Multivariable DELTA (δ) Operator Control in a Plant Physiology Experiment

Copyright © IFAC Mathematical and Control Applications in Agriculture and Horticulture. Hannover. Germany. 1997

MULTIVARIABLE DELTA (0) OPERATOR CONTROL IN A PLANT PHYSIOLOGY EXPERIMENT

P.G. McKenna, A. Jarvis, A. Chotai & P.C. Young

Centre for Research on Environmental Systems and Statistics (CRES), Institute of Environmental and Natural Sciences (lENS), Lancaster University, Lancaster, LAI 4YQ, UK.

Abstract: This paper describes the use of the True Digital Control (TDC) approach (Young, et al., 1987) in the design, simulation and implementation of ProportionalIntegral-Plus (PIP) control of the temperature and relative humidity within an enclosed plant physiology cabinet. Control of the temperature and relative humidity was originally carried out using separate single-input single-output (SISO) controllers running simultaneously (McKenna, et al., 1997). Due to the high degree of cross coupling which exists between these two channels it is more appropriate to use an optimal linear quadratic (LQ) design philosophy based upon the estimation of a multi-input multi-output (MIMO) model. Keywords: closed loop-control, discrete-time systems, state space, MIMO, optimal control. temperature control.

design through closed loop pole assignment or optimal linear quadratic (LQ) control. the theory is also easily extended to deal with multi-input multioutput (MIMO) systems (Chotai, et at .. 1997) and stochastic systems (Taylor, et al., 1996) .

I. INTRODUCTION The concept of True Digital Control (TDC) (Young, et aI., 1987) has been developed to incorporate all aspects of the control design process from model identification and estimation, through control design and simulation to implementation upon real systems. The TDC theory was initially formulated in terms of the backward shift or Z-I operator in order to fully utilise the power of digital computing, in this form the control design is based upon the definition of a non-minimal state space (NMSS) description of the system, in which the state consists of present and past values of the sampled input and output signals, all directly measured. A state variable feedback (SVF) control law is then developed; by the inclusion of an integral of error state in the NMSS description, the resulting Proportional-Integral-Plus or PIP control law exhibits type I servo-mechanism performance. The SVF nature of PIP control enables

In recent years the concept of the discrete differential or delta (0) operator has been revived by Goodwin. et aI., (1988) to bridge the gap between traditional control theory expressed in terms of the continuous time derivative or s operator and modern control design implemented in the backward shift operator. The PIP control design philosophy has been extended from backward shift to a delta operator NMSS representation (Young. et al.. 1991) based upon the definition of a state vector where the states are the input and output signals together with their discrete derivatives and an integral of error term. the control law is then implemented in terms or realisable delta operator filters. To date few

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applications of delta operator control have been produced. In this paper, the design a multi variable delta operator PIP controller, using a new filter form (McKenna, 1997) of the original delta operator state space (Young, et al., 1991) within a plant physiology experiment is presented.

and it is assumed that the row degrees of A(o) are is the delta operator which, for a equal. Here, sampling interval of /).t time units, is defined as follows in terms of the forward shift operator ;,.

°

o=z-\ i.e. &(k)

x(k+l)-x(k):

/).t

A series of experiments, being carried out within a cabinet, aim to explore the mechanisms by which plants respond to changes in the external environment. Plants vary the resistance to the gaseous diffusion of water vapour and carbon dioxide across the leaf surface by altering the aperture of stomatal pores such that the amount of water vapour lost from the leaf is minimised and the amount of carbon dioxide gained is maximised. Predicting how an individual or community may alter gas exchange in response to changes in the above and below ground environment has become increasingly important in the light of concerns over global climate change.

while A i and Bi (i=I.2, ... ,n) are matrices.

