Decentralized Robust Control of MIMO Systems: Quadruple Tank Case Study

Proceedings of the 9th IFAC Symposium Advances in Control Education The International Federation of Automatic Control Nizhny Novgorod, Russia, June 19-21, 2012

Decentralized Robust Control of MIMO Systems: Quadruple Tank Case Study Danica Rosinová. Alena Kozáková Institute of Control and Industrial Informatics, Slovak University of Technology Faculty of Electrical Engineering and Information Technology: Bratislava, Slovak Republic (e-mail: [email protected]; [email protected] ) Abstract: This paper presents part of material concerning decentralized control, taught in graduate course Control of Complex Systems. We present some basic aspects of decentralized control design concerning stability and performance and illustrate them on a case study: virtual model of a quadruple tank process. Control structure selection based on performance relative gain array (PRGA, Hovd, Skogestad, 1992) is used and its ability to evaluate the achievable performance is discussed. Robust stability condition for decentralized control is considered, which provides the upper level on subsystems, thus limiting the performance. Keywords: Decentralized control, Robust stability, Robust performance, Virtual reality. reality model of this process, built under Matlab by one of our student. We begin with control structure selection, i.e. choice of appropriate input-output pairing for decentralized control. Further step is independent single loops design so that it guarantees stability as well as required performance of the overall system including interactions. Two alternatives of stability condition for decentralized control structure are used: one based on small gain theorem for complementary sensitivity function and one for systems with no RHP (right half plane) zeros. To evaluate the achievable performance under decentralized control, Performance Relative Gain Array PRGA, (Hovd, Skogestad, 1992) is used. The application of these design tools is shown on case study. Presented material provides simple illustration of stability and performance issues in decentralized control design and can be used in teaching complex systems control.

1. INTRODUCTION Robust stability of uncertain dynamic systems has major importance when real world system models are to be controlled. Uncertainties due to inherent modelling/ identification inaccuracies in any physical plant model specify a certain uncertainty domain, e.g. as a set of linearized models obtained in different working points of the considered plant. Thus, a basic required property of the system is its stability within the whole uncertainty domain denoted as robust stability. Robust control theory provides analysis and synthesis approaches and tools applicable for various kinds of processes, including multi input – multi output (MIMO) dynamic systems. To reduce multivariable control problem complexity, MIMO systems are often considered as interconnections of a finite number of subsystems. This approach enables to employ decentralized control structure with subsystems having their local control loops. Compared with centralized MIMO controller systems, decentralized control structure yields certain performance deterioration, however weighted against by important benefits, such as design simplicity, hardware, operation and reliability improvement. Robustness is one of attractive qualities of a decentralized control scheme, since such control structure can be inherently resistant to a wide range of uncertainties both in subsystems and interconnections.

2. DECENTRALIZED CONTROL STRATEGY - BASIC STEPS Control design of MIMO system (plant) comprises several steps and tasks (Skogestad and Postlethwaite, 2009): a) study the plant and formulate the control objective; b) find a plant model, simplify it if necessary; c) analyze the model properties, scale the variables; d) decide which variables are to be controlled and which variables are to be the manipulated ones; e) select the control configuration: for decentralized control structure it means to choose the input – output pairing; f) specify the performance requirements respective to the control objective; g) determine the type of controller and design its parameters; h) examine the resulting control system, if the specified requirements are not met, redesign; i) analyze simulation results, if necessary repeat the whole procedure; j) realize the designed controller.

In this paper we concentrate on key aspects of decentralized control design which are taught in subject Control of Complex Systems (in Master’s study programme Cybernetics). The main aim of the considered decentralized control design strategy is to keep the overall system robust stability and to achieve required performance specifications. We study the basic steps of decentralized control design on quadruple tank process case study with 2 inputs and 2 outputs (Johansson et al., 1999; Johansson, 2000), since it includes both minimum and non-minimum phase configurations and attractive physical interpretation. In teaching we use virtual 978-3-902823-01-4/12/$20.00 © 2012 IFAC

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9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

We assume that we have MIMO system model linearized around the working point and we concentrate on points e), g), h), which are crucial for successful control design.

