minimum effort dead-beat control of linear servomechanisms with ripple-free response

*Correspondence to: Jir\ mH Mos\ na, Department of Cybernetics, University of West Bohemia, UniverzitnmH 8, 306 14 Plzen\ ,Czech Republic.

�E-mail: [email protected]

Contract/grant sponsor: Ministry of Education of Czech Republic; contract/grant number: MSM 2352 00004

Received 15 May 2000Copyright � 2001 John Wiley & Sons, Ltd. Revised 16 July 2001

OPTIMAL CONTROL APPLICATIONS AND METHODSOptim. Control Appl. Meth. 2001; 22: 127}144 (DOI: 10.1002/oca.688)

Minimum e!ort dead-beat control of linear servomechanismswith ripple-free response

Jir\ mH Mos\ na,*�� Jir\ mH Melichar and Pavel Pes\ ek

Department of Cybernetics, University of West Bohemia, Univerzitnn& 8, 306 14 Plzen\ , Czech Republic

SUMMARY

A new and systematic approach to the problem of minimum e!ort ripple-free dead-beat (EFRFDB) controlof the step response of a linear servomechanism is presented. There is speci"ed a set of admissible discreteerror feedback controllers, complying with general conditions for the design of ripple-free dead-beat (RFDB)controllers, regardless of the introduced degree of freedom, de"ned as the number of steps exceeding theirminimum number. The solution is unique for the minimum number of steps, while their increase enables oneto make an optimal choice from a competitive set of controllers via their parametrization in a "nite-dimensional space. As an objective function, Chebyshev's norm of an arbitrarily chosen linear projection ofthe control variable was chosen. There has been elaborated a new, e$cient algorithm for all stable systems ofthe given class with an arbitrary degree of freedom. A parametrized solution in a "nite space of polynomialsis obtained through the solution of a standard problem of mathematical programming which simulta-neously yields the solution of a total position change maximization of servomechanism provided thata required number of steps and control e!ort limitation are given. A problem formulated in this way isconsecutively used in solving the time-optimal (minimum-step) control of a servomechanism to a givensteady-state position with a speci"ed limitation on control e!ort. The e!ect of EFRFDB control on theexample of a linear servomechanism with torsion spring shaft, with the criterions of control e!ort andcontrol di!erence e!ort, is illustrated and analysed. Copyright � 2001 John Wiley & Sons, Ltd.

KEY WORDS: dead-beat; optimal control; time-optimal control; mathematical programming

1. INTRODUCTION

Historically, the solution of the dead-beat control problem [1, 2] is connected with two types ofcontrollers the designs of which di!er according to the requirements posed on a steady-state errorbehaviour after a "nite settling time. If only the zero steady-state error at sampling time instants isrequired, the design leads to the dead-beat (DB) controller with disputable practical applicability[3], due to the often appearing error intersample ripple, analysed e.g. in References [4]. Necessaryand su$cient conditions for the ripple-free dead-beat (RFDB) control with a general type ofreference signal are derived for error feedback controllers e.g. in References [5, 6] and for statefeedback controllers in Reference [4]. The increased number of steps has introduced an addi-tional degree of freedom into dead-beat control problems, thus enabling the formulation of

