Proceedings of the ASME 2013 Dynamic Systems and Control Conference DSCC2013 October 21-23, 2013, Palo Alto, California, USA
DSCC2013-4097
COMPARISON OF SEVERAL SELF-OPTIMIZING CONTROL METHODS FOR EFFICIENT OPERATION FOR A CHILLED WATER PLANT Baojie Mu Department of Electrical Engineering The University of Texas at Dallas Richardson, TX 75080 E-mail:
[email protected]
Yaoyu Li Department of Mechanical Engineering The University of Texas at Dallas Richardson, TX 75080 E-mail:
[email protected]
John E. Seem Building Efficiency Research Group Johnson Controls Inc. Milwaukee, WI 53201 E-mail:
[email protected]
In the past decades, there have been different algorithms of extremum seeking control developed based on different fashions of extracting the gradient information for real-time search of the optimum input. This study considers three kinds of ESC method: the Dither ESC, Switching ESC and Slidingmode ESC. The dither (or perturbation) based ESC (dither ESC, dESC) derives the gradient estimation by injecting small dither signal into the system input estimate [4-7]. The switching method ESC (swESC) is a classic version of ESC, in which the system hunts continuously about the extremum by directly obtaining the slope information [8, 9]. The slidingmode ESC (smESC) tracks the extremum of the system output by regulating the system opereation along a sliding manifold designed [10, 11]. In comparison, the SPSA is a type of stochastic searching method where the gradient is estimated by evaluting certain loss function via random perturbations [3, 12, 13]. The SPSA has shown some successful applications of high-dimensional real-time optimization problems. Among the different variations of SPSA, the one-measurement form of SPSA (SPSA1) first introduced by Spall [14] has proved to be asymptotically superior to the standard two-measurement form SPSA (SPSA2) [12, 13] for some certain class of problems. For system with slow dynamics, SPSA1 could save searching time. Further, Abdulla et al. [15] proposed another form of onemeasurement form SPSA that uses a similar analysis as Spall [14] where the proposed method has less sensitivity for the measurement of loss function. In the same publication, the authors also proposed an adaptive form of one-measurement SPSA where the Hessian matrix is estimated to accelerate the convergence speed.
ABSTRACT Self-optimizing control methods have received significant attention recently, due to the merit of nearly model-free capability of real-time optimization. Of particular interest in our study are two classes of self-optimizing control strategies, i.e. the Extremum Seeking Control (ESC) and Simultaneous Perturbation Stochastic Approximation (SPSA). Six algorithms, including dither ESC, adaptive dither ESC, switching ESC, one-measurement SPSA, and adaptive one-measurement SPSA are compared based on simulation study with a Modelcia based virtual plant of chiller-tower plant. The integral performance indices are evaluated to incorporate both transient and steadystate characteristics. Some design procedures are summarized for these self-optimizing control algorithms. INTRODUCTION Different from the objectives of regulation and stabilization about some known reference for traditional control system design, self-optimizing control deals with the real-time optimization problem of finding an optimum input to maintain the output at the extremum (maximum/minimum) value with least amount system knowledge [1, 2]. This study is to compare several methods in the two major classes of self-optimizing control algorithms, the Extremum Seeking Control (ESC) and the Simultaneous Perturbation Stochastic Approximation (SPSA) [3]. There have been ever growing applications of these algorithms to various engineering problems, e.g. aerospace, vehicle, HVAC, and renewable energy systems. The goal of this study is to compare several algorithms of ESC and SPSA for a practical application ± efficient operation of a chiller-tower chilled water plant.
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5 6
(Þ L :* %ÞÍ * %Þ E ÜÞ +ã ; * % á %Þ L ÞÁÖ7->ÁÖ * ÚÖ
6 ÖÖ ¿Öá-
CÜÞ kààÞ o L
E:
ÚÖ
Input (u)
(23)
Þ>5
Where áÞ L 5 > *
300
(22)
250 200
;Í ?, ÜCÞ L CÜÞ5 kààÞ o-CÜÞ kààÞ o,
ÖÖ ¿Öá66 ; ì: Ö66 ;?ì: Ö7, ÖÖ :¿Öá- >¿Öá. ;
CÜÞ5 kààÞ o L
0
66 ì: Ö66 ;?ì: Ö7; ÖÖ ¿Öá.