After completing step 1, the inputs or outputs can be reordered so that the respective transfer system matrix G with reordered columns or rows has the paired elements on the main diagonal. Then the decentralized controller can be represented by the diagonal matrix C( s ) = diag( Ci ) (see Fig.2). To find C( s ) (step 2), the so called independent design is considered, where individual loops are designed “independently” (simultaneously). Local controllers C i ( s ) are designed so that they:

The important task in MIMO systems is to decide on control configuration, i.e. the decomposition of the controller. One possible choice of appropriate control configuration, which substantially simplifies both control design and implementation issues, is decentralized control. In decentralized control, the whole system is considered as a set of interconnected subsystems with defined inputs, outputs and interconnections, and the decentralized controller has the respective diagonal or block diagonal structure corresponding to chosen input – output pairing, see Fig.1, Fig.2.

CS1

CS2

a) stabilize individual loops b) satisfy the overall system stability condition c) satisfy the bounds obtained from performance requirements. Note that conditions b) and c) are often contradictory.

CSn

In the following, sensitivity is denoted as −1 S ( s ) = ( I + G( s )C( s )) and closed loop transfer function (complementary sensitivity) is denoted as −1 T ( s ) = G( s )C( s )( I + G( s )C( s )) .

controlled system

2.2 Control configuration (pairing) selection Fig. 1. Decentralized control structure

To choose appropriate pairing, several interaction measures have been proposed in literature (RGA, dRGA, PRGA, etc.), more details can be found e.g. in (Schmidt, 2002). Relative Gain Array (RGA), frequently used in practice, is defined as

(

)

T

RGA( G ) = G( s ) o G( s )−1 (2) where o is entrywise matrix product (Hadamard product). Individual subsystems are then specified by the chosen pairing, their transfer functions are placed in the diagonal of the transfer function matrix. Structural stabilizability for the chosen pairing can be checked by the Niederlinski index:

Fig. 2. Decentralized control: G denotes controlled system, diag(Ci) is a decentralized controller, Gz corresponds to a disturbance z.

det (G( 0 )) Π( diag( G( 0 ))

2.1 Decentralized control problem formulation

NI =

Consider a MIMO plant described by linear model

If NI < 0 , the system cannot be stabilized using the chosen pairing and the pairing must be modified.

y( s ) = G( s )u( s )

(1)

(3)

where complex vectors y( s ), u( s ) are Laplace images of output and input signal of dimensions p and m respectively, G( s ) is transfer function matrix of dimensions p × m . In the following we assume the square system, i.e. p = m and stable plant G. Argument s is often omitted for better readability.

It must be noted that RGA index provides limited information, e.g. for the system with one way interconnections (when the transfer function matrix is upper or lower triangular). To better evaluate system structure and performance, PRGA index has been introduced (Section 2.4).

Our aim is to design appropriate decentralized control, so that the overall system stability is kept (including possible uncertainties) and the required performance is achieved.

After the appropriate pairing has been determined, the decentralized control law is to be designed. There are various approaches to find the respective diagonal controller matrix C(s). We adopt independent design as a simple possibility to design single loops so that the overall stability and performance requirements are kept, i.e. that interactions do not introduce instability and do not significantly deteriorate performance. Let us turn to stability condition for system with decentralized control. Matrix G(s) can be splitted into its diagonal and off-diagonal parts: G( s ) = G D ( s ) + G M ( s ) .

2.3 Stability condition for decentralized control

We focus on two most important steps in decentralized control design: 1. the determining of appropriate input-output pairing; 2. the respective single control loops design so that the overall requirements are kept.

