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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

Decentralized PI Controller Design Based on PhaseMargin Specifications

Petr Hušek

Abstract—An effective method for design and tuning of

decentralized PI controllers for stable multi-input, multi-outputsystems is presented in this brief. The direct Nyquist array isapplied to shape the Gershgorin bands for each loop separatelysuch that they pass through a specified point corresponding tophase margin specification that is used as a single tuning para-meter. The procedure is applied on the control of a laboratorymodel of quadruple-tank system.

Index Terms— Decentralized control, multivariate systems,Nyquist stability analysis, phase margin, quadruple-tank process.

I. INTRODUCTION

DESPITE the significant progress in applicability of advanced control techniques resulting in a growing

number of industrial implementations, the proportional-integral-differential (PID) controllers remain the most popularcontrollers used in process control. The reason for that consistsin their simple implementation, high reliability and robustness,and sufficient performance in the cases when the closed-looprequirements are not too strong[1]. Unfortunately, even thoughconsisting of three parameters only their tuning is not trivialeven for single-input, single-output (SISO) systems [2], [3]and consequently plenty of them are tuned poorly.

However, most of the plants encountered in industry containmore closed loops. The multivariate systems can be success-fully treated by classical control approaches. The resultedstate space or full transfer function matrix controllers are

nevertheless of a high order and thus difficult for final tuning.For severe interactions, the decoupling control schemes arepreferable [4]–[7]. When the interactions among differentloops are moderate, decentralized PID control achieves satis-factory behavior. Moreover, even if advanced strategies, suchas increasingly applied model predictive control are used, PIDcontrollers are still implemented at the lowest level loops. Eventhough the SISO design procedures cannot be applied directlydue to process interactions analysis and synthesis becomemuch easier than in the case of full dimensional controller. Acomprehensive survey of design methods can be found in [8].

For multiloop PID control design, several methods havebeen developed based on frequency-domain analysis. A com-

mon practice consists in using SISO techniques for diagonalelements disregarding the other ones followed by a detuning—decreasing the coefficients by some ad hoc factor such that

Manuscript received January 28, 2012; accepted February 4, 2013.Manuscript received in final form February 16, 2013. Recommended byAssociate Editor C. Lagoa.

The author is with the Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Prague 16627, Czech Republic (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2013.2248060

the influence of the other loops will not affect the main

loops too much [9]. The most well-known method usingdetuning control scheme is the biggest log modulus tun-ing [10]. If a model of the controlled MIMO plant is notavailable, methods adopting relay feedback tests resulting inmodifications of the Ziegler–Nichols (ZN) tuning rules canbe applied [11]–[14]. Chen and Seborg [15] modified theZN rules using localization of the stability region in the con-troller parameter space. A detailed analysis of generalizationof ultimate quantities to MIMO systems was elaborated byCampestrini et al. [16].

Another approach relies on extension of Nyquist stabilitycriterion to MIMO systems [17], [18]. The direct Nyquistarray ensures closed-loop stability and performance by anappropriate shaping of the Gershgorin bands, the MIMOversion of the Nyquist curve. For a second-order plus dead-time model, Ho et al. [19], [20] derived analytic formulas toguarantee gain and phase margin specifications for Gershgorinbands. An iterative tuning using the structural decomposi-tion of n × n MIMO system to n SISO ones is used in[21] and [22]. A similar idea allowing independent designof diagonal controllers called equivalent subsystems methodis presented in [23]. Garcia et al. [24] minimized a frequencycriterion for each independent loop weighting the deviations of the infinity norms of sensitivity and complementary sensitivityfunctions as well as crossover frequency from the specifiedvalues.

