An algorithm for design of deadbeat controllersof spatially invariant systems

Petr AugustaInstitute of Information Theory and Automation

Academy of Sciences of the Czech RepublicPrague, Czech Republic

Email: augusta@utia.cas.cz

Krzysztof GakowskiInstitute of Control and Computation Engineering

University of Zielona GoraZielona Gora, Poland

Email: k.galkowski@issi.uz.zgora.pl

AbstractIn the paper we deal with control of spatiallyinvariant systems described by parabolic partial differentialequations with one temporal and one spatial variables. Wepropose an algorithm for design of deadbeat controllers. Thealgorithm is based on polynomial methods. It consists in solvinga linear equation with bi-variate polynomials. Numerical exampleand simulations are included.

I. INTRODUCTIONWe shall concentrate on linear time-invariant spatially in-

variant systems. In particular, in the paper we consider systemsdescribed by parabolic partial differential equations (PDEs)with constant coefficients with one temporal variable and onespatial variable. The original PDE can be transformed to thedescription in the form of state-space equations or transferfunction. Then, the coefficients in the state-space matricesand the coefficients in the transfer functions are elements ofa ring. Hence, these systems can be studied within a classof linear systems over rings. Papers [1][5] belongs to thepioneer publications in this area. Comprehensive methodologyfor analysis and synthesis of spatially distributed systems isgiven in [6]. After several years of pause, papers, e. g., [7][9] found the topics on systems over rings being interesting.These papers propose new applications of spatially distributedsystems. The control action is based on an array of actuatorsand sensors. Spatial symmetry of systems is assumed. Theso-called spatially localised controllers are defined based ononly a finite number of error and actuator signals at nearbyactuator cells. A related fields are repetitive systems [10] andthe iterative learning control (ILC). Use of ILC for control ofspatio-temporal dynamics is demonstrated in [11], [12].

In the case of one temporal and one spatial variables, theparabolic PDE has the form

u

t

2u

x2 u

x u = f, (1)

where , , , are positive constants, u(t, x) is a solutionand f(t, x) is the right-hand side. A particular example of (1)is a heat equation

u

t

2u

x2= f, (2)

where u denotes a solution output, temperature (C), f theinput heat (C s1), t is the time (s), x is the spatial coordinate

(m) and is a constant (m2 s1) equal to

= cp

,

where is the thermal conductivity (W m1 K1), isthe density (kg m3) and cp is the heat capacity per unitmass (J K1 kg1). Initial and boundary conditions can beconsidered in the form u(0, x), u(t, 0), u(t, L) with L denotingdimension in the x direction of area, where the heat isconducted. A heat equation has applications in various fields ofthe industry. One of the examples is a multizone crystal growthfurnace, described in [13], [14]. The furnace is depicted inFig. 1. Block diagram of the closed-loop system is in Fig. 2.The problem consists in control of N zone temperatures byN energy inputs. The aim is to produce the temperatureprofile within the furnace, prescribed by N command signalsr1, r2, . . . , rN .

This paper proposes an algorithm for the design of deadbeatcontroller of a spatially invariant system. The goal of thedeadbeat control is to bring the system output to the steadystate in the smallest number of time steps. It supposes thedigital implementation of control and design method basedon the discrete-time model. Furthermore, the control action

Fig. 1. A sketch of a multizone crystal growth furnace

zone zone zone zone zone

controller controller controller controller controller

. . .1 2 3 N 1 N

r1 r2 r2 rN1 rN

Fig. 2. Block diagram of the closed-loop system.

22978-1-4799-5081-2/14/$31.00 2014 IEEE

applied by an array of actuators and sensors naturally callsfor the model discrete in the space. So, the model discretein both time and space must be derived. Modelling as well ascontrol design and simulations will be demonstrated by meansof the example of a heat conduction in a rod. This systems isdescribed by the heat equation (2) and its structure is depictedin Fig. 3.

The paper is organised as follows. The next section dealswith modelling, including the discretisation, the z-transformand deriving the transfer function. An algorithm for thedeadbeat control design is proposed in Sec. III. Sec. IV showsthe simulations of a heat conduction in a rod controlled by thedeadbeat controller. The Sec. V concludes the paper.

II. MODELLING

The model of a spatially invariant system will be derivedin the form of the transfer function. The derivation consists inthree steps, Crank-Nicolson discretisation, von Neumanns analysis of stability, the z-transform and manipulations.

A. Crank-Nicolson discretisation

The discretisation of the original PDE uses well-knownfinite difference methods, described in, for instance, [16].Roughly speaking, the principle consists in replacing deriva-tives by differences applying so-called difference scheme. Thisresults in recurrent equation in the case of parabolic PDE. Therecurrent equation can be solved or simulated by the iterativemethod. The difference scheme can be explicit or implicit [16].The explicit scheme is easier for numerical simulations, butit is conditionally stable. The condition to be satisfied is arelation between sampling time and sampling space periods.The implicit scheme results in unconditionally numericallystable approximation of system dynamics, i. e., both spaceand time sampling periods can be chosen arbitrarily. However,more than one unknown values of the solution have to becomputed at new time level. Moreover, all values on newlevel must be obtained simultaneously, what leads to solving

Fig. 3. [15] Distributed control of a distributed parameter system: a rod withan array of heaters and temperature sensors and a distributed controller (anarray of controllers)

a system of equations at each time level. This can be doneusing the numerical iterative methods proposed by [17].

