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Iterative Procedure for Tuning Decentralized PID Controllers
9th International Symposium on Advanced Control of Chemical Processes 9th International Symposium on Advanced Chemical Processes June 7-10, 2015. Whistler, British Columbia,Control Canadaof 9th International International Symposium on Advanced Advanced Control of Chemical Processes 9th Symposium on June 7-10, 2015. Whistler, British Columbia,Control Canadaof Chemical Processes Available online at www.sciencedirect.com June June 7-10, 7-10, 2015. 2015. Whistler, Whistler, British British Columbia, Columbia, Canada Canada
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Iterative Procedure for Tuning Iterative Procedure for Tuning Iterative Procedure for Iterative Procedure for Tuning Tuning Decentralized PID Controllers Decentralized PID Controllers Decentralized PID Controllers Decentralized PID Controllers
Thiago A.M. Euz´ ebio ∗∗ P´ ericles R. Barros ∗∗ Thiago A.M. Euz´ e bio e ricles R. Barros ∗∗ ∗ P´ ∗ P´ Thiago A.M. Euz´ e bio e Thiago A.M. Euz´ ebio P´ ericles ricles R. R. Barros Barros ∗ ∗ Department of Electrical Engineering, Federal University of Campina ∗ Department of Electrical Engineering, Federal University of Campina ∗ Department of Engineering, Federal Grande, Brazil (e-mail: [email protected], Department of Electrical Electrical Engineering, Federal University University of of Campina Campina Grande, Brazil (e-mail: [email protected], Grande, Brazil (e-mail: [email protected], [email protected]). Grande, [email protected]). (e-mail: [email protected], [email protected]). [email protected]). Abstract: An iterative procedure is proposed to design decentralized PID controllers for multiAbstract: An procedure is proposed to design decentralized PID for Abstract: An iterative iterative procedure is proposed to designthe decentralized PID controllers controllers for multimultiloop processes. Each SISO loop is is designed at to a time, controller parameters are computed Abstract: An iterative procedure proposed design decentralized PID controllers for multiloop processes. Each SISO loop is designed at a time, the controller parameters are computed loop Each SISO loop is designed at aa time, the controller parameters are computed by a processes. convex optimization problem with constraints on stability margins. Despite the SISO loop processes. Each SISO loop is designed at time, the controller parameters are computed by a convex optimization problem with constraints on stability margins. Despite the SISO by a optimization with stability margins. the SISO approach, loops interactionsproblem are taken intoconstraints account by on Gershgorin bands and Despite Equivalent Openby a convex convex optimization problem with constraints on stability bands margins. Despite the OpenSISO approach, loops interactions are taken into account by Gershgorin and Equivalent approach, loops interactions are taken into account by Gershgorin bands and Equivalent Openloop Process (EOP). Two simulation examples are presented to compare the performance with approach, loops interactions are takenexamples into account by Gershgorin bands and Equivalent Openloop Process (EOP). Two simulation are presented to compare the performance loop Process (EOP). Two Two simulation simulation examples examples are are presented presented to to compare compare the the performance performance with with related techniques. loop Process (EOP). with related techniques. related techniques. related © 2015, techniques. IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: PID, MIMO systems, Optimization, Gain and phase margins, Process control. Keywords: Keywords: PID, PID, MIMO MIMO systems, systems, Optimization, Optimization, Gain Gain and and phase phase margins, margins, Process Process control. control. Keywords: PID, MIMO systems, Optimization, Gain and phase margins, Process control. 1. INTRODUCTION The second step, which design is now based on EOPs, each 1. INTRODUCTION The which design based on each 1. INTRODUCTION The second secondisstep, step, whichiteratively design is is now now based on EOPs, EOPs, each controller re-tuned untilbased controller parame1. INTRODUCTION The second step, which design is now on EOPs, each controller is re-tuned iteratively until controller paramecontroller is re-tuned iteratively until controller parameters converge. The SISO controlleruntil design methodparameused in controller is re-tuned iteratively controller converge. The design method used The basic regulatory control layer of process plants largely ters terssteps converge. The SISO SISOascontroller controller design methodproblem used in in all is formulated a convex design optimization ters converge. The SISO controller method used in The basic regulatory control layer of process plants largely all steps is formulated as a convex optimization problem The basic regulatory control layer of process plants largely consists ofregulatory decentralized SISO PID controllers. Thelargely main which all steps is formulated as a convex optimization problem minimizes the diagonal loop integral error subject The basic control layer of process plants all steps is formulated as a convex optimization problem consists of SISO PID The main minimizes the loop consists are: of decentralized decentralized SISOstructure, PID controllers. controllers. The main which reasons relatively simple easy to The maintain, which minimizes the diagonal diagonal loop integral integral error error subject subject to minimum stability margins. loop consists of decentralized SISO PID controllers. main which minimizes the diagonal integral error subject reasons are: relatively simple structure, easy to maintain, to minimum stability margins. reasonsparameters are: relatively relatively simplethan structure, easy to to maintain, maintain, fewer to tune full multivariable con- to minimum stability margins. reasons are: simple structure, easy to minimum stability margins. fewer parameters to tune than full multivariable conWith this proposed procedure some advantages are achieved fewer parameters parameters to tune tune than full full multivariable controllers, and loop failure tolerance. Despite the SISOconap- With procedure some advantages are fewer to than multivariable trollers, and loop failure tolerance. Despite the SISO apWith this this proposed proposed procedure some advantages are achieved achieved compared to relatedprocedure methods. some First,advantages full Gershgorin bands trollers, and loop failure tolerance. Despite the SISO approach, tuning of each loop cannot be done independently With this proposed are achieved compared to related methods. First, full Gershgorin bands trollers, and loop failure tolerance. Despite the SISO approach, tuning of each loop cannot be done independently compared to related related methods. First, full Gershgorin bands are considered, different from Ho andfull Xu Gershgorin (1998), andbands Chen proach, tuning of each loop cannot be done independently due to loops interactions. Applying the tuning methods compared to methods. First, are considered, different from Ho and Xu (1998), and Chen proach, tuninginteractions. of each loop Applying cannot bethe done independently due to loops tuning methods are considered, different from Ho and Xu (1998), and Chen and Seborg (2001), that are based on no more than dueSISO to loops loops interactions. Applying the the tuning methods for systems ignoring interactions often leadsmethods to poor are considered, differentthat fromare Ho based and Xuon (1998), and Chen and Seborg (2001), no more than due to interactions. Applying tuning for SISO systems ignoring interactions often leads to poor and Seborg (2001), that are based on no more than two Gershgorin circles, which does not guarantee overall for SISO SISO systems systems ignoring interactions interactions often often leads leads to to poor poor two performance and stability. and Gershgorin Seborg (2001), that aredoes based on no moreoverall than circles, which not guarantee for ignoring performance and stability. two Gershgorin circles, which does not guarantee overall system stability. Second, this procedure is not restricted performance and stability. two Gershgorin circles, which does not guarantee overall stability. Second, this is not performance stability. One way to and consider loop interactions during controller system system stability. Second,processes, this procedure procedure is not restricted restricted to diagonally dominant model is reduction is not system stability. Second, this procedure not restricted One way to consider loop interactions during controller to diagonally dominant processes, model reduction is not One way way consider loop interactions interactions during controller design is to use the Gershgorin bands during (see e.g.controller Ho and necessary, to diagonally dominant processes, model reduction is not and high dimensional processes are not an issue. One to consider loop to diagonally dominant processes, model reduction is not design is to use the Gershgorin bands (see e.g. Ho and necessary, and high dimensional processes are not an issue. design is to use the Gershgorin bands (see e.g. Ho and Xu (1998), Chen and Seborg (2001), and Garcia et al. necessary, and high dimensional processes are not an issue. Third, lower bound on stability margins can be specified design is to use the Gershgorin bands (see e.g. Ho and necessary, andbound high dimensional processes are not specified an issue. Xu Chen Seborg (2001), and et lower on stability margins can Xu (1998), (1998), Chen and andstability Seborg can (2001), and Garcia Garcia et al. al. Third, (2005)). Closed-loop be ensured by shaping Third, lower bound on stability margins can be be specified specified to each lower diagonal loopon transfer function, independently how Xu (1998), Chen and Seborg (2001), and Garcia et al. Third, bound stability margins can be (2005)). Closed-loop stability can be ensured by shaping to each diagonal loop transfer function, independently (2005)). Closed-loop stability can be ensured ensured by shaping shaping the Gershgorin bandsstability in suchcan a way that they do not interactions to each each diagonal diagonal loop transfer transfer function, independently how how are considered in function, design method. (2005)). Closed-loop be