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9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

There is close relationship between PRGA and closed loop system performance specified by bounds on control error (offset) and disturbance

The uncertainties can be included into G M ( s ) . For stable open loop system G(s)C(s), the closed loop system stability condition based on small gain theorem is given in the next Lemma (Veselý, Harsányi, 2008).

ei ( jω)/r j ( jω) = S ij ( jω) < 1/ wri ( jω)

Lemma 1

ei ( jω)/z k ( jω) = [ SG z ]ik ( jω) < 1/ wzi ( jω)

Consider stable system G(s) with decentralized controller C(s). The respective closed loop system T(s) is stable if

G D−1W GM < 1 1 GM

∀ω, i,k

(9b)

where r j denotes j setpoint change, S ij is the respective

(4)

element of sensitivity function S, z k is expected disturbance

(5)

For frequencies, where a feedback is effective ( ω < ωB , ωB

and G z its transfer function; wri , wzi are performance weights for control error and disturbance respectively. denotes bandwith), it is assumed S = (I + GC ) ≈ (GC ) yielding the following bounds for individual loops −1

where matrix W is given by C −1 + G D = G DW −1 Inequality (5) can be reformulated into

GD−1TD < M 0 =

(9a)

th

or

GD−1W <

∀ω, i, j

g ii (jω)Ci ( jω) > ηij wri ( jω)

1 GM

(6)

where TD = G D C( I + G D C ) −1 .

(10a)

ηij are elements of PRGA index Γ g ii ( jω)Ci ( jω) > δik wzi ( jω)

Condition (6) can be used for stable system without or with RHP zeros (both for minimum and non-minimum phase case). However, the above condition can be rather limiting in low frequencies, where TD ≈ 1 , for stable system with no

∀ ω < ω B , ∀i,j ,

−1

∀ω < ωB , ∀i,k

(10b)

δik are elements of ΓGz.

RHP zeros this may be too restrictive. The alternative condition for this case is in Lemma 2.

Inequalities (10a), (10b) determine performance limits lower bounds on single loop modules to achieve the required control error and disturbance attenuation, the former is discussed in control design stage.

Lemma 2 (Skogestad and Postlethwaite, 2009)

3. CASE STUDY – QUADRUPLE TANK PROCESS

Consider stable system G(s) with decentralized controller C(s). Assuming that neither G nor GD has RHP zeros, the −1 overall closed loop system is stable if and only if (I − ES D ) is stable, where

This section demonstrates the proposed approach for decentralized PID controller design on a case study. The quadruple-tank process has been introduced in (Johansson et al., 1999; Johansson, 2000) and provides a case study to analyze both minimum and non-minimum phase MIMO systems on the same plant.

E = ( G − G D )G −1 = G M G −1 , S D = ( I + G D C ) −1 . The above condition can be reformulated:

(I − ES D )−1

stable

means det (I − ES D ) ≠ 0 . The sufficient stability condition −1

is then

G −1 S D

ES D < 1 , or 1 < M0 = . GM

3 (7)

Either of alternatives (6) or (7) must be satisfied for all frequencies.

4

(9) v1

1

2

v2

2.4 Performance margins for decentralized control system The Performance Relative Gain Array PRGA has been introduced (Hovd and Skogestad, 1992) and shown to provide information for appropriate pairing, but also performance limits for system with decentralized control.

Fig. 3. Quadruple tank process scheme.

PRGA is defined as

PRGA( G ) = Γ = G D ( s ) G( s ) −1

(8)

In teaching, we use visualized nonlinear model shown in Fig.3 - created in Matlab Virtual toolbox by a student of our Institute, within his final thesis, (Vincel, 2007). The inputs v1 and v 2 are pump 1 and 2 flows respectively, the aim is to control outputs y1 and y2 , i.e. levels in lower tanks 1 and 2.