°

x(k)=

[0 n·/ y' (k) .... ,y' (k),o n·/u ' (k) , ···, u ' (k) . z( k)

r (4)

In these equations, u'(k) and y ' (k) are the input and output signals filtered by the polynomial t(O) . such that, u'(k)= u(k) and y ' (k)= y(k) t (o) t(o )

(5)

where t(o) is a scalar nth order polynomial . chosen as a part of the control design and takes the general form, t(o) =0 n +tlO ,,-I + .. .+t"

(6)

and z(k) is the vector of integral of error states. which is defined as follows in terms of the inverse delta operator (0-1), or digital integrator . i.e_,

2. TIlE DELTA OPERATOR NMSS MODEL

(7)

In order to demonstrate the general NMSS approach to control system design, consider the following rinput. p-output, 0 operator system represented in terms of the left matrix fraction description (LMFD) form (e.g. Kailath, 1980),

in which yAk) is the reference or command input vector to the servomechanism system. Here. for Simplicity and brevity, but without any lack of generality. we let p=r and the NMSS representation is then defined directly in terms of the following 8 operator discrete-time state equations,

(1)

or, y(k)=(A(o

pXp and pXr

The most obvious delta operator Non-minimal State Space (NMSS) representation obtainable from this LMFD model is formulated in terms of the di scretetime derivatives of the output and input signals. This form (Chotai, et al.. 1997) has become the standard delta operator NMSS , however in thi s paper we choose to present a new form in which the states themselves are filtered by a delta operator polynomial, t(o),the characteristics of which are selected as part of the control design. In thi s filter form delta operator NMSS representation, the nonminimal state vector, x(k) is defined as ,

The mc philosophy has been used in this experiment to fulfil the design objective of Controlling the temperature and relative humidity within the cabinet by modulation of the signals applied to a variety of actuators. In earlier work (McKenna, et al., 1997) the approach taken to simultaneously control the temperature and relative humidity within the cabinet was to carry out the modelling by collecting open loop responses to step changes in the relevant inputs to each channel independently. The design of single-input singleoutput (SISO) PIP controllers for each channel was completed and tested by implementing each separately, this produced good results. However, there is a high degree of cross coupling between the temperature and relative humidity, therefore in order to improve the outputs when responding to simultaneous changes in the demand signals for both channels, a design based upon a mUlti-input multioutput (MIMO) model would be favourable.

A(o )y(k) =B(o )u(k)

(3 )

/).t

)r B(o )u(k) l

ox(k) =Fx(k)+Gu(k)+Dy " (k)

Where, A(o)=Io " +A,on-'+.. .+An B(o )=B,on-' +B20 n-1 + ...+Bn

y(k)=Hx(k)

(2) where the matrices F. G, D and H are defined in (9) .

20

-AI

-An

B,

B2

' B n_,

Bn

0 0

0 0

0 0

0 0

0 0

0 0

0

0

0

0

-~

-T2

-Tn_,

-Tn

Ir 0

0 I,

0 0

0 0

0

0

I,

0

-B,

-B2

-Bn_,

-Bn

B,

B2

-An_,

-A2

Ir

0

0

Ir

0 0 0 0

0 0 0 0

Ir 0 0 0

0 0 0 0

0

0

0

0

F=

A , -T, H =[~ -AI

A n-l -T"-1

A 2-T2 T2 -A2

An-Tn

Tn_, -An_,

Tn-An

In the above formulation, the block matrices Ir denote rXr identity matrices; and the matrices T; (i=1 ,2, ... ,n), are rXr diagonal matrices with identical diagonal entries, defined such that, t(O)

o T(O)=I,t(O)=: [ o o

0

...

t(O) .. · : ".

0

.. .

0

.. .

o

~1

B n_,

0 0 0

0 0 0

0 0 0 0

U

0 0

0

0

G= Ir

D= 0

0 0

0 0

0 0

0

0 0

I,

0]

Bn

(9)

is the SVF control gain matrix, in which L,. M , (i=1,2, ... ,n), and K, are rXr matrices. The negative sign associated with K, is introduced simply to allow the integral feedback term to take on the form normally used in more conventional PI control system design . The presence of the filter polynomial t(O) ensures that SVF control law can be implemented straightforwardly in the 0 operator domain and the resulting closed loop system block diagram takes the multi variable PIP form shown in Fig. I .

(l0)

t(O)

where,

and t(O) is defined in equation (6). The t(O) polynomial is introduced as a filter which provides a realisable form of the NMSS control law and also gives an extra degree of design freedom which can be selected by the designer to avoid the explicit design of either a state reconstruction filter (observer), or its stochastic equivalent, an optimal state estimator (Kalman filter). The type-l servomechanism requirement is automatically accommodated in (8) by the introduction of the multi variable integral of error vector, z(k) . Consequently, provided the closed-loop system is stable, then steady state decoupling is inherent in the design.