by to loop independently how the Gershgorin bands in such a way that they do not the Gershgorin Gershgorin bands in (−1, such 0) way encircle that they they do not not overlap the critical point and it the ap- interactions interactions are are considered considered in in design design method. method. the bands in such aa and way that do interactions are considered in design method. overlap the critical point (−1, 0) encircle it the apoverlap the critical point (−1, 0) and encircle it the appropriate number of times, in accordance with the generoverlap the critical point (−1, 0) and encircle it the ap2. PROBLEM STATEMENT propriate number of in accordance with the generpropriate number of times, times, inadvantage accordance with the gener2. alized Nyquist theorem. Anin of with this the method is propriate number of times, accordance gener2. PROBLEM PROBLEM STATEMENT STATEMENT alized Nyquist theorem. An advantage of this method is 2. PROBLEM STATEMENT alized Nyquist theorem. An advantage of this method is that knowledge of controller parameters from other loops alizedknowledge Nyquist theorem. An parameters advantage offrom thisother method is Consider the closed loop system of Figure 1, where G(s) = that of controller loops that knowledge of of controller parametersoffrom from other loops is unnecessary. However, effectiveness this other method is Consider the closed loop system of Figure 1, where G(s) = that knowledge controller parameters loops is However, effectiveness of method Consider the closed loop system of=Figure Figure 1,1 (s), where G(s) = [g is aclosed n×n loop process, C(s)of diag{c1, . . . ,G(s) cn (s)} ij (s)]n×n the system where = is unnecessary. unnecessary. However, effectiveness of this thisprocesses. method is is Consider restricted to diagonally dominant multi-loop [g ij (s)]n×n is a n×n process, C(s) = diag{c1 (s), . . . , cn (s)} is unnecessary. However, effectiveness of this method is restricted to diagonally dominant multi-loop processes. [g (s)] is a n×n process, C(s) = diag{c (s), . . . , c (s)} ij 1 n (s)c (s)] is a decentralized controller, and L(s) = [g n×n is a n×n process, C(s) = diag{c1 (s), ij . . j. , cn (s)} [gija(s)] restricted to to diagonally diagonally dominant dominant multi-loop multi-loop processes. processes. n×n n×n decentralized controller, and L(s) = [g restricted ij (s)cj (s)]n×n Another way to take loop interactions into account to is (s)] is aathe decentralized controller, and L(s) ij (s)c multivariable loop-gain. The r, = u, [g and y jj are the n×n (s)c (s)] is decentralized controller, and L(s) = [g Another way to take loop interactions into account to ij n×n the multivariable loop-gain. The r, u, and y are the Another way way to istake take loop interactions into account account to is controller design to use theinteractions concept of equivalent openis the multivariable loop-gain. The r, u, and y are Another to loop into to set-point, manipulated, and controlled variable vectors. controller design is to use the concept of equivalent openis the multivariable loop-gain. The r, variable u, and vectors. y are the the set-point, manipulated, and controlled controller design is to use the concept of equivalent openloop process (EOPs) (see e.g. Huang et al. (2003), Vu set-point, manipulated, and controlled variable vectors. controller design is to use the concept ofetequivalent openloop process (EOPs) (see e.g. Huang al. (2003), Vu set-point, manipulated, and controlled variable vectors. loop Lee process (EOPs) (see e.g.al.Huang Huang et This al. (2003), (2003), Vu and (2010), and (see Nie et (2011)). procedure loop process (EOPs) e.g. et al. Vu y and (2010), and et This procedure u r and Lee Lee initial (2010), and Nie Nieparameters et al. al. (2011)). (2011)). Thisloop. procedure y requires controller for each After u G(s) r and Lee (2010), and Nie et al. (2011)). This procedure y u r requires initial controller parameters for each loop. After G(s) y u r requires initial controller parameters for each loop. After all loops have been closed, the controller will then be reG(s) requires initial controller parameters for each loop. be After G(s) all loops have been closed, the controller will then reall loops have been closed, the controller will then be retuned one after the other with all other loops being closed C(s) all loops have been other closed, the controller will then closed be retuned one after with other loops C(s) tunedthe onecontrollers after the the other other with all all otherprevious loops being being closed C(s) with obtained in the step.closed This tuned one after the with all other loops being C(s) with the controllers obtained in the previous step. This with the the controllers controllers obtained in the theparameters previous step. step. This procedure will go on until controllers converge. with obtained in previous This Fig. 1. Decentralized control system. procedure will go on until controllers parameters converge. procedure will go go on model until controllers controllers parameters converge. Most methods need order reduction whenconverge. compute Fig. 1. Decentralized control system. procedure will on until parameters Fig. 1. Decentralized control system. Most methods need model order reduction when compute Most methods need model order reduction when compute Fig. 1. Decentralized control system. the EOP. MostEOP. methods need model order reduction when compute the the EOP. EOP. The PID controller transfer function from the j-th diagothe In this paper, an iterative procedure to tune decentralized The PID controller transfer function from the j-th diagoTheelement PID controller controller transfer function from from the the j-th j-th diagodiagoIn this paper, an iterative procedure to tune decentralized nal of C(s) transfer is The PID function In this this paper, an anisiterative iterative procedure to tune tune decentralized PID controllers proposed by taking otherdecentralized loops inter- nal element of C(s) is In paper, procedure to nal element of C(s) is PID controllers is proposed by taking other loops internal element of C(s) is PID controllers controllers is proposed proposed by taking other other loops interactions into account both by by Gershgorin bands as EOPs. PID is taking loops interactions into both by bands as kd s ki actions into account account both by Gershgorin Gershgorin bands as EOPs. EOPs. The procedure consists of two main steps. bands In the as first step, actions into account both by Gershgorin EOPs. k (1) cj (s) = kp + k The procedure consists of two main steps. In the first step, i + ds k k ss 1 The procedure consists of two main steps. In the first step, i d+ initial controllers are designed based on Gershgorin bands. + (1) (s) = k + c s T s j p k k f The procedure consists of two main steps. In the first step, i + (1) (s) = k + c initial controllers are designed based on Gershgorin bands. j (s) = kp + s + Tf sd+ 1 (1) c j p initial controllers are designed based on Gershgorin bands. s T s + 1 f initial controllers are designed based on Gershgorin bands. s T s+1
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f
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where kp , ki and kd are the controller parameters and Tf is the time constant of noise filter, which is assumed known. This PID formulation could be written as:
r1
c1(s)
r2
c2(s)
-
(2) cj (s) = ρT φ(s) s ]. Furwhere ρ = [ kp ki kd ] and φT (s) = [ 1 1s Tf s+1 thermore, the dynamic of each element of G(s) can be captured by a finite number of N frequency points, gij (jωk ) k = 1, . . . , N . With this parameterization, every point on the Nyquist diagram of ljj (jωk ) could be written as a linear function of the controller parameters:
rn
T
(3) ljj (jωk ) = ρT ℜjj (ωk ) + jρT ℑjj (ωk ), where ℜjj (ωk ) and ℑjj (ωk ) are, respectively, the real and the imaginary parts of φ(jωk )gjj (jωk ).
Consider the Nyquist plot of ljj (jω) = gjj (jω)cj (jω) with a circle of the radius (4)
i=1,i�=j
the Gershgorin circle centered on ljj . When these Gershgorin circles are superimposed on the diagonal elements of the Nyquist array, they form the Gershgorin bands. In the Direct Nyquist Array method (Rosenbrock (1970)), controllers are designed by shaping the Gershgorin bands using a trial and error graphical approach. The following theorem states the stability of the closed loop. Theorem 1. (Rosenbrock (1970)) Suppose the Gershgorin bands centered on the diagonal elements ljj (jω) of L(jω), and j = 1, . . . , n exclude the point (−1, 0). Let the jth Gershgorin band encircles the point (−1, 0) Nj times anticlockwise. Then, the closed-loop system is stable if, and only if, n
N j = p0
eq
y2
eq
yn
u2
g22(s)
un
gnn(s)
are closed. The EOP differs from the original open-loop transfer function by the interactions effects of the coupled loops, thus the EOP corresponds to the actual openloop transfer function under multi-loop cases. Applying the EOP concept, the block diagram in Figure 1 can be decomposed into a set of n equivalent SISO systems in Figure 2.
eq gii (s) = gii (s) −
n gij (s)cj (s)gji (s) j=1
1 + gjj (s)cj (s)
+
gii (s)ci (s)gii (s) . 1 + gii (s)ci (s)
(6)
4. CONTROLLER DESIGN BY LINEAR PROGRAMMING
3.1 The Gershgorin bands
|gij (jω)cj (jω)| ,
y1
g11(s)
Fig. 2. SISO Systems with the corresponding EOP.
Despite each cj (s) controller is designed by a SISO tuning method, the multi-loop interactions are taken into account to guarantee closed-loop stability. The proposed procedure is based on two approaches to consider loops interactions: Gershgorin bands and Equivalent Open-loop Process (EOPs).
Rj (ω) =
cn(s)
-
eq
u1
The equation that defines the equivalent open-loop process for each loop i with all other loops closed can be given as:
3. THE MULTI-LOOP INTERACTIONS
n
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Consider the SISO controller design method of Karimi et al. (2007), which computes controller parameters by a convex optimization problem that minimizes the disturbance effects subject to stability margins constraints. The proposed method in this article modifies the original SISO method to also take loops interactions into account in the controller design by Gershgorin bands and EOPs. 4.1 The Cost Function Load disturbance rejection can be expressed in terms of the integrated absolute error due to a unit step load disturbance at process input, IAE =
∞
∞
|e(t)|dt.