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9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

The state equations of nonlinear model of quadruple tank are

Quadruple tank process – uncertainty domain

a dh1 a γk = − 1 2 gh1 + 3 2 gh3 + 1 1 v1 dt A1 A1 A1

For quadruple tank system (13), we consider the uncertainty to be a change of valve position, i.e. change of γ 1 and γ 2 , uncertainty domain is specified by three working points:

dh2 a =− 2 dt A2

2 gh2 +

a4 A2

dh3 a =− 3 dt A3

( 1 − γ 2 )k 2 2 gh3 + v2 A3

2 gh4 +

γ 2k2 v2 A2

(11)

where Ti =

Ai ai

  γ1k1 0   A1  x   1 0 0     0   x3   ⋅ +  − 1 A4   x2   0 T2 T4 A2   x    4   −1 ( 1 − γ1 )k1 0   T4   A4 0

   ( 1 − γ 2 )k2   u1  A3 ⋅ γ 2k2  u    2 A1   0  

WP2: γ 1 = 0.8, γ 2 = 0.4; WP3: γ 1 = 0.8, γ 2 = 0.8 in non-minimum phase region: WP1:

dh4 a ( 1 − γ 1 )k1 = − 4 2 gh4 + v1 dt A4 A4 where for ith tank: Ai is cross-section, ai is cross-section of the outlet, hi is water level i; g is acceleration of gravity, the flow corresponding to pump i is kivi. Parameter γ 1 denotes position of the valve dividing the pump 1 flow into the lower tank 1: γ 1k1v1 and related upper tank 4: ( 1 − γ 1 )k1v1 and analogically γ 2 divides flow from pump 2 to tanks 2 and 3. The nonlinear model (11) can be linearized around the working point given by the water levels in tanks h10 ,h20 ,h30 , h40 : this is one of the first students’ tasks.  − 1 A3 T T A 1 3 1  x&1   −1 0 &   T3  x3  =   x&2   0   0  x&4   0 0 

in minimum phase region: WP1: γ 1 = 0.4, γ 2 = 0.8; (14a)

γ 1 = 0.1, γ 2 = 0.3; (14b)

WP2: γ 1 = 0.3, γ 2 = 0.1; WP3: γ 1 = 0.1, γ 2 = 0.1.

γ2 1 γ2

WP 1

WP 3

WP 2

0

1 WP 1

WP

γ1

3 0

1

a) minimum phase config.

0

(12)

WP 2

γ1

1

b) nonminimum phase config.

Fig. 4. Uncertainty domain specified by working points The nominal model G 0 ( s ) obtained as a model of mean parameter values is used for control design. 4. DECENTRALIZED CONTROLLER DESIGN FOR QUADRUPLE TANK

2hi 0 , i = 1,...,4 ; g

Students learn material outlined in Section 2, during labs they experiment on virtual model, find parameters of the linearized model in working points, choose appropriate pairing, design decentralized controller and verify their design and its qualities by applying it on virtual model control; possible results of control design are shown below.

deviation state variables are xi = hi − hi 0 and the respective control variables are ui = vi − vi 0 ; the argument t has been omitted; the state variables corresponding to levels in tanks 2 and 3 have been interchanged in state vector so that subsystems respective to input u1 from pump 1 (tanks 1 and 3) and u2 from pump 2 (tanks 2 and 4) are more apparent.

4.1 Model parameters

The respective transfer function matrix for inputs v1 and v2 and outputs y1 and y2 is

We consider quadruple tank linearized model (13) with parameters: A1 = A3 = 30 [ cm 2 ]; A2 = A4 = 35 [ cm 2 ];

c1 γ 1   T1 s + 1 G( s ) =  c 2 (1 − γ1 )   ( T4 s + 1 )( T2 s + 1 )

a1 = a 3 = 0.0977 [ cm 2 ]; a 2 = a 4 = 0.0785 [ cm 2 ] ; h10 = h20 = 20 [ cm ]; h30 = 2.75 [ cm ]; h40 = 2.22 [ cm ] ;

Tk where ci = i i Ai

c1 ( 1 − γ 2 )  ( T3 s + 1 )( T1 s + 1 )   c2 γ 2   T2 s + 1

(13)

g = 981 [ cm / s 2 ]; k1 = 1.790; k 2 = 1.827 . From three plant models (13) evaluated in working points taken from uncertainty regions (14a), (14b) for the minimum and non-minimum phase cases respectively, we obtain the resulting nominal (mean) models below.

