuned according to the control performance requirements given in Table 1. The simulation results of the closed-loop system response subject to 5%/min ramp unit load setpoint change from 80% to 90% along with the initial, the IFT tuned, and the manually tuned PID controllers are shown in Fig. 11. The ‘‘initial’’, ‘‘Manual’’ and ‘‘IFT’’ in this figure mean that the simulation results are obtained with the PID controller parameters set to the original, manually tuned and the IFT tuned values, respectively. The manually tuned PID parameters were provided by some experienced field engineer. The simulation results summarized in Table 2 show that the minimum for the criterion is achieved after eight iterations, with the reduction of the value of J from 87.2583 to 7.9814. In the ninth iteration, the criterion value starts increasing. Therefore, after nine H53ðsÞ ¼ 1 144s2 þ24sþ1 , H53ðsÞ ¼ 1 144s2 þ24sþ1 , H61ðsÞ ¼ 31:5s6 À317:7s5 þ1445s4 À3429s3 þ3591s2 À283:5s 4:347 Â 106 s10 þ4:505 Â 107 s9 þ2:116 Â 108 s8 þ5:297 Â 108 s7 þ6:32 Â 108 s6 þ1:886 Â 108 s5 þ2:323 Â 107 s4 þ1:305 Â 106 s3 þ2:94 Â 104 s2 þ105s , H62ðsÞ ¼ 5:088 Â 10À6 s5 À3:828 Â 10À6 s4 þ1:308 Â 10À6 s3 À2:455 Â 10À7 s2 þ2:412 Â 10À8 sÀ9:045 Â 10À10 1575s10 þ1498s9 þ653:9s8 þ166:9s7 þ26:18s6 þ2:458s5 þ0:1267s4 þ0:003011s3 þ2:015 Â 10À5 s2 þ6:844 Â 10À8 s , H64ðsÞ ¼ À0:0875sþ0:0075 945s5 þ828s4 þ246s3 þ28s2 þs , H74ðsÞ ¼ 0:5 9s2 þ6sþ1 : Table 2 Iteration data. Iteration Criteria, J k11 k12 k21 k22 k23 k31 0 87.2583 1 0.1 0.05 0.00001 2 4 1 47.1814 1.1401 0.1014 0.0541 0.00001 5.7345 3.8590 2 29.9074 1.1372 0.1016 0.0719 0.00001 8.6881 3.5960 3 21.2190 1.0811 0.1014 0.1025 0.00001 10.0482 3.2901 4 15.8913 1.0141 0.1011 0.1415 0.00001 10.7007 2.9827 5 12.1564 0.9504 0.1007 0.1902 0.00001 10.9887 2.6593 6 9.5995 0.8904 0.1001 0.2511 0.00001 11.0151 2.3146 7 8.2462 0.8349 0.0995 0.3485 0.00001 10.8606 1.9491 8 7.9814 0.7853 0.0987 0.3642 0.00001 10.6147 1.5766 9 8.1092 0.7444 0.0981 0.3777 0.00001 10.3517 1.2295 Iteration k32 k41 k42 k51 k52 k61 k62 0 0.1 4 0.4 1 0.0001 20 0.6 1 0.0988 3.8563 0.3666 0.9904 9.9744eÀ5 19.8587 0.5851 2 0.0919 3.5887 0.2465 0.9362 9.9753eÀ5 19.5989 0.5594 3 0.0858 3.2754 0.1873 0.8690 9.9795eÀ5 19.2954 0.5306 4 0.0821 2.9609 0.1794 0.8118 9.9838eÀ5 18.9878 0.5027 5 0.0793 2.6295 0.1826 0.7699 9.9870eÀ5 18.6603 0.4745 6 0.0772 2.2740 0.1903 0.7389 9.9891eÀ5 18.3050 0.4456 7 0.0754 1.8897 0.2003 0.7130 9.9905eÀ5 17.9140 0.4156 8 0.0739 1.4767 0.2126 0.6889 9.9917eÀ5 17.4811 0.3845 9 0.0730 1.0449 0.2265 0.6688 9.9928eÀ5 17.0111 0.3536 S. Zhang et al. / ISA Transactions 51 (2012) 609–621 619
  12. 12. steps iterations are stopped and the controller parameters in the eighth iteration are adopted as the optimal ones and substituted directly into the original nonlinear PID cluster to test its performance. If we use the manually tuned PID parameters, the J value is 55.5546 compared to 7.9814 by IFT as shown in Table 3. It can also be seen from the plots in Fig. 11 that the system dynamic performance corresponding to ramp unit load setpoint change has improved significantly with the IFT optimized PID controller parameters, especially the throttle pressure that has much less oscillation and settles down much faster compared to manual tunings. While the excess oxygen time response with IFT tuned parameters, although satisfactory, is not superior to those with manually tuned gains, we can improve its performance by putting a greater corresponding weight matrix in the control criterion Eq. (2.10). 