(7)
0
This criterion is difficult to deal with analytically because the evaluation requires the computation of time functions. The integrated error defined by
(5)
IE =
e(t)dt,
(8)
0
j=1
where p0 is the number of unstable poles of L(jω). Since most industrial process are open-loop stable, the controller design procedure assumes that p0 = 0. Thus, Gershgorin bands must not encircle nor include the critical point (−1, 0) to ensure stability. 3.2 The Equivalent Open-loop Process Consider an equivalent open-loop process (EOP) as the transfer function that describes the effective dynamics of each open-loop from ui to yi while all other loops
is more convenient. In ˚ Astr¨om et al. (2006) it is shown that IE = 1/ki . Thus, minimizing IE is equivalent to maximize ki . If the system is well damped, the quantities of IE and IAE are approximately the same. 4.2 The Loop Transfer Function Constraint Consider a straight line rn which crosses the negative real axis between 0 and −1 with a distance ℓ from the critical point. The angle between rn and the real axis is α which is a value between 0◦ and 90◦ , see Figure 3. This straight line divides the complex plane into two regions. The purpose
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is to maintain the plot of the loop transfer function at the region that does not contains the critical point.
l
-1 lg
l
Im ii(j )
rn
Im ii(j )
rg
g -1
Re
lii(j )
l
Re
lii(j ) Ri( )
l
ii
(j )
Fig. 3. The straight line rn and the plot of the loop transfer function.
Fig. 4. The straight line rg and the Gershgorin Bands.
The tuning parameters ℓ and α establishes robustness margins to the diagonal single loop. The higher the values of ℓ and α, the greater the distance of the plot of loop transfer function from the critical point. Also, it should be mentioned that ℓ and α are directly related with the well known stability margins, the gain margin:
In order to formulate the optimization problem as a linear one, consider a perpendicular line to rg passing through the Gershgorin circle center B as illustrated in Figure 5. Each Gershgorin circle is reduced to the intersection of these two lines, the point A. As point B is given by (ρT ℜjj (ωk ), ρT ℑjj (ωk )), point A can be computed as
gm ≥ (1 − ℓ)−1 , and the phase margin:
Φm ≥ arccos (1 − ℓ) sin2 α + cos α
π − αg , ρT ℜjj (ωk ) − Rj (ωk ) cos 2 π − αg . ρT ℑjj (ωk ) + Rj (ωk ) sin 2
(9)
1 − (1 − ℓ)2 sin2 α .
(10)
Im lii(j )
Therefore, minimum bounds in gain and phase margins could be guaranteed by appropriate values of α and ℓ.
-1
According to equation (3), the plot of loop transfer function is composed by N complex points (ρT ℜjj (ωk ), ρT ℑjj (ωk )). Therefore, to maintain these N points to the right side of rn , the following inequality should be satisfied:
Re
A
ρT (cot αℑjj (ωk ) − ℜjj (ωk )) + ℓ ≤ 1 ∀ωk . (11) This is done by geometry analysis as showed in Oliveira and Karimi (2012).
lii(j )
rg
Ri( )
R i ( ) s e n ( g )
B
Similar analysis can be done using equivalent open-loop eq process, leq (jωk ) = gjj (jωk )cj (jωk ). Now the inequality is given by:
R i ( ) c o s ( g )
Fig. 5. Gershgorin Bands Constraint. eq ρT cot αℑeq (12) jj (ωk ) − ℜjj (ωk ) + ℓ ≤ 1 ∀ωk , eq eq where ℜjj (ωk ) and ℑjj (ωk ) are, respectively, the real and eq (jωk ). Note that with the imaginary parts of φ(jωk )gjj this approach, inequality (12) imposes stability restrictions under the actual open-loop transfer function between uj and yj .
Therefore, to maintain these N points to the right side of rg , the following inequality should be satisfied: π − αg cot (αg )ρT ℑjj (ωk ) + cot (αg )Rj (ωk ) sin 2 π T − αg + ℓg ≤ 1 ∀ωk . − ρ ℜjj (ωk ) − Rj (ωk ) cos 2 (13)
4.3 The Gershgorin Bands Constraint Define now a straight line rg that crosses the negative real axis between 0 and −1 with a distance ℓg from the critical point. The angle between rg and the real axis is αg which is a value between 0◦ and 90◦ , see Figure 4. This straight line divides the complex plane into two regions. If the Gershgorin bands are kept to the right side of the rg line, it is ensured that no Gershgorin circle overlaps the critical point.
The inequality (13) must be written as a linear function of ρ. For this reason, the equation (4) is rewritten as:
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Rj (ωk ) =
n
i=1,i�=j
|lij (jωk )| =
n T ρ φ(jωk )gij (jωk ) .
i=1,i�=j
(14)
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constraint on loop-gain function, and another constraint on Gershgorin bands:
� � The product �ρT φ(jωk )gij (jωk )� can be written as: � (1) � (1) � ℜij (ωk ) + jℑij (ωk ) �� � � � (2) �[ kp ki kd ] ℜ(2) ij (ωk ) + jℑij (ωk ) �� � (3) (3) � ℜ (ωk ) + jℑ (ωk ) �
(m)
ij (m) ℑij (ωk )
1183
(15)
maximize ρ
ki (ρ)
(21) inequality (11) inequality (20). The second optimization problem maximizes ki subject to the constraint on loop-gain transfer function, and a constraint on the loop-gain transfer function of the equivalent open-loop process: maximize ki (ρ) subject to
ij
where ℜij (ωk ) and are, respectively, the real part and the imaginary part of the m-th row of vector φ(jωk )gij (jωk ). Thus, � � � � � � (1) (1) (2) (2) |lij (jωk )| = �kp ℜij + jℑij + ki ℜij + jℑij � �� (3) (3) � (16) + kd ℜij + jℑij � .
Note that equation (16) (the modulus of complex numbers sum) is still a nonlinear function to controller parameters. To get linearity, an upper bound is used: � � � � � (2) � (1) (2) � (1) � |lij (jωk )| ≤ |kp | �ℜij + jℑij � + |ki | �ℜij + jℑij � � � � (3) (3) � + |kd | �ℜij + jℑij � . (17)
Considering that the controller parameters are positive, equation (17) is rewritten as:
ρ
subject to
(22)
inequality (11) inequality (12).
The procedure to use these two optimization problems is summarized in an algorithm, Figure 6. The first step is to tuning all n loops based on Gershgorin bands, first optimization problem. If the stability margins results are close to the design specifications, the tuning procedure is finished. Otherwise, the second optimization problem is applied for each loop considering the EOPs with the controllers obtained in the previous step. This procedure will go on until the controller parameters converge. Start Apply Optimization method based on Gershgorin bands to the n loops
(1) (1) |ℜ (ω ) + jℑ (ω )| n k k ij ij � (2) |lij (jωk )| ≤ Rj∗ = ρT |ℜ(2) ij (ωk ) + jℑij (ωk )| . (3) (3) i=1,i�=j |ℜij (ωk ) + jℑij (ωk )| (18) where ℜij (ωk ) and ℑij (ωk ) are respectively the real and imaginary column vectors of φgij .
Are stability margins close to design specifications?
Yes
No Apply Optimization method based on EOPs to the n loops
For sake of simplicity, consider
∆(ωk ) =
(1) |ℜij (ωk ) (2) |ℜij (ωk ) (3) |ℜij (ωk )
+ + +
(1) jℑij (ωk )| (2) jℑij (ωk )| (3) jℑij (ωk )|
so the inequality in (13) becomes:
,
No
Do controller parameters converge?
Yes
(19)
End
Fig. 6. Flow Diagram of the Proposed Algorithm
� � �π � − αg ∆(ωk ) ρT cot (αg )ℑjj (ωk ) + cot (αg ) sin � � �2 �π − αg ∆(ωk ) + ℓg ≤ 1 ∀ωk . −ρT ℜjj (ωk ) − cos 2 (20) The value of Rj∗ is always greater or equal to the exact value of the Gershgorin ratio Rj . Thus, the optimization problem that uses Rj∗ can be conservative. However, for higher frequencies (inside the unitary circle) the difference between Rj and Rj∗ is negligible.
6. SIMULATION RESULTS Simulations examples are implemented to evaluate effectiveness of the proposed decentralized PID controller design method. The examples shares some features. The IE and IAE are computed due a unit step disturbance in process input. The constraint specifications on Gershgorin bands are the same for all examples, αg = 48.90◦ and lg = 0.17. 6.1 Example 1 Consider the Wood and Berry binary distillation column model:
5. PROPOSED PROCEDURE Based on the cost function and inequalities constraints formulated, two optimization problems are considered to be used in a systematic procedure. First optimization problem maximizes the integral gain ki subject to the
1184
�
y1 y2
�
=
�
−18.9e−3s 12.8e−s (16.7s+1) (21s+1) −19.4e−3s 6.6e−7s (10.9s+1) (14.4s+1)
��
� � u1 + u2
3.8e−8.1s (14.9s+1) 4.9e−3.4s (13.2s+1)
�
d.