2hi 0 , i = 1,2 . g

The plant can be shifted from minimum to non-minimum phase configuration and vice versa simply by changing a valve controlling the flow ratios γ 1 and γ 2 between lower and upper tanks. The minimum-phase configuration corresponds to 1 < γ 1 + γ 2 < 2 and the non-minimum-phase

4.2 Control configuration (pairing) selection The steady state RGA(0) is considered for a linearized model of quadruple tank (13), to choose appropriate pairing.

[

RGA( 0 ) = G (0)* G (0 )

one to 0 < γ 1 + γ 2 < 1 .

75

]

−1 T

1 − λ  λ =  λ  1 − λ

(15)

9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

where λ =

γ1γ 2 depends on valve parameters γ 1 and γ1 + γ 2 − 1

Performance bounds: upper bounds on weights wri ( jϖ) obtained from (10a) are depicted in Fig.6.

γ 2 exclusively. The pairing is chosen respectively to RGA elements positive and closest possible to 1. The diagonal elements λ are positive for 1 < γ 1 + γ 2 < 2 (minimum phase

stability condition: norm(...)
1

system) and the respective pairing is v1 − y1 , v 2 − y 2 . For

0 < γ 1 + γ 2 < 1 (non-minimum phase system), the opposite

0.5

pairing v1 − y 2 , v 2 − y1 is indicated. This result is approved by the computation of Niederlinski index (3), which is positive in both cases.

0

M0 norm(Td) norm(Sd)

-0.5

Minimum phase model

2.4667   62s + 1 G0 ( s ) =  1.5667   ( 30s + 1 )( 90s + 1 )

1.2333  ( 23s + 1 )( 62s + 1 )   3.1333   90s + 1

10

(16)

-3

10 frequency w[rad/s]

-2

10

-1

Fig.5. Robust stability condition for decentralized control 1 performance bound through PRGA11 100

performance bound through PRGA12 100

for pairing v1 − y1 , v 2 − y 2 . 50

0.6167   62s + 1  3.9170  ( 30s + 1 )( 90s + 1 ) 

0 -4 10

(17)

-2

10

0

10

performance bound through PRGA22 100

50

0 -4 10

In both minimum and non-minimum phase cases, decentralized (two loops) PID controller (18) is designed independently for both loops, e.g. using a SISO design method based on inverse model, and it is afterwards detuned, if necessary, basically by decreasing gains.

0 -4 10

0

10

50

4.3 Decentralized control design strategy

where Ci ( s ) = Pi + I i / s + Di s

-2

10

performance bound through PRGA21 100

columns of the transfer function matrix are interchanged respective to the opposite pairing v1 − y 2 , v 2 − y1 .

-2

10 frequency w[rad/s]

0 -4 10

0

10

-2

10 frequency w[rad/s]

0

10

Fig.6. Bounds on weighting functions for reference value step response: y1

30

(18) i=1, 2.

Decentralized control design strategy is following: when designing parameters of Ci ( s ) , in the first step we consider stability criterion (6) or (7) as a bound on loops responses, next step is to shape the loops responses within stability bounds to achieve the required performance specifications “measured” by performance bound (10a).

y1,w1

0  C1 ( s ) C( s ) =   C 2 ( s )  0

50

25

25

20

20

15

15

10

10

5

5

0

0

500 1000 time[s]

step response: y2

30

y2,w2

Non-minimum phase model 3.0830   ( 23s + 1 )( 62s + 1 ) G0 ( s ) =  0.7833   90s + 1

1500

0

0

500 1000 time[s]

1500

Fig.7. Step responses: minimum phase stable system Decentralized control for non-minimum phase case This configuration is characterized by the existence of transient RHP zeros (while individual transfer functions have no RHP zeros), which complicates the decentralized controller design. We illustrate the impact of interactions on two different designs of control loops.

Decentralized control for minimum phase case The PI controller parameters designed for individual loops are: P1=1.30, I1=0.053; P2=1.38, I2=0.049. Characteristics of the designed decentralized control system are in Figs 5-7. Fig. 5 shows that stability condition (6) is not satisfied in this case – blue line is for low frequencies above the red one, however, the overall stability is guaranteed by (for this case) the less restrictive condition (7), which is satisfied since green line is below the red one for all frequencies.