,

I

+

2

+...

L ) 1/

•

M ' -__I_(MIUs:. n-i + M ,Us:. n-2 +...M) , r(o)

-

,

Fig. 1. The multivariable filter form 0 operator PIP servomechanism system When presented in this form , the PIP controller can clearly be related structurally to traditional designs. such as multi variable PI controller. However. because it exploits fully the power of SVF within the NMSS setting, it is inherently much more tlexible and sophisticated, allowing for well known SVF control strategies such as closed loop pole assignment, with complete (or partial) decoupling control, and multivariable LQ optimal control. In addition, it has recently been shown (Tych. et af. 1996) how this formulation can form the basis for multi-objective optimal LQ control system design. where the off-diagonal elements of the optimal weighting matrix in the LQ cost function are exploited to allow more easily for multi-objective goal attainment (Chotai , et al., 1997), The NMSS

3. THE SVF PIP CONTROLLER In the usual manner, the State Variable Feedback (SVF) control law associated with the NMSS model (8) is defined in terms of the state variables and a matrix of control gains, i.e., u(k)=-Kx(k) =-L,On-Iy' (k)- .. .-Lny· (k)

1 (L Us:. n-' L Us:. n-2

L = r(o)

(12)

-M,On-'U' (k)- ... -Mnu· (k)+K,z(k) where,

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considered here since it has been successfully controlled using a SISO PIP design, as shown in an earlier publication (McKenna, et al., 1997).

formulation also facilitates consideration of other optimal control strategies based on state space concepts, such as the Linear-Exponential-ofQuadratic (LEQG) control with risk averse or risk preferring designs proposed by Whittle (1991); and

5. MODEL ESTIMATION

the related robust Hoo designs that have received so much attention in recent years (e.g. Doyle, et ai., 1989). Initial research on LEQG-PIP control in the backward shift {' operator case has shown how this approach to control system design compares with the more conventional LQ and LQG methods (Taylor, et al., 1996) and it is clear that this same approach can be extended to 0 operator models.

A mUlti-input single-output (MISO) delta operator transfer function (TF) model was identified for each of the two variables within the cabinet using the Simplified Refined Instrumental Variable (SRIV) approach to system identification and estimation (Young, et al., 1991). Output datasets were collected in response to step changes in the inputs to each of the actuators as shown in Fig. 3.

4. TIIE PLANT PHYSIOLOGY STUDY SYSTEM

The best identified models are selected on the basis of a compromise between model fit and parametric efficiency, using the coefficient of determination (Ri) and the YIC identification statistic (see e.g .. Young, 1989). The resulting 2-input 2-output model is shown in equation (14), where Yr(k) is the temperature in the cabinet, y,,,(k) is the relative humidity, u,(k) is the input to the heater and lI ,.,,( k ) is the input to the spray er, both input signals being in terms of the length of the pulse of heat or spray applied every 10 seconds. The sampling interval (Ill) is 10 seconds, expressed in minutes (i.e. '/6 minutes) . The TF model coefficients are expressed in the terms in which they were identified, these numbers relate to the outputs from the AID card rather than units with physical meaning. The real temperature and relative humidity have been obtained by using calibration data to produce the accurately scaled graphs shown in Fig. 3.

A 40 litre perspex cabinet, shown schematically in Fig. 2, has been constructed in the Biological Sciences department of Lancaster University specifically to carry out a number of experiments in plant physiology.

Cold waler radiator

Tcmper~urc .

hwnidity&

co! ,,'fIlcc:nlCation .'iOlsocpnlbe

Fig. 2. Schematic representation physiology study cabinet.

of

the

plant y( k)

The cabinet is a closed system in which the air circulates through a duct, being drawn out at the base and reintroduced at the top, sensors are located in the middle of the cabinet at plant height. On leaving the cabinet, the air in the duct is first cooled by passing through the fins of a radiator filled with cold running water, this produces a flow of air through the duct with a constant temperature and vapour pressure. The air is then heated by passing over a number of electric heater filaments, these act as the actuator for the temperature channel with the control actuated by pulse width modulation. The control law determines the length of the time power is supplied to the heater every ten seconds, the sensor readings are also recorded every ten seconds. Immediately before the air is reintroduced to the cabinet, carbon dioxide is introduced from a bottle of I % CO 2, and water droplets are sprayed into the air flow, these are the control inputs for the carbon dioxide concentration and humidity and are also pulse width modulated. The CO 2 channel is not