IFAC ADCHEM 2015 1184 June 7-10, 2015. Whistler, BC, Canada Thiago A.M. Euzébio et al. / IFAC-PapersOnLine 48-8 (2015) 1180–1185
Table 1. Controller parameters in Example 1. kp 0.440 0.152 0.554 −0.100 −0.100 −0.102
ki 0.091 0.033 0.083 −0.009 −0.017 −0.016
Tf 0.0 1.0 1.0 0.0 0.1 0.1
1.2 1 0.8 0.6 0.4
Nyquist Diagram
0.2
0.2
0.2
0
0
0
Imaginary Axis
Imaginary Axis
Nyquist Diagram
kd 0.000 0.800 0.147 0.000 −0.203 −0.147
1
c2 (s)
Method Chen-Seborg Proposed (GB) Proposed (EOP) Chen-Seborg Proposed (GB) Proposed (EOP)
y
Controller c1 (s)
−0.2 −0.4
−0.2
−0.6
−0.8
−0.8
−1
−0.5 Real Axis
50
100 150 Time (min)
200
0
50
100 150 Time (min)
200
−0.4
−0.6
−1
0
−1
0
−1
0
2
(b) Loop 2
1.5
−0.5 Real Axis
y
2
(a) Loop 1
2.5
1
eq Fig. 7. Nyquist plot of gjj cj (solid blue line) and gjj cj (dashed black line) for the Proposed (EOP) PID controller in Example 1. Straight line rn in red line.
0.5 0
Table 2. Robustness and performance measures in Example 1. Controller c1 (s) c2 (s)
Method Chen-Seborg Proposed (GB) Proposed (EOP) Chen-Seborg Proposed (GB) Proposed (EOP)
gm 4.33 3.04 3.22 3.79 3.32 4.00
Φm 48.71◦ 69.74◦ 59.22◦ 60.45◦ 50.62◦ 50.08◦
IE 11.03 30.10 12.06 115.82 60.34 63.74
IAE 11.68 48.25 12.06 115.82 85.18 83.63
During the proposed procedure, the corresponding design specifications for both Loop 1 and Loop 2 are α = 62.14◦ and ℓ = 0.67. These specifications correspond to a lower bound both on gain margin and phase margin of 3.0 and 45◦ , respectively. Initially, the decentralized PID controller is designed using the proposed method based on Gershgorin bands (GB). Then, the proposed method based on Equivalent Openloop Process (EOP) is applied. After three iterations, the final parameters are obtained. In Table 1, the controller parameters for each method are shown (Proposed (GB) and Proposed (EOP)), it is also shown the PI controller computed by Chen and Seborg (2001). eq The Nyquist plots of gjj (jω)cj (jω) and gjj (jω)cj (jω) are shown in Figure 7. It can be seen that the designed systems fulfill the design specifications. Some performance and robustness measures are summarized in Table 2.
The resultant closed-loop output response to unit step setpoint change and unit step disturbance change are shown in Figure 8. Although the same design specifications, note that Loop 1 output response is smoother for the Proposed (GB) controller than for Proposed (EOP) controller. The Loop 2 output response for a unit step disturbance is faster for both proposed controllers than Chen-Seborg controller. 6.2 Example 2 Consider a binary ethanol-water system of a pilot-plant distillation column proposed by Ogunnaike and Ray:
Fig. 8. Step responses of the first and second outputs (solid blue line: Proposed (EOP), dashed black line: Proposed (GB), and dashed green line: Chen-Seborg). −2.6s −3.5s −s G(s) =
0.66e (6.7s+1) 1.11e−6.5s (3.25s+1) −34.68e−9.2s (8.15s+1)
−0.61e (8.64s+1) −2.36e−3s (5s+1) 46.2e−9.4s (10.9s+1)
−0.0049e (9.06s+1) −0.01e−1.2s (7.09s+1) 0.87(11.61s+1)e−s (3.89s+1)(18.8s+1)
.
The design specifications were chosen to guarantee minimum stability margins for each loop. This means a minimum gain margin of 4.5, 4.8, and 3.0, as well as a minimum phase margin of 55◦ , 36◦ and, 35◦ for loops 1, 2 and, 3, respectively. Because the system is not diagonally dominant, the respective Gershgorin bands are too wide, which implies in margins results too far from design specifications. So, the parameters result from the proposed method based on Gershgorin bands are just used as an initial controller to the proposed method that uses EOPs. As can be seen in Figure 9, and Table 4, the margins results fulfill the design specifications. Controller parameters obtained using the proposed approach is show in Table 3, also results from Huang et al. (2003) (HJCP) and the well known Biggest Log-modulus (BLT) (Luyben (1986)) are presented. The resultant closed-loop output responses to unit step set-point change are shown in Figure 10. 7. CONCLUSION This paper has presented an iterative procedure for decentralized PID controller design. Each SISO loop is designed at a time by solving an optimization problem that is efficiently solved by a linear programming approach. The loop interactions are taken into account by Gershgorin bands
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Table 3. Controller parameters in Example 2.
c3 (s)
kp 2.00 1.51 1.27 −0.42 −0.30 −0.16 2.83 2.63 3.94
ki 0.46 0.09 0.33 −0.10 −0.02 −0.08 0.67 0.40 1.77
Nyquist Diagram
kd 7.50 0.00 1.69 −1.29 0.00 −0.31 0.02 0.00 0.91
Tf 4.6 0.0 0.1 4.4 0.0 0.1 0.0 0.0 0.1
1
1
c2 (s)
Method HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP)
y
Controller c1 (s)
0
0
0
−0.2 −0.4
−0.2
150
200 250 Time (min)
300
350
400
450
0
50
100
150
200 250 Time (min)
300
350
400
450
0
50
100
150
200 250 Time (min)
300
350
400
450
0.5 0
−0.6
−0.8
−0.8
−1
−1
−1
−0.5 Real Axis
0
5
(b) Loop 2
0 3
(a) Loop 1
y
Nyquist Diagram
−5 −10
0.2 0
Imaginary Axis
100
−0.4
−0.6
0
y
Imaginary Axis
Imaginary Axis
50
1 2
0.2
−0.5 Real Axis
0
Nyquist Diagram
0.2
−1
0.5
−15
−0.2 −0.4 −0.6 −0.8 −1
−1
−0.5 Real Axis
Fig. 10. Step responses of the first, second, and third outputs (solid blue line: Proposed (EOP), dashed magenta line: HJCP, dashed black line: BLT).
0
(c) Loop 3 eq Fig. 9. Nyquist plot of gjj cj (solid blue line) and gjj cj (dashed black line) for the Proposed (EOP) PID controller in Example 2. Straight line rn in red line.
Table 4. Robustness and performance measures in Example 2. Controller c1 (s) c2 (s) c3 (s)
Method HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP)
gm 4.56 4.50 5.16 4.80 4.52 6.97 4.01 4.48 3.15
Φm 33.53◦ 100.0◦ 65.02◦ 36.38◦ 109.0◦ 44.46◦ 65.02◦ 79.52◦ 44.67◦
IE 2.17 10.81 3.05 10.17 61.81 12.03 1.51 2.53 0.57
IAE 2.58 10.81 3.93 11.41 61.81 20.29 1.51 2.54 0.68
and EOPs. If the process is diagonally dominant, only a single-iteration is sufficient to get stability margins results close to design specifications. The proposed procedure does not require work on model order reductions, indeed can be applied to any complex dynamic and multiple time delays. Simulation examples illustrate the effectiveness of the method, some robustness and performance indexes are compared to related techniques. REFERENCES ˚ Astr¨om, K. J., and H¨ agglund, T. (2006) Advanced PID Control. Instrument Soc. Amer., 2006. Chen, D., and Seborg, D.E. (2001) Multiloop PI/PID controller design based on Gershgorin bands. Proceedings of the American Control Conference, 2001. Garcia, D., Karimi, A., and Longchamp, R. (2005) PID Controller Design for Multivariable Systems Using Ger-
shgorin Bands. Proc. 16th IFAC World Congress, pp. 183-188, 2005. Ho, W.K., and Xu, W. (1998) Multivariable PID Controller Design Based on the Direct Nyquist Array Method. Proceedings of the American Control Conference, pp. 3524-3528, 1998. Huang, H. , Jeng, J. , Chiang, C. , and Pan, W. (2003) A direct method for multi-loop PI/PID controller design. Journal of Process Control, 13, pp. 769-786, 2003. Karimi, A. , Kunze, M. , and Longchamp, R. (2007) Robust controller design by linear programming with application to a double-axis positioning system. Control Eng. Practice, 15, pp. 197-208, 2007. Luyben, W.L. (1986) Simple method for tuning SISO controllers in multivariable systems. Industrial Eng. and Chem. Process Design Development,25, pp. 654-660, 1986. Nie, Z. , Wang, Q., Wu, M., and He, Y. (2011) Tuning of multi-loop PI controllers based on gain and phase margin specifications. Journal of Process Control, 21, pp. 1287-1295, 2011. Oliveira, V., and Karimi, A. (2012) Robust and gainscheduled PID controller design for condensing boilers by linear programming. IFAC Conference in Advances in PID Control, 2012. Rosenbrock, H.H. (1970) State-space and multivariable theory. London Nelson, 1970. Vu, T.N.L. , and Lee, M. (2010) Independent design of multi-loop PI/PID controllers for interacting multivariable processes. Journal of Process Control, 20, pp. 922933, 2010.