optimization problems with objective functions usually evaluating error, control or outputsequences [7, 8]; however, a general solution of RFDB control was not obtained. The design ofthe DB controller with an optimal response to a more general type of reference signal is solved inReferences [9, 10]. As an optimality criterion, a weighed sum of squared response errors toprototype reference signals is chosen. The solution leads to a constrained static optimization. Thedesign of the energy-optimal RFDB controller for a linear servomechanism can be found inReference [11]. In Reference [12] the problem of the minimum e!ort riple-free dead-beat control(EFRFDB) is formulated. The measure of e!ort is de"ned by the maximum magnitude of thecontrol action that may be interpreted as Chebyshev's norm of the respective control segment.A starting point for the solution is the speci"cation of control sequence coe$cients which areexpressed as a function of impulse function coe$cients of the closed-loop transfer function andsimultaneously build the optimality criterion. The authors obtained by a step-by-step method ananalytic solution for the "rst-order plant with an arbitrary degree of freedom. A new approach tothe design of EFRFDB controllers, used in this paper, is focused on the generalization andalgorithmic e$ciency of the solution. The obtained results hold for a whole class of linearservosystems with an optional degree of freedom. The optimality criterion is generalized andconsidered as Chebyshev's norm of a speci"ed linear projection of the control segment realizinga desired position change. The speci"cation of the weighting matrix in the linear projection makesit possible to formulate and solve a wide variety of problems covering e.g. the minimum e!ortRFDB position control without a priori speci"ed limitation on the control signal or themaximization of a total position change of a servomechanism, provided that the control e!ort orother technologic limitations are speci"ed. Further, it can be shown that the solution to theproblem formulated in this way can also be used in solving the problem of the time-optimal(minimum-step) control with a control signal limitation. The approach consequently arises fromthe speci"cation of an admissible set of RFDB controllers and a competitive set of controllersfrom which the optimal controller is determined [13, 14]. The optimization is conditioned bya parametrization of a competitive set of controllers in a "nite-dimensional space of polynomials,that transforms the minimum-step control sequence into the control sequence corresponding tothe controller with a given degree of freedom. It is shown that the optimization over thecompetitive set can be transformed into a classic mathematical programming and this fact stressesthe e$ciency of a new, elaborated algorithm for the design of EFRFDB controllers.

2. STATEMENT OF THE PROBLEM

Consider a linear position servomechanism described by the transfer function

� (s)

u(s)"G(s)"

b (s)

s a (s)(1)

where � is the position of the servomechanism and u is input, a(s) and b (s) are coprimepolynomials with a (0)O0, b (0)O0 and �b)�a (�"degree). Suppose that the linear servo-mechanism in cascade with the zero-order hold is controlled with a sampling period ¹ at timeinstants t

�, k"0, 1,2 Our basic aim is to transfer the system from a steady-state starting

position w��

with a "nite number of control steps N to the given steady-state "nal position,

128 J. MOS[ NA, J. MELICHAR AND P. PES[ EK

Optim. Control Appl. Meth. 2001; 22: 127}144Copyright � 2001 John Wiley & Sons, Ltd.

de"ned by the relation

�(t)"w��

"const; t3Re and t*t�

(2)

Supposing that the sampling frequency is not in resonance with any of damped naturalfrequencies of the plant, the aim can be attained if and only if

� (t�)"w

��"const and u

�"0; k*N (3)

where u�is the value of the manipulated variable at time t

�. Expressing condition (2) in the

equivalent form (3) the dead-beat control (DB) can be transformed into the ripple-free dead-beatcontrol (RFDB) eliminating the intersample activity. Consider the discrete model of a servo-mechanism with the sampling period ¹ in the form

�(z)

u (z)"G

�(z)"

b�(z)

(z!1) a�(z)

(4)

where a�(z), b

�(z) are coprime polynomials and a

�(z) is monic. Since a (0)O0, b (0)O0 we get

a�(1)O0, b

�(1)O0. Generally, �b

�(z)"�a

�(z))�a (s) holds. In the case that the control fre-

quency is not in resonance with damped natural frequencies of the servosystem, this inequalitybecomes an equality. The resonance is indicated through a lower order of the transfer function ofthe discrete model (4) in view of transfer function (1).

Our goal is to design an error feedback controller ensuring the attainment of a desiredsteady-state (3) from an initial steady-state position

� (t��

)"w��

"const and u�"0 k(0 (5)

in the prescribed number of steps N in such a way that the row-vector control sequence

u"[u�, u

�,2, u

�,2, u

��] (6)

minimizes the control e!ort criterion (EFRFDB control)

J(u)" max��2��

�u�� (7)

or the control di!erence e!ort criterion (�EFRFDB control)

J (u)" max��2�

�u�!u

�� (8)

with boundary conditions u��

"u�"0.

MINIMUM EFFORT DEAD-BEAT CONTROL 129


Figure 1. Block diagram of the digital control loop.