This brief presents a multiloop PI control tuning method thatshapes the Gershgorin bands such that they do not contain thepoint e− jM where the angle M is a tuning parameter. Themotivation comes from SISO systems controller design wheresuch angle corresponds to phase margin that often serves as aclosed-loop step response tuning parameter since it is relatedto its damping. All the PI controller candidates for each loopare plotted as a curve in the k P − k I plane and the ones withthe highest integral part are chosen since they minimize thesum of integral errors. The presented method is implementedon control of a laboratory model of quadruple-tank process[25], [26].

The main advantage of the presented procedure consists inits simplicity—it contains only one tuning parameter and thecomputation is one-step and simple. On the other hand, themethod does not allow fine tuning and is not suitable forsystems with strong interactions.

II . DIRECT NYQUIST ARRAY DESIGN

Consider an n × n system G(s) = [gi j (s)]n×n controlled by adecentralized controller C (s) = diag{c1(s), c2(s) , . . . , cn(s)},see Fig. 1.

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2 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

Fig. 1. Decentralized control scheme.

The individual control loops are paired according to relativegain analysis (RGA) [18]. The following denote:

L(s) = G(s)C (s) (1)

H (s) = ( I + G(s)C (s))−1 G(s)C (s) (2)

the open-loop and closed-loop transfer matrix, respectively.Consider the Nyquist plot of gkk ( jω)ck ( jω) with a superim-posed circle of the radius

n

k =1,k =m

|gkm ( jω)cm ( jω)|. (3)

This circle is referred to as the Gershgorin circle. The wholeband composed of the circles for all ω > 0 is called theGershgorin band. Stability of the closed loop can be testedby the following theorem.

Theorem 1 (Direct Nyquist Array (DNA) [17], [18]):

Let the Gershgorin bands be centered on the diagonalelements lmm ( jω) of L( jω), and m = 1, . . . , n exclude thepoint (−1 + j0) (the transfer matrix L(s) is called columndiagonally dominant). Let the i -th Gershgorin band encirclesthe point (−1 + j0) N i times anticlockwise. Then, theclosed-loop transfer matrix H (s) is stable if and only if

ni=1

N i = p0 (4)

where p0 is the number of unstable poles of L(s).Since most practical processes are open-loop stable,

p0 = 0 is assumed in this brief. In that case the closed-loop transfer matrix H (s) with column diagonally dominantopen-loop transfer matrix L(s) is stable if and only if Nyquistplots of lmm ( jω) do not encircle the point (−1 + j0) for allm = 1, . . . , n.

Remark 1: The stability analysis by direct Nyquist arrayrepresents sufficient condition only, i.e., if some of the Ger-shgorin bands contains the critical point nothing can be said

about stability or instability of the closed loop. Remark 2: To diminish the interactions as much as possible,a constant decoupling compensator can be inserted betweenthe controller and the plant. The frequency on which thecompensator will act is usually chosen zero or equal to thebandwidth of the closed loop.

III. PHASE MARGIN SPECIFICATION REGION FOR

DECENTRALIZED PI CONTROL

In compliance with the phase margin definition for SISOsystems, let us try to shape the Gershgorin bands such thatthey pass through the point e− jφM in the complex plane

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Re

I m

Aφ

M

−1

lmm

(j ω )

Fig. 2. Phase margin for Gershgorin band.

(point A in Fig. 2) where the value of φM is usually chosenbetween 0 and π/2.

Let us remind that the extension of phase margin to MIMOsystems is not straightforward because of complexity of matrixperturbations that result in many different definitions. The def-inition employed here [27] corresponding to perturbations of characteristic loci is simple for computation but it requires thediagonal dominance of the system. The most general definitionformulated by Ba-ron and Jonckheere [28] considers arbitraryunitary matrix perturbations in the feedback path. However,

from practical reasons, it is not used for a controller designbecause it is tough to imagine its relation to phase changes inindividual loops. The most useful definition, proposed in [29],takes into account only diagonal phase perturbations but evenfor this simplification, the phase margin computation requiressolving a constraint optimization task.