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Iterative Procedure for Tuning Iterative Procedure for Tuning Iterative Procedure for Iterative Procedure for Tuning Tuning Decentralized PID Controllers Decentralized PID Controllers Decentralized PID Controllers Decentralized PID Controllers
Thiago A.M. Euz´ ebio ∗∗ P´ ericles R. Barros ∗∗ Thiago A.M. Euz´ e bio e ricles R. Barros ∗∗ ∗ P´ ∗ P´ Thiago A.M. Euz´ e bio e Thiago A.M. Euz´ ebio P´ ericles ricles R. R. Barros Barros ∗ ∗ Department of Electrical Engineering, Federal University of Campina ∗ Department of Electrical Engineering, Federal University of Campina ∗ Department of Engineering, Federal Grande, Brazil (e-mail: [email protected], Department of Electrical Electrical Engineering, Federal University University of of Campina Campina Grande, Brazil (e-mail: [email protected], Grande, Brazil (e-mail: [email protected], [email protected]). Grande, [email protected]). (e-mail: [email protected], [email protected]). [email protected]). Abstract: An iterative procedure is proposed to design decentralized PID controllers for multiAbstract: An procedure is proposed to design decentralized PID for Abstract: An iterative iterative procedure is proposed to designthe decentralized PID controllers controllers for multimultiloop processes. Each SISO loop is is designed at to a time, controller parameters are computed Abstract: An iterative procedure proposed design decentralized PID controllers for multiloop processes. Each SISO loop is designed at a time, the controller parameters are computed loop Each SISO loop is designed at aa time, the controller parameters are computed by a processes. convex optimization problem with constraints on stability margins. Despite the SISO loop processes. Each SISO loop is designed at time, the controller parameters are computed by a convex optimization problem with constraints on stability margins. Despite the SISO by a optimization with stability margins. the SISO approach, loops interactionsproblem are taken intoconstraints account by on Gershgorin bands and Despite Equivalent Openby a convex convex optimization problem with constraints on stability bands margins. Despite the OpenSISO approach, loops interactions are taken into account by Gershgorin and Equivalent approach, loops interactions are taken into account by Gershgorin bands and Equivalent Openloop Process (EOP). Two simulation examples are presented to compare the performance with approach, loops interactions are takenexamples into account by Gershgorin bands and Equivalent Openloop Process (EOP). Two simulation are presented to compare the performance loop Process (EOP). Two Two simulation simulation examples examples are are presented presented to to compare compare the the performance performance with with related techniques. loop Process (EOP). with related techniques. related techniques. related © 2015, techniques. IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: PID, MIMO systems, Optimization, Gain and phase margins, Process control. Keywords: Keywords: PID, PID, MIMO MIMO systems, systems, Optimization, Optimization, Gain Gain and and phase phase margins, margins, Process Process control. control. Keywords: PID, MIMO systems, Optimization, Gain and phase margins, Process control. 1. INTRODUCTION The second step, which design is now based on EOPs, each 1. INTRODUCTION The which design based on each 1. INTRODUCTION The second secondisstep, step, whichiteratively design is is now now based on EOPs, EOPs, each controller re-tuned untilbased controller parame1. INTRODUCTION The second step, which design is now on EOPs, each controller is re-tuned iteratively until controller paramecontroller is re-tuned iteratively until controller parameters converge. The SISO controlleruntil design methodparameused in controller is re-tuned iteratively controller converge. The design method used The basic regulatory control layer of process plants largely ters terssteps converge. The SISO SISOascontroller controller design methodproblem used in in all is formulated a convex design optimization ters converge. The SISO controller method used in The basic regulatory control layer of process plants largely all steps is formulated as a convex optimization problem The basic regulatory control layer of process plants largely consists ofregulatory decentralized SISO PID controllers. Thelargely main which all steps is formulated as a convex optimization problem minimizes the diagonal loop integral error subject The basic control layer of process plants all steps is formulated as a convex optimization problem consists of SISO PID The main minimizes the loop consists are: of decentralized decentralized SISOstructure, PID controllers. controllers. The main which reasons relatively simple easy to The maintain, which minimizes the diagonal diagonal loop integral integral error error subject subject to minimum stability margins. loop consists of decentralized SISO PID controllers. main which minimizes the diagonal integral error subject reasons are: relatively simple structure, easy to maintain, to minimum stability margins. reasonsparameters are: relatively relatively simplethan structure, easy to to maintain, maintain, fewer to tune full multivariable con- to minimum stability margins. reasons are: simple structure, easy to minimum stability margins. fewer parameters to tune than full multivariable conWith this proposed procedure some advantages are achieved fewer parameters parameters to tune tune than full full multivariable controllers, and loop failure tolerance. Despite the SISOconap- With procedure some advantages are fewer to than multivariable trollers, and loop failure tolerance. Despite the SISO apWith this this proposed proposed procedure some advantages are achieved achieved compared to relatedprocedure methods. some First,advantages full Gershgorin bands trollers, and loop failure tolerance. Despite the SISO approach, tuning of each loop cannot be done independently With this proposed are achieved compared to related methods. First, full Gershgorin bands trollers, and loop failure tolerance. Despite the SISO approach, tuning of each loop cannot be done independently compared to related related methods. First, full Gershgorin bands are considered, different from Ho andfull Xu Gershgorin (1998), andbands Chen proach, tuning of each loop cannot be done independently due to loops interactions. Applying the tuning methods compared to methods. First, are considered, different from Ho and Xu (1998), and Chen proach, tuninginteractions. of each loop Applying cannot bethe done independently due to loops tuning methods are considered, different from Ho and Xu (1998), and Chen and Seborg (2001), that are based on no more than dueSISO to loops loops interactions. Applying the the tuning methods for systems ignoring interactions often leadsmethods to poor are considered, differentthat fromare Ho based and Xuon (1998), and Chen and Seborg (2001), no more than due to interactions. Applying tuning for SISO systems ignoring interactions often leads to poor and Seborg (2001), that are based on no more than two Gershgorin circles, which does not guarantee overall for SISO SISO systems systems ignoring interactions interactions often often leads leads to to poor poor two performance and stability. and Gershgorin Seborg (2001), that aredoes based on no moreoverall than circles, which not guarantee for ignoring performance and stability. two Gershgorin circles, which does not guarantee overall system stability. Second, this procedure is not restricted performance and stability. two Gershgorin circles, which does not guarantee overall stability. Second, this is not performance stability. One way to and consider loop interactions during controller system system stability. Second,processes, this procedure procedure is not restricted restricted to diagonally dominant model is reduction is not system stability. Second, this procedure not restricted One way to consider loop interactions during controller to diagonally dominant processes, model reduction is not One way way consider loop interactions interactions during controller design is to use the Gershgorin bands during (see e.g.controller Ho and necessary, to diagonally dominant processes, model reduction is not and high dimensional processes are not an issue. One to consider loop to diagonally dominant processes, model reduction is not design is to use the Gershgorin bands (see e.g. Ho and necessary, and high dimensional processes are not an issue. design is to use the Gershgorin bands (see e.g. Ho and Xu (1998), Chen and Seborg (2001), and Garcia et al. necessary, and high dimensional processes are not an issue. Third, lower bound on stability margins can be specified design is to use the Gershgorin bands (see e.g. Ho and necessary, andbound high dimensional processes are not specified an issue. Xu Chen Seborg (2001), and et lower on stability margins can Xu (1998), (1998), Chen and andstability Seborg can (2001), and Garcia Garcia et al. al. Third, (2005)). Closed-loop be ensured by shaping Third, lower bound on stability margins can be be specified specified to each lower diagonal loopon transfer function, independently how Xu (1998), Chen and Seborg (2001), and Garcia et al. Third, bound stability margins can be (2005)). Closed-loop stability can be ensured by shaping to each diagonal loop transfer function, independently (2005)). Closed-loop stability can be ensured ensured by shaping shaping the Gershgorin bandsstability in suchcan a way that they do not interactions to each each diagonal diagonal loop transfer transfer function, independently how how are considered in function, design method. (2005)). Closed-loop be by to loop independently how the Gershgorin bands in such a way that they do not the Gershgorin Gershgorin bands in (−1, such 0) way encircle that they they do not not overlap the critical point and it the ap- interactions interactions are are considered considered in in design design method. method. the