0 1000 2000 0 5 10 15 Time (s) r1(MegawattOutput/MW) Setpoint 0 1000 2000 -2 -1 0 1 2 x 105 Time (s) r2(ThrottlePressure/Pa) Setpoint 0 1000 2000 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (s) r4(ExcessOxygen/%) Setpoint 0 1000 2000 -0.01 0 0.01 Time (s) r6(DrumLevel/m) Setpoint 0 1000 2000 0 5 10 15 Time (s) y1(MegawattOutput/MW) Initial 0 1000 2000 -2 -1 0 1 2 x 105 Time(s) y2(ThrottlePressure/Pa) Initial 0 1000 2000 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (s) y4(ExcessOxygen/%) Initial 0 1000 2000 -0.01 0 0.01 Time (s) y6(DrumLevel/m) Initial 0 1000 2000 0 5 10 15 Time (s) y1(MegawattOutput/MW) Manual 0 1000 2000 -2 -1 0 1 x 105 Time(s) y2(ThrottlePressure/Pa) Manual 0 1000 2000 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time(s) y4(ExcessOxygen/%) Manual 0 1000 2000 -0.01 0 0.01 Time (s) y6(DrumLevel/m) Manual 0 1000 2000 0 5 10 15 Time (s) y1(MegawattOutput/MW) IFT 0 1000 2000 -2 -1 0 1 2 x 105 Time(s) y2(ThrottlePressure/Pa) IFT 0 1000 2000 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (s) y4(ExcessOxygen/%) IFT 0 1000 2000 -0.01 0 0.01 Time (s) y6(DrumLevel/m) IFT Fig. 11. Comparison of closed-loop responses under 5%/min ramp unit load setpoint change with the original, manually tuned, and IFT tuned PID controllers. The units are: megawatts for y1 and r1, Pa for y2 and r2, % for y4 and r4, and meters for y6 and r6. Table 3 Comparison of initial, manually tuned, and IFT tuned PID gains in the nonlinear cluster used in the closed-loop simulation and their corresponding objective function values. k11 k12 k21 k22 k23 k31 k32 Initial 1 0.1 0.05 0.00001 2 4 0.1 Manual 0.7 0.06 0.035 0.00004 6 4 0.1 IFT 0.7853 0.0987 0.3642 0.00001 10.6147 1.5766 0.0739 k41 k42 k51 k52 k61 k62 J Initial 4 0.4 1 0.0001 20 0.6 81.7085 Manual 4 0.55 1.3 0.005 10 1 52.5163 IFT 1.4767 0.2126 0.6889 9.9905eÀ5 17.4811 0.3845 5.2768 S. Zhang et al. / ISA Transactions 51 (2012) 609–621620
  13. 13. 5. Conclusions The general area of automatic multi-loop PID tuning shows an unexpectedly strong promise. The basic IFT configuration, one of the simplest techniques in this area, has demonstrated excellent tuning capability for a nonlinear MIMO PID cluster of the inter- mediate complexity controlling a simplified nonlinear model representing power plant dynamics at a single operating point. The technique shows good flexibility in shaping the time domain responses of system components according to specifications through the appropriately chosen static and/or dynamic weight- ing of the individual terms in the performance index. While a comparison was done against an empirically tuned system, a multi-loop tuning method may provide substantial benefits even if its performance is the same as or only slightly better than an empirically tuned system. 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