Although a closed-loop system satisfying phase marginspecification used in this brief may not be stable, it iswidely accepted by the practitioners since such a case doesis unlikely to occur in practical applications. The conditioncan be expressed as

|e jφM

+ gmm ( jω)cm ( jω)| ≥

nk =1,k =m

|gkm ( jω)cm ( jω)| ∀ω, m (5)

with the equality sign holding for just one ω = ωgm for eachm = 1, . . . , n corresponding to the touching point.

Since, according to the assumption, the diagonal elementsare all stable, one can always find a sufficiently small pro-portional controller such that inequality 5 holds for every ω.Since 0 < M ≤ π/2 for a stable gmm (s) of at leastsecond order, one can find a frequency ω = ωgm suchthat gmm ( jωgm ) = (−e jM) = −π + M and thus byan appropriate proportional controller, one can make theleft-hand side of (5) arbitrarily small (even zero) to force the

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HUŠEK: DECENTRALIZED PI CONTROLLER DESIGN 3

Fig. 3. Principal scheme of quadruple-tank system.

TABLE I

NONMINIMUM PHASE SYSTEM PARAMETER VALUES

A1, A3 [cm2] 25

A2, A4 [cm2] 30

a1, a3 [cm2] 0.084

a2, a4 [cm2] 0.065

g [cm/s2] 981

h01, h0

2 [cm] (11.4 12)

h03, h0

4 [cm] (4.99 4.72)

u

0

1, u

0

2 [V] (4.15 4.41)k 1, k 2 [cm3/Vs] (2.32 2.58)

γ1, γ2 [−] (0.35 0.27)

equality. For a first order gmm (s), a purely integral controllerwill do the same job. Thus, at least one PI controller solving(5) exists for every 0 < M ≤ π/2 and each m = 1, . . . , n.

Let us write each element of the plant transfer matrixgkm ( jω) (possibly with time delay) as

gkm ( jω) = akm (ω) + jbkm (ω) = r km (ω)e jφkm (ω) (6)

and use a decentralized PI controller C (s) = diag{c1(s) , . . . , cn (s)}

cm (s) = k Pm +k Im

s, m = 1, . . . , n (7)

to control the plant.With

Rm (ω) =

nk =1,k =m

|gkm ( jω)|, m = 1, . . . , n (8)

and after some algebraic manipulations, the inequality

0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3

kP

k I

φM

=0°

φM

=15°

φM

=25°

φM

=40°

Fig. 4. PI controllers with phase margin specifications.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

x 10−3

kP

k I

φM

=0°

φM=15

°

φM

=25°

φM

=40°

Fig. 5. PI controllers with phase margin specifications.

TABLE II

DECENTRALIZED PI C ONTROLLERS CONSTANTS

φM[°] k P1 k I1 k P2 k I20 0.5342 0.0082 0.2754 0.0032

15 0.3944 0.0050 0.2351 0.002325 0.3301 0.0037 0.2119 0.001740 0.2651 0.0023 0.1786 0.0012

CS [15] 0.4958 0.0040 0.2667 0.0016

(5) becomes

k 2Pm

r 2mm (ω) − R2

m (ω)

+ k 2Im

r 2mm (ω) − R2m (ω)

ω2

+ 2k Pm (amm (ω) cos φM + bmm (ω) sin φM)

+ 2k Imamm (ω) sin φM − bmm (ω) cos φM

ω+ 1 ≥ 0. (9)

All the PI controllers with k Pm > 0, k Im > 0 of themth loop satisfying (9) for 0 ≤ ω < ∞ form a region ink P − k I plane for which the corresponding Gershgorin banddoes not contain the point e− jφM . The controllers lying on theboundary of that region guarantee that the Gershgorin bandpasses through that point.

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0 500 1000 1500 2000 2500 3000 3500 400011.4

11.5

11.6

11.7

11.8

11.9

12

12.1

12.2

12.3

t[s]

y 1

[ c m ]

φM

= 0°

φM

= 15°

φM

= 25°

φM

= 40°

Fig. 6. Model responses of level in tank 1 for step setpoint changes.