bands in such aa and way that do interactions are considered in design method. overlap the critical point (−1, 0) encircle it the apoverlap the critical point (−1, 0) and encircle it the appropriate number of times, in accordance with the generoverlap the critical point (−1, 0) and encircle it the ap2. PROBLEM STATEMENT propriate number of in accordance with the generpropriate number of times, times, inadvantage accordance with the gener2. alized Nyquist theorem. Anin of with this the method is propriate number of times, accordance gener2. PROBLEM PROBLEM STATEMENT STATEMENT alized Nyquist theorem. An advantage of this method is 2. PROBLEM STATEMENT alized Nyquist theorem. An advantage of this method is that knowledge of controller parameters from other loops alizedknowledge Nyquist theorem. An parameters advantage offrom thisother method is Consider the closed loop system of Figure 1, where G(s) = that of controller loops that knowledge of of controller parametersoffrom from other loops is unnecessary. However, effectiveness this other method is Consider the closed loop system of Figure 1, where G(s) = that knowledge controller parameters loops is However, effectiveness of method Consider the closed loop system of=Figure Figure 1,1 (s), where G(s) = [g is aclosed n×n loop process, C(s)of diag{c1, . . . ,G(s) cn (s)} ij (s)]n×n the system where = is unnecessary. unnecessary. However, effectiveness of this thisprocesses. method is is Consider restricted to diagonally dominant multi-loop [g ij (s)]n×n is a n×n process, C(s) = diag{c1 (s), . . . , cn (s)} is unnecessary. However, effectiveness of this method is restricted to diagonally dominant multi-loop processes. [g (s)] is a n×n process, C(s) = diag{c (s), . . . , c (s)} ij 1 n (s)c (s)] is a decentralized controller, and L(s) = [g n×n is a n×n process, C(s) = diag{c1 (s), ij . . j. , cn (s)} [gija(s)] restricted to to diagonally diagonally dominant dominant multi-loop multi-loop processes. processes. n×n n×n decentralized controller, and L(s) = [g restricted ij (s)cj (s)]n×n Another way to take loop interactions into account to is (s)] is aathe decentralized controller, and L(s) ij (s)c multivariable loop-gain. The r, = u, [g and y jj are the n×n (s)c (s)] is decentralized controller, and L(s) = [g Another way to take loop interactions into account to ij n×n the multivariable loop-gain. The r, u, and y are the Another way way to istake take loop interactions into account account to is controller design to use theinteractions concept of equivalent openis the multivariable loop-gain. The r, u, and y are Another to loop into to set-point, manipulated, and controlled variable vectors. controller design is to use the concept of equivalent openis the multivariable loop-gain. The r, variable u, and vectors. y are the the set-point, manipulated, and controlled controller design is to use the concept of equivalent openloop process (EOPs) (see e.g. Huang et al. (2003), Vu set-point, manipulated, and controlled variable vectors. controller design is to use the concept ofetequivalent openloop process (EOPs) (see e.g. Huang al. (2003), Vu set-point, manipulated, and controlled variable vectors. loop Lee process (EOPs) (see e.g.al.Huang Huang et This al. (2003), (2003), Vu and (2010), and (see Nie et (2011)). procedure loop process (EOPs) e.g. et al. Vu y and (2010), and et This procedure u r and Lee Lee initial (2010), and Nie Nieparameters et al. al. (2011)). (2011)). Thisloop. procedure y requires controller for each After u G(s) r and Lee (2010), and Nie et al. (2011)). This procedure y u r requires initial controller parameters for each loop. After G(s) y u r requires initial controller parameters for each loop. After all loops have been closed, the controller will then be reG(s) requires initial controller parameters for each loop. be After G(s) all loops have been closed, the controller will then reall loops have been closed, the controller will then be retuned one after the other with all other loops being closed C(s) all loops have been other closed, the controller will then closed be retuned one after with other loops C(s) tunedthe onecontrollers after the the other other with all all otherprevious loops being being closed C(s) with obtained in the step.closed This tuned one after the with all other loops being C(s) with the controllers obtained in the previous step. This with the the controllers controllers obtained in the theparameters previous step. step. This procedure will go on until controllers converge. with obtained in previous This Fig. 1. Decentralized control system. procedure will go on until controllers parameters converge. procedure will go go on model until controllers controllers parameters converge. Most methods need order reduction whenconverge. compute Fig. 1. Decentralized control system. procedure will on until parameters Fig. 1. Decentralized control system. Most methods need model order reduction when compute Most methods need model order reduction when compute Fig. 1. Decentralized control system. the EOP. MostEOP. methods need model order reduction when compute the the EOP. EOP. The PID controller transfer function from the j-th diagothe In this paper, an iterative procedure to tune decentralized The PID controller transfer function from the j-th diagoTheelement PID controller controller transfer function from from the the j-th j-th diagodiagoIn this paper, an iterative procedure to tune decentralized nal of C(s) transfer is The PID function In this this paper, an anisiterative iterative procedure to tune tune decentralized PID controllers proposed by taking otherdecentralized loops inter- nal element of C(s) is In paper, procedure to nal element of C(s) is PID controllers is proposed by taking other loops internal element of C(s) is PID controllers controllers is proposed proposed by taking other other loops interactions into account both by by Gershgorin bands as EOPs. PID is taking loops interactions into both by bands as kd s ki actions into account account both by Gershgorin Gershgorin bands as EOPs. EOPs. The procedure consists of two main steps. bands In the as first step, actions into account both by Gershgorin EOPs. k (1) cj (s) = kp + k The procedure consists of two main steps. In the first step, i + ds k k ss 1 The procedure consists of two main steps. In the first step, i d+ initial controllers are designed based on Gershgorin bands. + (1) (s) = k + c s T s j p k k f The procedure consists of two main steps. In the first step, i + (1) (s) = k + c initial controllers are designed based on Gershgorin bands. j (s) = kp + s + Tf sd+ 1 (1) c j p initial controllers are designed based on Gershgorin bands. s T s + 1 f initial controllers are designed based on Gershgorin bands. s T s+1
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f
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where kp , ki and kd are the controller parameters and Tf is the time constant of noise filter, which is assumed known. This PID formulation could be written as:
r1
c1(s)
r2
c2(s)
-
(2) cj (s) = ρT φ(s) s ]. Furwhere ρ = [ kp ki kd ] and φT (s) = [ 1 1s Tf s+1 thermore, the dynamic of each element of G(s) can be captured by a finite number of N frequency points, gij (jωk ) k = 1, . . . , N . With this parameterization, every point on the Nyquist diagram of ljj (jωk ) could be written as a linear function of the controller parameters:
rn
T
(3) ljj (jωk ) = ρT ℜjj (ωk ) + jρT ℑjj (ωk ), where ℜjj (ωk ) and ℑjj (ωk ) are, respectively, the real and the imaginary parts of φ(jωk )gjj (jωk ).
Consider the Nyquist plot of ljj (jω) = gjj (jω)cj (jω) with a circle of the radius (4)
i=1,i�=j
the Gershgorin circle centered on ljj . When these Gershgorin circles are superimposed on the diagonal elements of the Nyquist array, they form the Gershgorin bands. In the Direct Nyquist Array method (Rosenbrock (1970)), controllers are designed by shaping the Gershgorin bands using a trial and error graphical approach. The following theorem states the stability of the closed loop. Theorem 1. (Rosenbrock (1970)) Suppose the Gershgorin bands centered on the diagonal elements ljj (jω) of L(jω), and j = 1, . . . , n exclude the point (−1, 0). Let the jth Gershgorin band encircles the point (−1, 0) Nj times anticlockwise. Then, the closed-loop system is stable if, and only if, n
N j = p0
eq
y2
eq
yn
u2
g22(s)
un
gnn(s)
are closed. The EOP differs from the original open-loop transfer function by the interactions effects of the coupled loops, thus the EOP corresponds to the actual openloop transfer function under multi-loop cases. Applying the EOP concept, the block diagram in Figure 1 can be decomposed into a set of n equivalent SISO systems in Figure 2.
eq gii (s) = gii (s) −
n gij (s)cj (s)gji (s) j=1
1 + gjj (s)cj (s)
+
gii (s)ci (s)gii (s) . 1 + gii (s)ci (s)
(6)
4. CONTROLLER DESIGN BY LINEAR PROGRAMMING
3.1 The Gershgorin bands
|gij (jω)cj (jω)| ,
y1
g11(s)
Fig. 2. SISO Systems with the corresponding EOP.
Despite each cj (s) controller is designed by a SISO tuning method, the multi-loop interactions are taken into account to guarantee closed-loop stability. The proposed procedure is based on two approaches to consider loops interactions: Gershgorin bands and Equivalent Open-loop Process (EOPs).
Rj (ω) =
cn(s)
-
eq
u1
The equation that defines the equivalent open-loop process for each loop i with all other loops closed can be given as:
3. THE MULTI-LOOP INTERACTIONS
n
1181
Consider the SISO controller design method of Karimi et al. (2007), which computes controller parameters by a convex optimization problem that minimizes the disturbance effects subject to stability margins constraints. The proposed method in this article modifies the original SISO method to also take loops interactions into account in the controller design by Gershgorin bands and EOPs. 4.1 The Cost Function Load disturbance rejection can be expressed in terms of the integrated absolute error due to a unit step load disturbance at process input, IAE =
∞
∞
|e(t)|dt.