1
2
3
Time (sec)
4 4
x 10
5
There are many candidates for such a BÞ , in the above (Þ L proposed aSPSA1, the BÞ was chosen such that * 5 6 Í %Þ * %Þ E ÜÞ +ã ; :* in this paper. For more choices of BÞ , the readers are referred to [17]. A general guideline for designing aSPSA1 controllers is summarized as 1) At P L P4 , Set counter index G L s , pick initial guess àà4 and nonnegative gain coefficients =á ? # Ù and Û ( Ù L räxrtá Û L räsrs are recommended) for sequences =Þ and ?Þ . 2) At P L P4 E G6 F ÜP, Evaluate the loss function U:àÞ>> ; 3) At P L P4 E :G E s;6 F ÜP, > Evaluate the loss function U:àÞ?5 ;; 4) At P Ð >P4 E :G E s;6 F ÜPá P4 E :G E s;6;, Generate updated Compute gradient approximation (Þ according to Eqs.(22, 23) CÜÞ kààÞ oand Hessian matrix * Generate updated à estimate: ààÞ>5 L ààÞ ?5 à (Þ =Þ CÜÞ kàÞ o* Generate updated iteration index: • L • E sá Generate updated sequences: =Þ L =Þ>5 á ?Þ L ?Þ>5 á ¿Þá5 L ¿Þ>5á5 á ¿Þá6 L ¿Þ>5á6 5), 6) same as iSPSA1.
Output (y)
2.5
x 10
2.4 2.3
0
1
2
3
Time (sec)
4 4
x 10
FIGURE 8. INPUT/OUTPUT TRAJECTORY OF dESC
Input (u)
300 250 200 0
1
2
3
Time (sec)
4 4
x 10
5
Output (y)
2.5
x 10
2.4 2.3
0
1
2
3
Time (sec)
SIMULATION STUDY The aforementioned six self-optimizing methods are employed to a Modelica based dynamic virtual chiller-tower plant of air conditioning systems in [16]. The map of the plant is shown in Fig. 7. As the Modelica models in the virtual plant are built upon the nonlinear differential equations, the simulated process variables are considered to reflect key characteristics of the actual process. The input/output performances of six methods are shown in Fig. 8 through Fig. 13.
4 4
x 10
FIGURE 9. INPUT/OUTPUT TRAJECTORY OF adESC
Input (u)
300 250 200 0
1
2
3
Time (sec)
4 4
x 10
5
Output (y)
2.5
x 10
2.4 2.3
0
1
2 Time (sec)
FIG. 7. STATIC MAP OF THE CHILLER-TOWER SYSTEM WITH INPUT OF COOLING TOWER FAN SPEED AND OUTPUT OF TOTAL POWER CONSUMPTION.
3
4 4
x 10
FIGURE 10. INPUT/OUTPUT TRAJECTORY OF swESC
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consider the effect of both time and error. The integral performance indices are computed by t (24) ISE e(t ) 2 dt
Input (u)
300 250
0
1
2
3
Time (sec)
IAE
4 4
x 10
5
0
1
2
3
Output (y)
Time (sec)
4 4
x 10
FIGURE 11. INPUT/OUTPUT TRAJECTORY OF smESC
Input (u)
200 0
1
2
3
Time (sec)
4 4
x 10
5
x 10
2.5 Output (y)
(25)
e(t ) dt t2
ITSE
³
ITAE
³
t1 t2
t1
te(t ) 2 dt
(26)
t e(t ) dt
(27)
ISE
IAE
ITSE
ITAE
dESC
0.1524
0.2427
0.0175
0.0546
adESC
0.0902
0.1848
0.0096
0.048
swESC
0.1189
0.1997
0.0109
0.0326
smESC
0.0465
0.1358
0.0069
0.0459
SPSA1
0.3108
0.4388
0.0581
0.1445
aSPSA1
0.342
0.4586
0.0732
0.1771
250
2.4 0
1
2
3
Time (sec)
TABLE 2. INTEGRAL PERFORMANCE INDICES OF SIX SELF-OPTIMIZING METHODS (P Ð >wrrrá twrrr?)
4 4
300 250 200 0
1
2
3
Time (sec)
4
x 10
2.4 2.3
0
1
2 Time (sec)
IAE
ITSE
ITAE
dESC
0.0824
0.1719
0.0091
0.0514
3
adESC
0.054
0.1456
0.0067
0.0461
swESC
0.0595
0.1084
0.0029
0.0145
smESC
0.0406
0.1595
0.0106
0.0656
SPSA1
0.1861
0.3404
0.0377
0.1269
aSPSA1
0.2317
0.4008
0.0639
0.1729
As shown in Table 1, the adESC has smaller performance indices than that of dESC, which indicates that adESC has better transient performance than that of dESC. The reason is that with same perturbed dither signal, adESC has a faster convergence speed. It can be shown that smESC has smaller indices than that of swESC except for ITAE. The reason is that smESC has larger steady oscillation but faster convergence speed. Even aSPSA1 has faster convergence speed, its steady state oscillation is larger than that of iSPSA1. Thus its performance indices are larger. The larger steady state oscillation of aSPSA1 is due to the introduction of two perturbation sequences while only one perturbation sequence is introduced for iSPSA1. With a larger time interval of 20,000 second where the steady state performance are weighted more than transient, Table 2
4
x 10
5
2.5
ISE
x 10
FIGURE 12. INPUT/OUTPUT TRAJECTORY OF iSPSA1
Input (u)
t2
t1
TABLE 1. INTEGRAL PERFORMANCE INDICES OF SIX SELF-OPTIMIZING METHODS (P Ð >wrrrá swrrr?)