Introducing a generalized criterion J (u)"�u ) Q��

as Chebyshev's norm of a linear projectionof the control segment u with a reasonable chosen weighting matrix Q of the dimension N�M, itis possible to express the value of the criterion J assigned to the control segment u as

J (u)" max��2�

�u ) q�� (9)

where q�denotes columns of matrix Q. Notice that the control e!ort criterion (7) can be viewed as

a special case of the generalized criterion with the identity weighting matrix of the dimensionN�N and the control di!erence e!ort criterion (8) as a special case with the appropriate structureof the weighting matrix of the dimension N�(N#1). Thus, a reasonable choice of the weightingmatrixQ enables one to respect various technologic limitations. It will be shown that the solutionof the minimum e!ort RFDB control of a servomechanism from an initial steady-state positionw��

to a required steady-state position w��

with the prescribed number of steps N simulta-neously yields also the solution of the total position change �w"w

��!w

��maximization,

attainable by the control sequence with the prescribed number of steps N and with the e!ortlimitation given by the inequality �u )Q�

�)m, where m is a constant. The step-by-step solution

of this maximization problem for di!erent number of steps gives the solution of the time-optimal(minimum-step) position control of a servomechanism respecting a control e!ort limitation.

3. ADMISSIBLE SET OF RFDB CONTROLLERS

In the "rst instance, the set of RFDB controllers that will take part in the optimization of thetransient response of a servomechanism will be determined. Consider the control-loop inFigure 1 with the transfer function of a controller

u(z)

e (z)"D(z)"

b�(z)

a�(z)

; �b�(z))�a

�(z) (10)

where e�"w

�!�

�is the position error of a servomechanism and w

�is a desired position at time

t�. Further, we suppose that a

�(z), b

�(z) are coprime polynomials and a

�(z) is monic. The

closed-loop transfer function has the form

� (z)

w (z)"F (z)"

b�(z) b

�(z)

a�(z)

(11)



where

a�(z)"(z!1) a

�(z) a

�(z)#b

�(z) b

�(z) (12)

The closed-loop system response to desired position changes of a servomechanism at time t�,

k*0, de"ned as �w�"w

�!w

��, can be described by transfer functions

e(z)

�w (z)"F

�(z)"

(z!1) a�(z) a

�(z)

a�(z)

(13)

u (z)

�w(z)"F

(z)"

F(z)

G�(z)

"

(z!1) a�(z) b

�(z)

a�(z)

(14)

The position of a servomechanism is transferred from an initial steady-state position w��

toa "nal steady-state position w

��in N steps after the change of desired value at time t

�if and only

if it holds that

F�(z)"

(z!1) b�(z)

z�and F

(z)"

(z!1) b(z)

z�(15)

where b�(z) and b

(z) are arbitrary polynomials with the degree N!1. It is well known that the

solution exists only for N*N�, whereN�"1#�a(s) and for N"N� this solution is unique. ForN'N� the solution is not unique and has N

"N!N� degrees of freedom. Solutions can be

parametrized in a space of polynomials � (z) of maximum degree Nand with the requirement

�(1)"1. To an arbitrarily chosen polynomial � (z) from this parametric space there exists just onecontroller with the transfer function D(z) given by the relation

b�(z)"�� (z) a

�(z) (16)

and, due to the cancellation of the polynomial a�(z) in (12)} (14), by the diophantine equation

(z!1) a�(z)#� � (z) b

�(z)"z� (17)

From the solutions a�(z), � of the diophantine equation we choose the unique solution

characterized by �"const and �a�(z)"N!1. Thus, the transfer function D�(z) of the mini-

mum-step RFDB controller is given by the solution b��, a�

�, �� of these polynomial equations for

�(z)"1.Now, the transfer functions (13) and (14) have the form

F�(z)"

(z!1) a�(z)

z�and F

(z)"

(z!1) b�(z)

z�(18)

Due to the cancellation of the polynomial a�(z) in closed-loop transfer functions, the corres-

ponding modes do not in#uence transient response; however, it will not hold for non-zero initial



conditions. Further, we conclude that the closed-loop system is stable if and only if thepolynomial a(s) in the transfer function of the servomechanism G(s) is stable.