(7)
0
This criterion is difficult to deal with analytically because the evaluation requires the computation of time functions. The integrated error defined by
(5)
IE =
e(t)dt,
(8)
0
j=1
where p0 is the number of unstable poles of L(jω). Since most industrial process are open-loop stable, the controller design procedure assumes that p0 = 0. Thus, Gershgorin bands must not encircle nor include the critical point (−1, 0) to ensure stability. 3.2 The Equivalent Open-loop Process Consider an equivalent open-loop process (EOP) as the transfer function that describes the effective dynamics of each open-loop from ui to yi while all other loops
is more convenient. In ˚ Astr¨om et al. (2006) it is shown that IE = 1/ki . Thus, minimizing IE is equivalent to maximize ki . If the system is well damped, the quantities of IE and IAE are approximately the same. 4.2 The Loop Transfer Function Constraint Consider a straight line rn which crosses the negative real axis between 0 and −1 with a distance ℓ from the critical point. The angle between rn and the real axis is α which is a value between 0◦ and 90◦ , see Figure 3. This straight line divides the complex plane into two regions. The purpose
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is to maintain the plot of the loop transfer function at the region that does not contains the critical point.
l
-1 lg
l
Im ii(j )
rn
Im ii(j )
rg
g -1
Re
lii(j )
l
Re
lii(j ) Ri( )
l
ii
(j )
Fig. 3. The straight line rn and the plot of the loop transfer function.
Fig. 4. The straight line rg and the Gershgorin Bands.
The tuning parameters ℓ and α establishes robustness margins to the diagonal single loop. The higher the values of ℓ and α, the greater the distance of the plot of loop transfer function from the critical point. Also, it should be mentioned that ℓ and α are directly related with the well known stability margins, the gain margin:
In order to formulate the optimization problem as a linear one, consider a perpendicular line to rg passing through the Gershgorin circle center B as illustrated in Figure 5. Each Gershgorin circle is reduced to the intersection of these two lines, the point A. As point B is given by (ρT ℜjj (ωk ), ρT ℑjj (ωk )), point A can be computed as
gm ≥ (1 − ℓ)−1 , and the phase margin:
Φm ≥ arccos (1 − ℓ) sin2 α + cos α
π − αg , ρT ℜjj (ωk ) − Rj (ωk ) cos 2 π − αg . ρT ℑjj (ωk ) + Rj (ωk ) sin 2
(9)
1 − (1 − ℓ)2 sin2 α .
(10)
Im lii(j )
Therefore, minimum bounds in gain and phase margins could be guaranteed by appropriate values of α and ℓ.
-1
According to equation (3), the plot of loop transfer function is composed by N complex points (ρT ℜjj (ωk ), ρT ℑjj (ωk )). Therefore, to maintain these N points to the right side of rn , the following inequality should be satisfied:
Re
A
ρT (cot αℑjj (ωk ) − ℜjj (ωk )) + ℓ ≤ 1 ∀ωk . (11) This is done by geometry analysis as showed in Oliveira and Karimi (2012).
lii(j )
rg
Ri( )
R i ( ) s e n ( g )
B
Similar analysis can be done using equivalent open-loop eq process, leq (jωk ) = gjj (jωk )cj (jωk ). Now the inequality is given by:
R i ( ) c o s ( g )
Fig. 5. Gershgorin Bands Constraint. eq ρT cot αℑeq (12) jj (ωk ) − ℜjj (ωk ) + ℓ ≤ 1 ∀ωk , eq eq where ℜjj (ωk ) and ℑjj (ωk ) are, respectively, the real and eq (jωk ). Note that with the imaginary parts of φ(jωk )gjj this approach, inequality (12) imposes stability restrictions under the actual open-loop transfer function between uj and yj .
Therefore, to maintain these N points to the right side of rg , the following inequality should be satisfied: π − αg cot (αg )ρT ℑjj (ωk ) + cot (αg )Rj (ωk ) sin 2 π T − αg + ℓg ≤ 1 ∀ωk . − ρ ℜjj (ωk ) − Rj (ωk ) cos 2 (13)
4.3 The Gershgorin Bands Constraint Define now a straight line rg that crosses the negative real axis between 0 and −1 with a distance ℓg from the critical point. The angle between rg and the real axis is αg which is a value between 0◦ and 90◦ , see Figure 4. This straight line divides the complex plane into two regions. If the Gershgorin bands are kept to the right side of the rg line, it is ensured that no Gershgorin circle overlaps the critical point.
The inequality (13) must be written as a linear function of ρ. For this reason, the equation (4) is rewritten as:
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Rj (ωk ) =
n
i=1,i�=j
|lij (jωk )| =
n T ρ φ(jωk )gij (jωk ) .
i=1,i�=j
(14)
IFAC ADCHEM 2015 June 7-10, 2015. Whistler, BC, Canada Thiago A.M. Euzébio et al. / IFAC-PapersOnLine 48-8 (2015) 1180–1185
constraint on loop-gain function, and another constraint on Gershgorin bands:
� � The product �ρT φ(jωk )gij (jωk )� can be written as: � (1) � (1) � ℜij (ωk ) + jℑij (ωk ) �� � � � (2) �[ kp ki kd ] ℜ(2) ij (ωk ) + jℑij (ωk ) �� � (3) (3) � ℜ (ωk ) + jℑ (ωk ) �
(m)
ij (m) ℑij (ωk )
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(15)
maximize ρ
ki (ρ)
(21) inequality (11) inequality (20). The second optimization problem maximizes ki subject to the constraint on loop-gain transfer function, and a constraint on the loop-gain transfer function of the equivalent open-loop process: maximize ki (ρ) subject to
ij
where ℜij (ωk ) and are, respectively, the real part and the imaginary part of the m-th row of vector φ(jωk )gij (jωk ). Thus, � � � � � � (1) (1) (2) (2) |lij (jωk )| = �kp ℜij + jℑij + ki ℜij + jℑij � �� (3) (3) � (16) + kd ℜij + jℑij � .
Note that equation (16) (the modulus of complex numbers sum) is still a nonlinear function to controller parameters. To get linearity, an upper bound is used: � � � � � (2) � (1) (2) � (1) � |lij (jωk )| ≤ |kp | �ℜij + jℑij � + |ki | �ℜij + jℑij � � � � (3) (3) � + |kd | �ℜij + jℑij � . (17)
Considering that the controller parameters are positive, equation (17) is rewritten as:
ρ
subject to
(22)
inequality (11) inequality (12).
The procedure to use these two optimization problems is summarized in an algorithm, Figure 6. The first step is to tuning all n loops based on Gershgorin bands, first optimization problem. If the stability margins results are close to the design specifications, the tuning procedure is finished. Otherwise, the second optimization problem is applied for each loop considering the EOPs with the controllers obtained in the previous step. This procedure will go on until the controller parameters converge. Start Apply Optimization method based on Gershgorin bands to the n loops
(1) (1) |ℜ (ω ) + jℑ (ω )| n k k ij ij � (2) |lij (jωk )| ≤ Rj∗ = ρT |ℜ(2) ij (ωk ) + jℑij (ωk )| . (3) (3) i=1,i�=j |ℜij (ωk ) + jℑij (ωk )| (18) where ℜij (ωk ) and ℑij (ωk ) are respectively the real and imaginary column vectors of φgij .
Are stability margins close to design specifications?
Yes
No Apply Optimization method based on EOPs to the n loops
For sake of simplicity, consider
∆(ωk ) =
(1) |ℜij (ωk ) (2) |ℜij (ωk ) (3) |ℜij (ωk )
+ + +
(1) jℑij (ωk )| (2) jℑij (ωk )| (3) jℑij (ωk )|
so the inequality in (13) becomes:
,
No
Do controller parameters converge?
Yes
(19)
End
Fig. 6. Flow Diagram of the Proposed Algorithm
� � �π � − αg ∆(ωk ) ρT cot (αg )ℑjj (ωk ) + cot (αg ) sin � � �2 �π − αg ∆(ωk ) + ℓg ≤ 1 ∀ωk . −ρT ℜjj (ωk ) − cos 2 (20) The value of Rj∗ is always greater or equal to the exact value of the Gershgorin ratio Rj . Thus, the optimization problem that uses Rj∗ can be conservative. However, for higher frequencies (inside the unitary circle) the difference between Rj and Rj∗ is negligible.
6. SIMULATION RESULTS Simulations examples are implemented to evaluate effectiveness of the proposed decentralized PID controller design method. The examples shares some features. The IE and IAE are computed due a unit step disturbance in process input. The constraint specifications on Gershgorin bands are the same for all examples, αg = 48.90◦ and lg = 0.17. 6.1 Example 1 Consider the Wood and Berry binary distillation column model:
5. PROPOSED PROCEDURE Based on the cost function and inequalities constraints formulated, two optimization problems are considered to be used in a systematic procedure. First optimization problem maximizes the integral gain ki subject to the
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�
y1 y2
�
=
�
−18.9e−3s 12.8e−s (16.7s+1) (21s+1) −19.4e−3s 6.6e−7s (10.9s+1) (14.4s+1)
��
� � u1 + u2
3.8e−8.1s (14.9s+1) 4.9e−3.4s (13.2s+1)
�
d.