Since we consider high values of the sampling time periodfor the purpose of the deadbeat control, we prefer an uncon-ditionally stable approximation by the implicit schemes. Oneof them is Crank-Nicolson discretisation. It was derived byCrank and Nicolson [18]. It replaces time derivative by

u

t uk+1,l uk,l

T(3)

and second order space derivative by

u2

x2 uk,l+1 2uk,l + uk,l1

2h2+

+uk+1,l+1 2uk+1,l + uk+1,l1

2h2,

(4)

where T is sampling time period (s) and h is sampling spaceperiod (m). In Fig. 4 we show how the stable region of s-plane (the left-half plane) is mapped to z-plane under theabove relation (3). We can see that all stable region of s-planemapped into z-plane does not lie within the stable region ofz-plane. In other words, stable continuous-time system can betransformed to unstable discrete-time system.

Discretisation consists in substituting (3) and (4) into theoriginal PDE. This results in a partial recurrent equation. Inthe case of (2) the partial recurrent equation reads

uk+1,l uk,lT

= uk,l+1 2uk,l + uk,l1

2h2+

+uk+1,l+1 2uk+1,l + uk+1,l1

2h2+ fk,l.

(5)

In Fig. 5 we show the computation mask associated with (5).Points needed for computation are plotted. Input mask ismarked by red points, the output values currently beingcomputed are marked by the white colour.

B. Von Neumanns analysis of stability

Von Neumanns analysis of the numerical stability, de-scribed in, e. g., [19], is a standard tool to analyse the con-vergence of discretised models. Despite the fact that (5) was

Re

Im

1 1

1

1

Fig. 4. An image of the stable region of s-plane under Crank-Nicolsondiscretisation (3)

23

k 1

k

k + 1

k + 2

l

2

l

1 l

l+

1

l+

2 x

t

Fig. 5. The computation mask associated with (5). Input mask is markedby red points, the output values currently being computed are marked by thewhite colour.

obtained using implicit difference scheme and is uncondition-ally stable, we analyse its convergence in this section. Theanalysis gives a relation between T and h, whose satisfactionguarantees the convergence of the discrete model to theoriginal one. In the case of (5), the analysis should resultin a condition, which is always satisfied. To proceed vonNeumanns analysis we consider the zero right-hand side, (5)becomes

uk+1,l uk,lT

=uk,l+1 2uk,l + uk,l1

2h2+

+uk+1,l+1 2uk+1,l + uk+1,l1

2h2.

Now, we substitute gk ej l for uk,l. The difference scheme isnumerically stable if and only if |g| 1. We havegk+1 ej l gk ej l

T=

2h2gk(ej(l+1) 2ej l + ej(l1))

+

2h2gk+1

(ej(l+1) 2ej l + ej(l1)) .

Manipulation and using Eulers relations leads to

g =h2 + T (cos 1)h2 T (cos 1) .

One can make sure that |g| 1 is always satisfied.

C. Transfer function

The transfer function of a system will be derived in the form

P (d,w) =b(d,w)

a(d,w), (6)

i. e., in the form of a fraction of two bi-variate polynomialsb and a. We have assumed the system is spatially invariant,that is the parameters of the system do not depend on loca-tion. Spatial invariance assumes the infinite domain, i. e., theboundaries at infinite distance. This assumption is not realistic,but it leads to a reasonable simplification of the control design.Simulations and verification of controllers should follow thedesign procedure.

With the above assumption, we can define the z-transformas follows. Let fk,l be a real-valued function and fk,l = 0 forall k < 0. The z-transform of fk,l is given as

F (d,w) =+k=0

+l=

fk,l wl dk. (7)

We consider u to be output, and f to be input in (6).Performing the z-transform (7) on the recurrence equations (5)and manipulating we obtain (6) with

b(d,w)=T d

a(d,w)= 1 T2h2

(w 2 + w1)

d T2h2

d(w 2 + w1) .

(8)

III. CONTROL DESIGN

The aim of this section is to design a controller which1) guarantees that output yk tracks a step reference rk,2) drives the control error ek to zero in a least number of

steps.For the deadbeat control we consider the scheme of Fig. 6. Aplant is given by the transfer function (6). Let the controllerbe series connection of two parts. The first part is fixed. Itguarantees the above requirement no. 1 and has the transferfunction

CI(d,w) =1

1 d . (9)

The second part of the controller is subject of design and hasthe transfer function

C(d,w) =n(d,w)

m(d,w), (10)

where m and n are bi-variate polynomials. The design proce-dure must take into account the above requirement no. 2. Itsdescription follows.