4. PARAMETRIZATION OF RFDB CONTROLLERS

In this section, the procedure for the choice of the optimal controller from the set of allcompetitive controllers, de"ned in the previous section, will be determined. Note that the transferfunction for the control sequence response F

(z) can be expressed through the transfer function

for the minimum-step control sequence F�(z) by the utilization of the parametric space of

polynomials � (z). After the substitution of (16) and (17) into (18), the transfer function for thecontrol sequence response F

(z) has the form

F(z)"

� (z)z�

(z!1) a�(z)

z�� "

� (z)z�

F�(z) (19)

Relation (19) enables one to obtain the control sequence response for an arbitrary controllerwith a "nite number of steps as a weighted linear combination of time-shifted responses of F�

(z).

Computations, pertinent to the optimal choice of the controller, are complicated by the fact thata parametric space of polynomials � (z) speci"ed as

�(z)"�

��

��z�, �

�3Re, � (1)"1 (20)

is not a linear space and optimization problems even with quadratic types of criteria leads toa constrained optimization, necessitating the use of the Lagrange method of undeterminedmultipliers. Therefore, it is desirable to transform this polynomial parametric space into a linearspace of all polynomials with the degree at most N

!1, speci"ed as

�(z)"��

��z�, �

�3Re (21)

by means of the bijective transformation

� (z)"1#(z!1)� (z) (22)

that transforms the constrained optimization to the unconstrained one and the task can bedirectly solved as a standard linear programming task. The polynomial � (z) is of the degreen)N

i! the corresponding polynomial � (z) is of the degree n!1. At the level of coe$cients the

mutual dependence between these polynomials can be expressed by the following relations:

��"1!

��

��, i"0, 1,2, n!1 (23)



and inversely

��"1!�

�(24)

��"�

��!�

�, i"1,2, n!1 (25)

After the substitution of (22) into (19) the transfer function for the control sequence response,parametrized by N

real coe$cients of the polynomial � (z), is given by

F(z)"z�� F�

(z)#

��

��z�� (z!1)F�

(z) (26)

After a rearrangement we get

F(z)"�z�� F�

(z)#

�

��

��

(g��

!g�)�w (27)

where �w"w��

!w��

and g�is the step response of the transfer function z�� F�

(z). This step

response is identical with the known step response of the minimum-step RFDB controllerdelayed by i steps. Since the control u

�can take up non-zero values at most at the "rst N steps, we

can con"ne ourselves to the analysis of the control segment

u"[u�, u

�,2, u

�,2, u

��] (28)

Let the segment of step response of the transfer function F�(z) be determined and arranged into

a row vector of the length N

g�"[g�

�, g�

�,2, g�

��

, 0,2, 0] (29)

The control segment can be obtained via the inverse Z-transform of F�(z) from (18) and thus the

polynomial b��(z) can be written in the form

b��(z)"g�

�z��#g�

�z��#2, g�

��

z#g��

��(30)

Let us denote

g��

"g�) Z��, i"0,2 (31)

where g�is a segment of the length N pertaining to the step response of the transfer function

z��F�(z) and Z�� is the square matrix of the order N with unit elements immediately above the



main diagonal realizing one-step delay

Z��"

0 1 0 2 0

0 0 1 2 0

�

0 0 0 2 1

0 0 0 2 0

(32)

After the substitution of (31) into (27) we get for �wO0

u

�w"g

�#�

�

��

��

) g�� ) (I!Z��) (33)

Arranging the coe$cients of the polynomial � (z) into the row vector � of the length Nwe get

�"[��

, ��

,2, ��, �

�] (34)

and arranging the row segments of step responses g�, i"0, 1,2,N

!1 into the matrix G of

dimension N�N we have

G"

g�

g�

g��

"

g��

g��

g�

2 g��

��0 2 0 0

0 g��

g��

2 g��

��g��

��2 0 0

2 2 2 2 2 �

0 0 0 2 2 2 2 g��

��0

(35)

Obviously,

�

��

��

) g��

"� )G (36)

holds and substituting (36) into (33) we obtain the control sequence for the transient response ofthe servomechanism, which is parametrized by the polynomial � (z)

u"�g�

#� )G ) (I!Z��) ) �wO�g�

#� ) H ) �w (37)

where the matrix H is of the dimension N�N and it is de"ned as

H"G ) (I!Z��) (38)



5. MINIMUM EFFORT RFDB CONTROLLER

The solution of this optimization task arises from the fact that the value of the optimalitycriterion J(u)"�u ) Q�

�is equal to the least m3Re for which it holds that �u ) Q�

�)m. The

value of the criterion for a given control segment u can thus be expressed as the solution of theproblem of mathematical programming

�u ) Q�)m ) 1 (39)

mPmin OJ(u) (40)

where 1 is an M-dimensional row vector of ones and � z � denotes the row vector of absolute valuesof its elements. The vector inequality (39) speci"es for the given Q and u the set of admissiblevalues of the variable m, the minimum of which gives the value of the optimality criterion J(u).Using (37) in (39) and (40), the value of the optimality criterion can be expressed inN

-dimensional parametric space of RFDB controllers as

�g�

) Q#� ) H ) Q � ) ��w�)m ) 1 (41)

mPmin OJ (�) (42)

Our aim is to "nd such a �H3Re� that, during the required position change, the controlminimizes the criterion J(�) with respect to all �3Re� . Dividing inequality (41) by a positivevalue of ��w� and substituting �"m/��w�, it is possible to reformulate the task as the problem ofmathematical programming de"ned for variables �3Re� and �3Re

�g�

) Q#� ) H ) Q �)� ) 1 (43)

�Pmin��O�H"

J(�H)

��w�(44)

After a simple rearrangement, the optimization problem can be transformed into the standardproblem of linear programming. Rewriting the absolute value in (43) we get after simplemanipulation the equivalent to (43) and (44) in the form

!� )1!� ) H ) Q)g�

) Q (45)

!� ) 1#� ) H ) Q)!g�

) Q (46)

�Pmin��O�H"

J (�H)

��w�(47)



The solution (�H, �H) gives for the optimality criterion speci"ed by the matrix Q and for theplant speci"ed by the matrix H the optimal controller, characterized by the vector parameter�H3Re� and by the value of the optimality criterion J(�H)"�H ��w�.

The overall design procedure is the following:

1. Discretize plant (1) with the zero-order hold for the chosen sampling period ¹ and obtainpolynomials a

�(z), b

�(z), de"ned by (4).

2. Find the minimum-step RFDB controller by solving the diophantine equation (17) withN"N�, and �(z)"1 for the unknown a

�(z) and � with the minimum degree. The poly-

nomial b��(z) is obtained after the substitution of �� into (16).

3. Using (18), "nd the step response g�of the minimum-step RFDB controller given by (29)

and thus the polynomial b��(z) given by (30). Knowing the row vector g

�, construct the

matrix G speci"ed by (35) for the chosen degree of freedom Nand compute the matrix

H according to (38).4. Solve the standard problem of linear programming formulated by (45)} (47). The optimal

controller is implicitly de"ned by the polynomial �H(z) and by the value of the optimalitycriterion.

5. Use (22) to obtain polynomial �H(z). Find the optimal RFDB controller by solving diophan-tine equation (17) with N"N�#N

and �H(z) for the unknown a

�(z) and � with the

minimum degree. Substituting �H into (16) we obtain bH�(z) and thus controller (10).

Now, it is easy to show that this solution simultaneously yields also the solution of the totalposition change �w"w

��!w

��maximization, attainable by the control sequence with

a chosen number of stepsN and with the e!ort limitation given by inequality �u ) Q��

)m, wherem is a speci"ed constant. Such a problem can be described by

�u(�) ) Q �)m ) 1 (48)

��w�Pmax�O��Hw� (49)

Substituting relation (37) into (48) we obtain the equivalent optimization task in the form(45)}(47) provided that ��Hw�"(1/�H) ) mO�H ) m. The value of ��Hw� depends, under the givenlimitation �u ) Q�

�)m and control period ¹, on the chosen number of steps N or alternatively,

on the settling time ¹��

"¹N. This dependence can generally be expressed as a function�H"�

�(¹

��).

The step-by-step solution of the total position change maximization for control stepsN"N

�,N

�#1,2 induces the solution of the time-optimal (minimum-step) position control of

the servomechanism from an initial steady-state position w��

to a "nite steady-state positionw��

, attainable by the control sequence with the e!ort limitation given by inequality�u ) Q�

�)m. We obtain the maximum attainable e!ect ��Hw�"�

�(¹N) of the control

sequence in dependence on the number of control steps. The desired position change �w isattainable in the minimum number of steps NH, given by the number of steps for which��w�)�

�(¹N)m holds. The ascending dependence �

�(¹N) is an advantage when the solution is

to be determined.