In the first case, taking the same decentralized controller as in minimum phase case, the overall stability condition is not satisfied, though the individual loops indicates stable performance. Step responses in Fig. 8 show the significant differences between individual loops (both are stable and damped) and the overall system, which is unstable. 76

9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012

30

0

50

0

40

1000 2000 3000 time[s] step response-loop2: y2

30

1000 2000 time[s]

3000

20

30

25

25

20

20

15

15

10

10

5

5

500

1000

t[s]

1500

2000

2500

0

0

500

1000

1500

2000

2500

t[s]

Fig.11. Step responses: non-minimum phase, stable system

30

5. CONCLUSION

25 0

30

0 0

35 y2,w2

y1,w1

20 10

1000 2000 3000 time[s] step response-loop1: y1

40

20

30 y1[cm ]

y2,w2

y1,w1

40

20

step response: y2

40

y2[cm]

step response: y1

50

0

1000 2000 time[s]

Decentralized control design strategy is illustrated on the quadruple tank case study. Pairing, robust stability and performance under decentralized control are studied. The difference between minimum and non-minimum phase cases can be seen from comparison of figures 5 with 9, 6 with 10, which indicates nearly ten times lower bandwidth for nonminimum phase case. This is approved by step responses in Fig. 7 and Fig. 11. In teaching, the case study is represented by a nonlinear virtual model shown in Fig. 3, control is designed for linearized model and verified in Simulink and then applied on virtual model.

3000

Fig.8. Step responses: non-minimum phase, unstable system The next (detuned) case: P1= 0.208, I1= 0.0039; P2=0.238, I2= 0.0030 shows that as soon as condition (6) is satisfied (Fig. 9: blue line below the red one), the overall system responses are similar to single loop ones – Fig. 11; performance indicators are still satisfactory for low frequencies (Fig. 10). stability condition: norm(T...)
M0 norm(T...)

2.5

Acknowledgment The work has been supported by the Slovak Scientific Grant Agency, Grant No. 1/1241/12 and by Slovak Research and Development Agency, Grant APVV-0211-10.

2 1.5

REFERENCES

1

Hovd, M. and Skogestad, S. (1992). Simple Frequencydependent Tools for Control System Analysis, Structure Selection and Design. Automatica, 28, no.5, pp. 989-996. Johansson, K.H. (2000). The Quadruple-Tank Process: A Multivariable Laboratory Process with an Adjustable Zero. IEEE Transactions on Control Systems Technology, Vol. 8, No. 3, pp.456-465. Johansson, K. H., Horch, A., Wijk, O. and Hansson, A. (1999). Teaching Multivariable Control Using the Quadruple-Tank Process. In: Proc. 38nd IEEE CDC, Phoenix, AZ. Rosinová, D. and Kozáková, A. (2009). Robust decentralized PID controller design: a case study. In: IEEE Sankt Peterburg, Russia Rosinová, D. and Markech, M. (2008). Robust Control of Quadruple – Tank process. ICIC Express Letters, 2, (No. 3), pp.231-238. Schmidt, H. (2002): Model based design of decentralized control configurations. Licentiate Thesis, Royal Institute of Technology, Stockholm, Sweden Skogestad, S. and Postlethwaite, I. (2009). Multivariable feedback control: analysis and design. John Wiley & Sons Ltd., Chichester, West Sussex, UK. Vincel, J. (2009). Setting a Mechanical System Computer Model Using Matlab/SimMechanics. Diploma thesis, Slovak University of Technology, FEI, Bratislava, Slovakia.

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Fig.9. Robust stability condition for decentralized control 2 performance bound through PRGA11 100

performance bound through PRGA12 100

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-4

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performance bound through PRGA21 100

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performance bound through PRGA22 100

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-4

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Fig.10. Bounds on weighting functions for reference value

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