300
Output (y)
³
where [ P5 á P6 ] is the selected time interval, A:P; is the difference between actual input/output and their corresponding desired values. Table 1 shows the integral performances indices with time interval of 10,000 second. Note that the performance indices are computed from the plant output where the time vector and output data are normalized to unity.
2.4
2.3
2
x 10
2.5
2.3
³
t1
200
4 4
x 10
FIGURE 13. INPUT/OUTPUT TRAJECTORY OF aSPSA1
The integral performance indices (i.e. ISE, IAE, ITSE,ITAE) have shown better performance among typical performance characteristics (i.e. steady state error, settling time) as they
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Proceedings of the ASME 2013 Dynamic Systems and Control Conference DSCC2013 October 21-23, 2013, Palo Alto, California, USA
DSCC2013-4097
COMPARISON OF SEVERAL SELF-OPTIMIZING CONTROL METHODS FOR EFFICIENT OPERATION FOR A CHILLED WATER PLANT Baojie Mu Department of Electrical Engineering The University of Texas at Dallas Richardson, TX 75080 E-mail:
[email protected]
Yaoyu Li Department of Mechanical Engineering The University of Texas at Dallas Richardson, TX 75080 E-mail:
[email protected]
John E. Seem Building Efficiency Research Group Johnson Controls Inc. Milwaukee, WI 53201 E-mail:
[email protected]
In the past decades, there have been different algorithms of extremum seeking control developed based on different fashions of extracting the gradient information for real-time search of the optimum input. This study considers three kinds of ESC method: the Dither ESC, Switching ESC and Slidingmode ESC. The dither (or perturbation) based ESC (dither ESC, dESC) derives the gradient estimation by injecting small dither signal into the system input estimate [4-7]. The switching method ESC (swESC) is a classic version of ESC, in which the system hunts continuously about the extremum by directly obtaining the slope information [8, 9]. The slidingmode ESC (smESC) tracks the extremum of the system output by regulating the system opereation along a sliding manifold designed [10, 11]. In comparison, the SPSA is a type of stochastic searching method where the gradient is estimated by evaluting certain loss function via random perturbations [3, 12, 13]. The SPSA has shown some successful applications of high-dimensional real-time optimization problems. Among the different variations of SPSA, the one-measurement form of SPSA (SPSA1) first introduced by Spall [14] has proved to be asymptotically superior to the standard two-measurement form SPSA (SPSA2) [12, 13] for some certain class of problems. For system with slow dynamics, SPSA1 could save searching time. Further, Abdulla et al. [15] proposed another form of onemeasurement form SPSA that uses a similar analysis as Spall [14] where the proposed method has less sensitivity for the measurement of loss function. In the same publication, the authors also proposed an adaptive form of one-measurement SPSA where the Hessian matrix is estimated to accelerate the convergence speed.
ABSTRACT Self-optimizing control methods have received significant attention recently, due to the merit of nearly model-free capability of real-time optimization. Of particular interest in our study are two classes of self-optimizing control strategies, i.e. the Extremum Seeking Control (ESC) and Simultaneous Perturbation Stochastic Approximation (SPSA). Six algorithms, including dither ESC, adaptive dither ESC, switching ESC, one-measurement SPSA, and adaptive one-measurement SPSA are compared based on simulation study with a Modelcia based virtual plant of chiller-tower plant. The integral performance indices are evaluated to incorporate both transient and steadystate characteristics. Some design procedures are summarized for these self-optimizing control algorithms. INTRODUCTION Different from the objectives of regulation and stabilization about some known reference for traditional control system design, self-optimizing control deals with the real-time optimization problem of finding an optimum input to maintain the output at the extremum (maximum/minimum) value with least amount system knowledge [1, 2]. This study is to compare several methods in the two major classes of self-optimizing control algorithms, the Extremum Seeking Control (ESC) and the Simultaneous Perturbation Stochastic Approximation (SPSA) [3]. There have been ever growing applications of these algorithms to various engineering problems, e.g. aerospace, vehicle, HVAC, and renewable energy systems. The goal of this study is to compare several algorithms of ESC and SPSA for a practical application ± efficient operation of a chiller-tower chilled water plant.
1
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[16] ; /L < /L - ( 6HHP 3 /L ³'\QDPLF 0RGHOLQJ and Self-optimizing Operation of Chilled Water Systems 8VLQJ ([WUHPXP 6HHNLQJ &RQWURO ´ Energy and Buildings, Vol. 58, March 2013, pp. 172-182. [17] J.C. Spall, 2000. "Adaptive Stochastic Approximation by the Simultaneous Perturbation Method". IEEE Transactions on Automatic Control, 45(10), pp. 18391853. [18] J.C. Spall, 2009. "Feedback and weighting mechanisms for improving jacobian estimates in the adaptive simultaneous perturbation algorithm,´ IEEE Transactions on Automatic Control, 54(6), pp. 1216-1229.
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