Figure 2. Step responses of the uncontrolled plant.

The obtained solution of the minimization task uniquely determines the parametric poly-nomial �H(z) and thus also the parametric polynomial �H(z)"(z!1)�H(z)#1. The optimalcontroller

u(z)

e(z)"DH(z)"

bH�(z)

aH�(z)

, �bH�(z))�aH

�(z) (50)

is determined by (16) and (17), and minimizes the chosen criterion that evaluates the N-steptransient response of the servomechanism.

6. EXAMPLES

The suggested procedure for the minimum e!ort RFDB control of the step response of a linearservomechanism will be illustrated on the plant described by the transfer function

� (s)

u (s)"G(s)"

10

s (0.2s#1) (0.001s#0.01s#1)(51)

The system is stable with two real poles ��"0, �

"!5 and two complex poles

��

"!5$31.2250i. Step responses of the uncontrolled plant are shown in Figure 2.At "rst, the minimum-step RFDB controller and then the minimum e!ort controllers will be

designed. Suppose that the desired settling time is ¹��

"0.8 s. Recalling that N�"1#�a (s)"4and ¹

��"¹N, the sampling period must be chosen as ¹"0.2 s. Hereafter, we will follow the

design procedure from the previous section:

1. Discretizing (1) with the zero-order hold, we get

b�(z)"0.6723 z#0.0978 z!0.3444 z#0.0802

a�(z)"1.0000 z!1.1031 z#0.4058 z!0.0498



2. Using the solution of the diophantine equation we obtain the minimum-step RFDB controller

D�(z)"1.977 z!2.181 z#0.8022 z!0.09842

z!0.3291 z!0.5224 z#0.1585

The control sequence realizing the step response of the unit position change reads

u"[1.9769 !2.1807 0.8022 !0.0984]

and yields the minimum of the e!ort criterion J"2.1807.

For the design of the minimum ewort RFDB controller with the settling time ¹��

"0.8 s, thereappears the question as to how to choose N

. Analysing the in#uence of N

on the normalized

e!ort criterion (see Figure 3(c)), it is obvious that in our case a reasonable choice is N"N�"4

because further increase does not bring an essential e!ect. As it holds that N"N#N�"8, the

corresponding sampling period is ¹"0.1 s.

1. We discretize the plant with this sampling period and with the zero-order hold to obtain thepolynomials

b�(z)"0.1148 z#0.5123 z#0.3434 z#0.0449

a�(z)"1.0000 z#0.6063 z!0.3677 z!0.2231

2. Using the solution of the corresponding diophantine equation, we obtain the minimum-stepRFDB controller D�(z)

D�(z)"0.9848 z#0.5971 z!0.3622 z!0.2197

z#0.8869 z#0.3824 z#0.04421

3. The closed-loop step response g�of the minimum-step RFDB controller can be taken from

the numerator of its transfer function

g�"[0.9848 0.5971 !0.3622 !0.2197]

Using g�we obtain the matrices G and H.

4. The standard problem of linear programming is solved. The solution JH"0.2293 speci"esthe optimal controller through the polynomial

�H(z)"0.2328 z#0.3245 z#0.5874 z#0.7465

5. Using the bijective transformation we obtain the polynomial

�H(z)"0.2328 z�#0.0917 z#0.2629 z#0.1591 z#0.2535



Figure 3. Minimum-step RFDB control and the optimal EFRFDB control: (a) step responses, (b) controlsequences, (c) dependence of the normalized control e!ort criterion on the settling time for ¹

��"0.2; 0.4; 0.6

and 0.8 s.

and the solution of the respective diophantine equation reads

�H"0.9848

aH�(z)"z�#0.9737 z #0.8458 z�#0.6911 z�

#0.4992 z#0.2973 z#0.104 z#0.01121



By means of �H and a�(z) we obtain

bH�(z)"0.2293 z�#0.2293 z #0.2293 z�#0.2293 z�

#0.2293 z#0.03597 z!0.1268 z!0.0557

The polynomial bH�resp. aH

�is the numerator resp. denominator of the transfer function of

the optimal EFRFDB controller. The generated control sequence realizing the step responseof the unit position change is directly given by the polynomial bH