= [

17.77 0+0.1317 31.210 - 26.21

-4.08} 0+0.1317 136.950 +24.31

(k )

0 2 + 13770 +0.0802 0 2+ 1.3770 +0.0802 ( 14)

where,

Y(k)=[Y r( k)] and U(k)=[Ur( k)] )',h (k) U rr ,( k)

( 15 )

This matrix transfer function (MTF) model can then be expressed in the LMFD (I) form where.

°

1 01- 2 [0.1317 0 1 [0 0 ] J + 1.377[+ 0 0.0802

A(o)= [ 0 I

17.77 -4.081 [ 0

°]

R(o)= [ 31.21 136.95[+ -26.21 24.31

(16)

The model has a very good fit to the data. as shown in Fig. 3, where Ri=O.987 for the temperature channel and 0.980 for the relative humidity . i.e. the model explains 98.7% and 98 .0* of these data respectively.

22

TeIl1'erature

6. PIP CONTROL DESIGN

~

28

uo

.

~

The delta operator NMSS representation of the MIMO model (14) is generated by placing the matrices in the LMFD form (16) into the state matrices defined in (9), whilst the T-polynomial is chosen heuristically as part of the control design procedure.

•

26

"

o

. _'

10

20

30

Relative humidity

0 en

2

5u

I

-0

20

10

This state space form is used to calculate the LQ optimal control gains (see for example, Chotai , et al., 1997) with the assumed diagonal NMSS state weighting matrix Q having diagonal elements [1,1 ,1,1,1 ,1,1 ,1,0.01 ,0.1] and the input weighting matrix R set to a 2x2 identity matrix. the Tpolynomial is chosen as,

30

Heater pulse width

~ 0

( 17) 0

en

2

5u

I

-0

10

20

30

The 0.01 and 0.1 elements in Q relate to the integral of error states and are selected to achieve the desired speed of response whilst generating pulse width inputs within the bounds defined by the 10 second sampling interval. This produces the following control gain matrices, as defined in the SVF control law, (12).

Spray er pulse width

~ 0 I--

o

'---

'--

10

20 30 Tirre / minutes Fig. 3. The delta operator MIMO modelling results: open loop output data shown as points, model output and the two inputs shown as lines.

1.2352

L\ = [ -0.2799

0.0500J L =[0.9931 0.0626] 0.2132 ' 2 -0.2250 0.2675 .

4.7420 0.7994] [-4.5207 0.4846] M\ = [ 0.5538 7.9808 ' M 2 = -2.1785 -1.9728 '

The cross-coupling evident in the data shown in Fig. 3 is more complex than originally expected. The introduction of heat to the system will inevitably lead to a fall in relative humidity, however on viewing the response to the heat input in the figure above it is clear that the immediate effect is to cause a rise in relative humidity, this seems counter-intuitive but can be explained by considering the mechanics of the actuator involved.

0.1957 K[ = [ -0.0443

0.0144J 0.0612 .

This control design is then implementable in a number of forms , one of which is shown in Fig. I.

7. IMPLEMENTATION

As shown in Fig. 2, the air is first heated in the duct outside the cabinet and a water mist in injected into the flow immediately prior to re-entry . If the pulse of water mist being injected into the flow is assumed constant, then the absolute humidity of the air entering the cabinet will be governed by the temperature of that air, thus an increase in temperature immediately leads to a corresponding increase in the efficiency of the relative humidity actuator. After a short while this effect becomes cancelled out as the air in the cabinet becomes well mixed and uniformly warmer and the sprayer efficiency stabilizes such that the relative humidity is seen to start falling . This relationship is evident in the model through the non-minimum phase character of transfer function between Yrh(k) and u,(k), the bottom left transfer function in (14).