The number of steps driving the control error to zero iscommensurable to the degree of transfer function from inputreference rk to control error ek. It follows from the schemeof Fig. 6 that this transfer function reads

e =1

1 +1

1 db

a

n

m

r =(1 d) am

(1 d) am+ b n r. (11)

Since we supposse the reference to be step, i. e.,

r(d,w) =1

1 d r(w),

rk ek vk

yk

C= nm1

1d P=ba

controller

Fig. 6. Control scheme, rk input reference, ek control error, vk controlaction, yk output of the plant

24

the control error (11) becomes

e =am

(1 d) am+ b n r. (12)

To guarantee the driving ek to zero in a least number of steps,(12) has to be a polynomial in the variable d. In other words,the denominator of (12) has to be equal to a constant s in thering R[w][d],

(1 d) am+ b n = s(w). (13)Now, the control error becomes

e = amr

s.

A crucial step of the controller design is to find a solutionto the bi-variate polynomial equation (13). It follows fromwell-known Hilberts nullstellensatz, see e. g. [20], that (13)is solvable if and only if (1 d) a and b vanish implies svanishes.

It is well known, see, e. g., [21], that if (13) has a solution,it has infinitely many solutions. To minimise the number ofsteps driving the control error to zero, a solution minimisinga degree in the variable d of m must be found. It was shownin [22] that the algorithms for the uni-variate polynomialequations can be applied to solve bi-variate polynomial equa-tions. In our case we consider polynomials a, b,m, n, s inthe variables d and w to be uni-variate polynomials in thevariable d with the non-constant coefficients being functionsof the variable w. Algorithms for the uni-variate polynomialequations minimising the degree of m can be found in,e. g., [23], [24]. The controller design can be summarised asfollows.

Algorithm 1 (Deadbeat controller): Input: polynomials a, bdescribing the plant. Output: polynomials m, n describing thecontroller

1) Choose s that (1 d) am+ b n = s is solvable.2) Solve (1 d) am+ b n = s for m,n. Take the solution

minimising degree of m.

The deadbeat controller is then given by substitution of m andn into (9) and (10).

As a numerical example we consider a system of a heatconduction in a rod described by (6) with polynomials a andb given by (8). Let the polynomial s be

s(w) = 2h2 K T (w 2 + w1).The polynomial equation (13) is solvable and the solutionminimising the degree of m is

m= 2h2

n=4K

h2 6T(2K

h2+ 3T

)(w + w1

)(4K

h2 2T

)d+

(2K

h2 1T

)d(w + w1

).

(14)

IV. SIMULATIONS

The above described concept is demonstrated by means ofa numerical example in this section. We consider the systemof a heat conduction in a rod described by (2). For the

00.5

11.5

2

x 105

02

4

68

1020

20.2

20.4

20.6

20.8

21

t (s)x (node)

refe

renc

e (C

)Fig. 7. Input reference rk

00.5

11.5

2

x 105

02

46

810

1219.6

19.8

20

20.2

20.4

20.6

20.8

21

21.2

21.4

t (s)x (node)

tem

pera

ture

(C)

Fig. 8. Output of the plant yk

00.5

11.5

2

x 105

02

4

68

101

0.5

0

0.5

1

t (s)x (node)

e (

C)

Fig. 9. Control error ek

25

00.5

11.5

2

x 105

02

4

68

100.02

0.015

0.01

0.005

0

0.005

0.01

0.015

0.02

t (s)x (node)

con

trol a

ctio

n

Fig. 10. Control action vk

purpose of the numerical simulation, we discretise (2) in thespatial variable only using (4). The time variable is continuous.We consider the controller given by the series connection oftransfer functions (9) and (10) with polynomials m and nof (14). The controller is discrete in time and space and isconnected with the plant via zero-order hold block.

We choose = 230, = 2700, cp = 900 and h = 1/12.Number of nodes is 10. Zero boundary and initial conditionsare considered. Let the sampling time period be T = 3600 s,that is one hour.

The closed-loop system is stable even though the value ofthe sampling time period T is large. The input reference andoutput of the plant are plotted in Figs. 7 and 8, respectively.One can see that output tracks the input reference. The controlerror goes to zero in a small number of steps, see Fig. 9.The control action is plotted in Fig. 10. It has a very smallamplitude, however, it operates during a long time period, so,small amplitude does not imply low overall input energy.

V. CONCLUSIONS

The algorithm for the deadbeat control design for spatiallyinvariant systems was proposed in the paper. The designmethod is based on solving a linear equation with bi-variatepolynomials. The design was illustrated by mean of numericalexamples and simulations. A heat conduction in a rod was con-trolled by the deadbeat controller. Systems with one temporaland one spatial variable were considered in the paper, however,the proposed algorithm can easily be extended to more thanone spatial variable.

ACKNOWLEDGMENT

The work of the first author was financially supported by theCzech Science Foundation under the project GPP103/12/P494.

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