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Table 1. Controller parameters in Example 1. kp 0.440 0.152 0.554 −0.100 −0.100 −0.102
ki 0.091 0.033 0.083 −0.009 −0.017 −0.016
Tf 0.0 1.0 1.0 0.0 0.1 0.1
1.2 1 0.8 0.6 0.4
Nyquist Diagram
0.2
0.2
0.2
0
0
0
Imaginary Axis
Imaginary Axis
Nyquist Diagram
kd 0.000 0.800 0.147 0.000 −0.203 −0.147
1
c2 (s)
Method Chen-Seborg Proposed (GB) Proposed (EOP) Chen-Seborg Proposed (GB) Proposed (EOP)
y
Controller c1 (s)
−0.2 −0.4
−0.2
−0.6
−0.8
−0.8
−1
−0.5 Real Axis
50
100 150 Time (min)
200
0
50
100 150 Time (min)
200
−0.4
−0.6
−1
0
−1
0
−1
0
2
(b) Loop 2
1.5
−0.5 Real Axis
y
2
(a) Loop 1
2.5
1
eq Fig. 7. Nyquist plot of gjj cj (solid blue line) and gjj cj (dashed black line) for the Proposed (EOP) PID controller in Example 1. Straight line rn in red line.
0.5 0
Table 2. Robustness and performance measures in Example 1. Controller c1 (s) c2 (s)
Method Chen-Seborg Proposed (GB) Proposed (EOP) Chen-Seborg Proposed (GB) Proposed (EOP)
gm 4.33 3.04 3.22 3.79 3.32 4.00
Φm 48.71◦ 69.74◦ 59.22◦ 60.45◦ 50.62◦ 50.08◦
IE 11.03 30.10 12.06 115.82 60.34 63.74
IAE 11.68 48.25 12.06 115.82 85.18 83.63
During the proposed procedure, the corresponding design specifications for both Loop 1 and Loop 2 are α = 62.14◦ and ℓ = 0.67. These specifications correspond to a lower bound both on gain margin and phase margin of 3.0 and 45◦ , respectively. Initially, the decentralized PID controller is designed using the proposed method based on Gershgorin bands (GB). Then, the proposed method based on Equivalent Openloop Process (EOP) is applied. After three iterations, the final parameters are obtained. In Table 1, the controller parameters for each method are shown (Proposed (GB) and Proposed (EOP)), it is also shown the PI controller computed by Chen and Seborg (2001). eq The Nyquist plots of gjj (jω)cj (jω) and gjj (jω)cj (jω) are shown in Figure 7. It can be seen that the designed systems fulfill the design specifications. Some performance and robustness measures are summarized in Table 2.
The resultant closed-loop output response to unit step setpoint change and unit step disturbance change are shown in Figure 8. Although the same design specifications, note that Loop 1 output response is smoother for the Proposed (GB) controller than for Proposed (EOP) controller. The Loop 2 output response for a unit step disturbance is faster for both proposed controllers than Chen-Seborg controller. 6.2 Example 2 Consider a binary ethanol-water system of a pilot-plant distillation column proposed by Ogunnaike and Ray:
Fig. 8. Step responses of the first and second outputs (solid blue line: Proposed (EOP), dashed black line: Proposed (GB), and dashed green line: Chen-Seborg). −2.6s −3.5s −s G(s) =
0.66e (6.7s+1) 1.11e−6.5s (3.25s+1) −34.68e−9.2s (8.15s+1)
−0.61e (8.64s+1) −2.36e−3s (5s+1) 46.2e−9.4s (10.9s+1)
−0.0049e (9.06s+1) −0.01e−1.2s (7.09s+1) 0.87(11.61s+1)e−s (3.89s+1)(18.8s+1)
.
The design specifications were chosen to guarantee minimum stability margins for each loop. This means a minimum gain margin of 4.5, 4.8, and 3.0, as well as a minimum phase margin of 55◦ , 36◦ and, 35◦ for loops 1, 2 and, 3, respectively. Because the system is not diagonally dominant, the respective Gershgorin bands are too wide, which implies in margins results too far from design specifications. So, the parameters result from the proposed method based on Gershgorin bands are just used as an initial controller to the proposed method that uses EOPs. As can be seen in Figure 9, and Table 4, the margins results fulfill the design specifications. Controller parameters obtained using the proposed approach is show in Table 3, also results from Huang et al. (2003) (HJCP) and the well known Biggest Log-modulus (BLT) (Luyben (1986)) are presented. The resultant closed-loop output responses to unit step set-point change are shown in Figure 10. 7. CONCLUSION This paper has presented an iterative procedure for decentralized PID controller design. Each SISO loop is designed at a time by solving an optimization problem that is efficiently solved by a linear programming approach. The loop interactions are taken into account by Gershgorin bands
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Table 3. Controller parameters in Example 2.
c3 (s)
kp 2.00 1.51 1.27 −0.42 −0.30 −0.16 2.83 2.63 3.94
ki 0.46 0.09 0.33 −0.10 −0.02 −0.08 0.67 0.40 1.77
Nyquist Diagram
kd 7.50 0.00 1.69 −1.29 0.00 −0.31 0.02 0.00 0.91
Tf 4.6 0.0 0.1 4.4 0.0 0.1 0.0 0.0 0.1
1
1
c2 (s)
Method HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP)
y
Controller c1 (s)
0
0
0
−0.2 −0.4
−0.2
150
200 250 Time (min)
300
350
400
450
0
50
100
150
200 250 Time (min)
300
350
400
450
0
50
100
150
200 250 Time (min)
300
350
400
450
0.5 0
−0.6
−0.8
−0.8
−1
−1
−1
−0.5 Real Axis
0
5
(b) Loop 2
0 3
(a) Loop 1
y
Nyquist Diagram
−5 −10
0.2 0
Imaginary Axis
100
−0.4
−0.6
0
y
Imaginary Axis
Imaginary Axis
50
1 2
0.2
−0.5 Real Axis
0
Nyquist Diagram
0.2
−1
0.5
−15
−0.2 −0.4 −0.6 −0.8 −1
−1
−0.5 Real Axis
Fig. 10. Step responses of the first, second, and third outputs (solid blue line: Proposed (EOP), dashed magenta line: HJCP, dashed black line: BLT).
0
(c) Loop 3 eq Fig. 9. Nyquist plot of gjj cj (solid blue line) and gjj cj (dashed black line) for the Proposed (EOP) PID controller in Example 2. Straight line rn in red line.
Table 4. Robustness and performance measures in Example 2. Controller c1 (s) c2 (s) c3 (s)
Method HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP) HJCP BLT Proposed (EOP)
gm 4.56 4.50 5.16 4.80 4.52 6.97 4.01 4.48 3.15
Φm 33.53◦ 100.0◦ 65.02◦ 36.38◦ 109.0◦ 44.46◦ 65.02◦ 79.52◦ 44.67◦
IE 2.17 10.81 3.05 10.17 61.81 12.03 1.51 2.53 0.57
IAE 2.58 10.81 3.93 11.41 61.81 20.29 1.51 2.54 0.68
and EOPs. If the process is diagonally dominant, only a single-iteration is sufficient to get stability margins results close to design specifications. The proposed procedure does not require work on model order reductions, indeed can be applied to any complex dynamic and multiple time delays. Simulation examples illustrate the effectiveness of the method, some robustness and performance indexes are compared to related techniques. REFERENCES ˚ Astr¨om, K. J., and H¨ agglund, T. (2006) Advanced PID Control. Instrument Soc. Amer., 2006. Chen, D., and Seborg, D.E. (2001) Multiloop PI/PID controller design based on Gershgorin bands. Proceedings of the American Control Conference, 2001. Garcia, D., Karimi, A., and Longchamp, R. (2005) PID Controller Design for Multivariable Systems Using Ger-
shgorin Bands. Proc. 16th IFAC World Congress, pp. 183-188, 2005. Ho, W.K., and Xu, W. (1998) Multivariable PID Controller Design Based on the Direct Nyquist Array Method. Proceedings of the American Control Conference, pp. 3524-3528, 1998. Huang, H. , Jeng, J. , Chiang, C. , and Pan, W. (2003) A direct method for multi-loop PI/PID controller design. Journal of Process Control, 13, pp. 769-786, 2003. Karimi, A. , Kunze, M. , and Longchamp, R. (2007) Robust controller design by linear programming with application to a double-axis positioning system. Control Eng. Practice, 15, pp. 197-208, 2007. Luyben, W.L. (1986) Simple method for tuning SISO controllers in multivariable systems. Industrial Eng. and Chem. Process Design Development,25, pp. 654-660, 1986. Nie, Z. , Wang, Q., Wu, M., and He, Y. (2011) Tuning of multi-loop PI controllers based on gain and phase margin specifications. Journal of Process Control, 21, pp. 1287-1295, 2011. Oliveira, V., and Karimi, A. (2012) Robust and gainscheduled PID controller design for condensing boilers by linear programming. IFAC Conference in Advances in PID Control, 2012. Rosenbrock, H.H. (1970) State-space and multivariable theory. London Nelson, 1970. Vu, T.N.L. , and Lee, M. (2010) Independent design of multi-loop PI/PID controllers for interacting multivariable processes. Journal of Process Control, 20, pp. 922933, 2010.
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