0 500 1000 1500 2000 2500 3000 3500 400011.8

11.9

12

12.1

12.2

12.3

12.4

12.5

12.6

12.7

12.8

t[s]

y 2

[ c m ]

φM

= 0°

φM

= 15°

φM

= 25°

φM

= 40°

Fig. 7. Model responses of level in tank 2 for step setpoint changes.

According to the DNA theorem, in order to achieve closed-loop stability, the frequency plots of the open-loop diago-nal elements lmm ( jω) should not encircle the critical point(−1 + j0). Solving the equation

lmm ( jω) = gmm ( jω)cm( jω)

=

k Pm +

k Im

jω

(amm (ω) + jbmm (ω))

= −1 (10)for real and imaginary part separately, one obtains the stabilityregion in k P − k I plane, which is delimited by

k Pm = −amm (ω)

r 2mm (ω), k Im = −ω

bmm (ω)

r 2mm (ω)(11)

plotted for 0 ≤ ω < ∞.The PI controllers guaranteeing the phase margin φM are

those lying on the boundary of the region characterized by(9) and inside the stability region (11). The region (9) canbe determined in a graphical way as the intersection of theellipses plotted for each 0 ≤ ω < ∞. Numerically, a range

0 500 1000 1500 2000 2500 3000 3500 400011.3

11.4

11.5

11.6

11.7

11.8

11.9

12

12.1

12.2

12.3

t[s]

y 1

[ c m ]

φM = 0°

φM

= 15°

φM

= 25°

φM

= 40°

Fig. 8. Real system responses of level in tank 1 for step setpoint changes.

0 500 1000 1500 2000 2500 3000 3500 400011.8

11.9

12

12.1

12.2

12.3

12.4

12.5

12.6

12.7

12.8

t[s]

y 2

[ c m ]

φM

= 0°

φM

= 15°

φM

= 25°

φM

= 40°

Fig. 9. Real system responses of level in tank 2 for step setpoint changes.

for k Pm should be specified at first and then the endpointsof the admissible range of k Im are chosen from solutions of quadratic equations for each value of k Pm and 0 ≤ ω < ∞.

Remark 3: As it was mentioned above, similarly to SISOcase, the controllers guaranteeing the phase margin specifi-cation may not guarantee overall system stability since theGershgorin band passing through the point e− jφM still maycontain the critical point (−1 + j0). Although such cases are

very rare, the inequality (9) with φM = 0° can be added toavoid the Gershgorin band to contain the critical point.From the whole set of controllers guaranteeing the phase

margin, we will choose those with maximum integral partfor each loop since they minimize the sum of integral errorsand with reasonable damping they generally produce the bestvalues for the speed of response and performance, see [30].

IV. CONTROL OF NONMINIMUM PHASE

QUADRUPLE-TANK PROCESS

The decentralized PI controller design procedure derivedabove will be implemented on a laboratory model of

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HUŠEK: DECENTRALIZED PI CONTROLLER DESIGN 5

quadruple-tank process in nonminimum phase configuration.A schematic diagram of the process is depicted in Fig. 3.

The process inputs u1 and u2 [V] are the voltages to thepumps and the outputs y1 and y2 [cm] are the levels in thelower tanks. After linearization around an operating point, weobtain a fourth-order state-space model

˙ x = A x + Bu y = C x + Du (12)

with ui = ui − u0i , yi = yi − y0

i , i = 1, 2, x i = hi − h0i ,

i = 1, . . . , 4 where hi [cm] denote the levels in the tanks and

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−1

T 10

A3

A1T 30

0 −1

T 20

A4

A2T 4

0 0 −1

T 30

0 0 0 −1

T 4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

γ1k 1

A20

0 γ2k 2

A2

0(1 − γ2)k 2

A3(1 − γ1)k 1

A40

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(13)