The delta operator PIP controller was designed as described above and then simulated uSing SIMULINKTM. The control laws were then generated in their incremental forms for implementation upon the real system. The results of simulating the closed loop responses to step changes in the demand level for each of the output signals are shown in Fig. 4. The simulation incorporates the addition of output noise to both the temperature and relative humidity signals, the magnitudes of which have been chosen to approximate those observed during the collection of the open loop data shown in Fig. 3. In addition to the inclusion of noise in the simulation, realistic bounds have been place upon the control inputs such that they may not be negative or greater than the ten second sampling interval.

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9. ACKNOWLEDGEMENTS

26 U 0

The authors wish to express their thanks to Lancaster University, the UK Engineering and Physical Sciences Research Council (EPSRC) and the Natural Environment Research Council (NERC) for supporting the research carried out in this paper.

25 0

5

10

15

20

25

30

80 ~

75

REFERENCES

70 10 15 0 5 3 Heater pulse width

20

25

30

20

25

30

Chotai, A., Young, P.e., McKenna, P.G ., and W . Tych (1997). Proportional-Integral-Plus (PIP) design for delta (0) operator systems : Part 2. MIMO systems. Submittedfor publication. Doyle, J.e., Glover, K., Khargoneker, P.P., and BA Francis (1989). State space solutions to standard H2 and H~ control problems. IEEE Trans. on Auto. Control , AC-34, pp. 831-847 . Goodwin, G.e. , R.H. Middleton and M . Salgado (1988). A unified approach to adaptive control. In: Implementation of self tuning controllers (K. Warwick (Ed.» , pp. 126-139. Peter Perigrinus, London. Kailath, T. (1980). Linear Systems , Englewood Cliffs, N.J.: Prentice Hall. McKenna, P.G. (1997). Delta operator modelling. forecasting and control, Ph.D. Thesis. Lancaster U ni versi ty . McKenna, P.G., Jarvis , A. , Chotai , A. and P.e. Young (1997). Delta operator PIP control of a plant physiology experiment, to appear. Proc. Third IFAC Symposium on Modelling and Control in Biomedical Systems (Including Biological Systems), Warwick. Taylor, C.J., Young, P.e. and A. Chotai (1996) . PIP optimal control with a risk sensitive criterion . lEE Conference Publication no. 427. UKACC International Conference CONTROL' 96 . pp.959-964. Whittle, P. (1991). Likelihood and cost as path integrals, 1nl Royal Stat. Soc., Series B (Methodological) , 53, pp. 505-538 . Young, P.e. , M.A. Behzadi, e.L. Wang and A. Chotai (1987). Direct digital control by inputout, state variable feedback pole ass ignment. Int. 1nl. of Control, 46, pp. 1867-1881 . Young, P.C. (1989). Recursive estimation . forecasting and adaptive control. In: Control and Dynamic Systems (e.T.Leondes (Ed.». pp. 119-166. Academic Press, San Diego. Young, P.C., A. Chotai and W. Tych ( 1991). Identification, estimation and control of continuous-time systems described by delta operator models. In : Identification of Continuous Time Systems (N .K. Sinha and G .P. Rao (Ed.» , pp. 363-41 8. Kluwer Academic Publishers, Dordrecht.

~2 c

81 a.l

'" 0 10 15 5 6 Spray er pulse width ~4 c

82 a.l

'" 0 15 20 25 30 Time I minutes Fig. 4. Implementation results for multi variable operator PIP controller 0

5

10

0

The graphs above show that the MIMO controller achieves an increase in temperature without any significant effect upon the relative humidity and similarly a step in the relative humidity causes little disturbance to the temperature output. The downward steps in demand produce responses which are delayed by the inability to apply non-realistic negative control inputs. Both demand levels are reached within the experimental requirements whilst remaining within the control input bounds.

8. CONCLUSIONS The use of a new filter form of the delta operator NMSS to design a multi variable PIP controller has been shown to produce steady-state decoupled control of the temperature and relative humidity within a 40 litre plant study cabinet. Through the use of techniques outlined in Chotai, et aI., (1997) the decoupling could be further improved. The delta operator design produces responses which are comparable with those obtained through a backward shift controller therefore even with coarsely sampled data the use 0 operator approach remains valid. Finally, since the 0 operator is the discrete time equivalent to the continuous time s operator, the procedures outlined in this paper can be readily adapted to continuous-time systems.

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