C =

1 0 0 00 1 0 0

D =

0 00 0

where the time constants are given as

T i = Ai

ai

2h0

i

g, i = 1, . . . , 4 (14)

with Ai [cm2] and ai [cm2], i = 1, . . . , 4, denoting thecross sections of the tanks and their outlets, respectively,k i [cm3V/s], i = 1, 2 being the pump constants and γi [−],

i = 1, 2, γi ∈ [0, 1] the relative valves constants dividing thepump flows into the upper and lower tanks. g[cm/s2] denotesthe gravitational constant.

The transfer matrix of the system is

G(s) =

⎡⎢⎢⎢⎣γ1c1

1 + sT 1

(1 − γ2)c1

(1 + sT 3)(1 + sT 1)

(1 − γ1)c2

(1 + sT 4)(1 + sT 2)

γ2c2

1 + sT 2

⎤⎥⎥⎥⎦ (15)

with ci = (T i k i / Ai ), i = 1, 2.An interesting feature of the system is that by setting

the valve constants γi , one can change the minimum phasesystem (if γ1 + γ2 > 1) to the nonminimum phase one (if γ1 + γ2 < 1), see [25]. For the controller design, we willapply the nonminimum phase configuration with the parametervalues given in Table I. It follows from the RGA analysis thatif taking into account interactions in closed loop, the more

0 500 1000 1500 2000 2500 3000 3500 400011.4

11.5

11.6

11.7

11.8

11.9

12

12.1

t[s]

y 1

[ c m ]

φM

= 25°

Chen

Fig. 10. Comparison of PI controller proposed and Chen and Seeborg [15].

0 500 1000 1500 2000 2500 3000 3500 400012

12.05

12.1

12.15

12.2

12.25

12.3

12.35

12.4

12.45

12.5

t[s]

y 2

[ c m ]

φM

= 25°

Chen

Fig. 11. Comparison of PI controller proposed and Chen and Seeborg [15].

suitable pairing is u1 − y2 and u2 − y1, hence we interchangethe columns in the plant transfer matrix (15).

The corresponding transfer matrix yields

G(s) =

⎡⎢⎢⎢⎣

0.81

1 + 47.7s

3.77

(1 + 37.7s)(1 + 80.1s)

3.59

(1 + 30s)(1 + 47.7s)

0.69

1 + 80.1s

⎤⎥⎥⎥⎦

.

Using the proposed procedure, the sets of all PIcontrollers satisfying phase margin specifications forφM = 0°, 15°, 25°, 40° for both loops are depicted inFigs. 5 and 4. The values of controller parameters withmaximum integral part on each curve, which will beconsidered for control, are summarized in Table II.

The courses of the levels in tanks 1 and 2 for step referencesignals for the model and the real laboratory system are plottedin Figs. 6–9, respectively. One can see that the phase marginφM represents a suitable parameter to tune the decentralizedcontrol ranging from an underdamped behavior for too lowvalues to an overdamped response if its value is too high.

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The comparison of the responses of decentralized PI controllerobtained using the proposed method for φM = 25° and theprocedure suggested by Chen and Seeborg in [15] (last row inTable II), based on modification of the Ziegler–Nichols rules,is shown in Figs. 10 and 11.

V. CONCLUSION

In this brief, a method for decentralized PI controllerdesign of MIMO systems was presented. The method isapplicable to a plant described by an arbitrary transfermatrix with time delays. The proposed controller satisfiesthe phase margin specification for the Gershgorin bands. Thecontrol experiments carried out on the laboratory model of quadruple-tank system confirmed that the phase margin is asuitable tuning parameter closely related to the damping of the closed-loop response. Moreover, the experiments revealedthat if appropriately tuned, the proposed design achievesbetter results than a popular method based on modificationof Ziegler–Nichols rules.

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