�as

u"[0.2293 0.2293 0.2293 0.2293 0.2293 0.03597 !0.1268 !0.0557]

Step responses of the minimum-step RFDB control (dashed line) and the optimal EFRFDBcontrol (solid line) are illustrated in Figure 3(a), the corresponding control sequences are shownin Figure 3(b) and the dependence of the normalized control e!ort criterion on the degree offreedomN

for the choice of settling time ¹

��"0.2, 0.4, 0.6 and 0.8 s is shown in Figure 3(c). The

elongation of the settling time results in principle in the decrease of the needful e!ort and the samee!ect has an increase in degrees of freedom. Notice that for a su$ciently large N

the value of the

e!ort criterion does not change much and the control sequence holds a constant absolute valuepractically for all control steps N"N�#N

and switches its sign N�!1 times. Thus, the

optimal EFRFDB control for a su$ciently large Ncan be viewed as a discrete approximation of

the time optimal control with the control signal limitation.The discrete time-optimal control is illustrated in Figure 4. At "rst, the dependence of the

maximum position change, characterized by the function ��(¹

��), on the settling time and for

control periods ¹"0.025 and 0.1 s is shown in Figure 4(a). Then, as an example of the maximumposition change e!ect, the desired position change �w"30, the control e!ort limitation �u�)1and the control period ¹"0.1 were chosen and the minimum-step controller with respect to thegiven e!ort limitation is to be found. The maximization task leads to the speci"cation ofthe minimum attainable settling time (the minimum number of steps). It is obvious that themaximization e!ect consists in evolving the maximum admissible control e!ort at the take-o!interval and the only problem is when to start the braking, so that the desired position should beattained at the shortest time possible*again with limitation on the admissible control e!ort.Position and velocity step responses and control sequences are illustrated in Figure 4(b) and 4(c).Notice that for the control period ¹"0.025 a shorter settling time is obtained and the admissiblee!ort limitation is better used during braking. If the evolved control lasts longer than the timeresponse of the uncontrolled plant, the rate of position change settles down at the value k m,where k is the gain and m is the control e!ort limitation. Thus, it should asymptotically hold that��(¹

��)"km ¹

��for a non-dynamic system. Due to the dynamics of the plant connected with

the accelerating and braking and due to the chosen control period too, the resulting maximiza-tion e!ect will be diminished and can be approximated by the relation �

�(¹

��)"km (¹

��!c

�),

where c�respects the control period and the dynamics of the plant. The algorithm thus works

perfectly when the resulting settling time ¹��

is shorter than the time response of the uncontrol-led plant; for longer settling times it is theoretically correct, however, that the number ofconstraints in the linear programming task increases as well as the controller's order. Therefore, itis better to solve this problem as a minimum e!ort &braking' control of the servomechanism



Figure 4. Discrete-time optimal control: (a) Dependence of the maximum position change ��on the settling

time ¹��

for control periods ¹"0.025 and 0.1 s. (b) Discrete-time optimal control for �w"30; �u�)1;¹"0.1 s. (c) Discrete-time optimal control for �w"30; �u�)1; ¹"0.025 s.



Figure 5. Comparison of EFRFDB and �EFRFDB control for N"4, ¹

��"0.8 s. (a) step responses,

(b) control sequences.

moving along a certain time interval with a constant velocity. The solution to such a problem isbeyond the scope of this paper and will be tackled in future.

In Figure 5 the comparison of both the control e!ort and the control e!ort di!erence criteriaare illustrated. For the choice of ¹

��"0.8 s and N

"4, position step responses of the optimal

EFRFDB control (dashed line) and �EFRFDB control (solid line) are shown in Figure 5(a). Thecorresponding control sequences are shown in Figure 5(b). The limitation on control di!erenceswith the growing number of steps verge on a &triangular' shape of the control variable and can betaken for a discrete approximation of the time-optimal control with the velocity limitation (this issimilar to the fact that the weighting of time derivatives in a criterion leads to an integrationcomponent in the controller).

7. CONCLUSION

The suggested procedure for the design of the minimum-step RFDB controllers and optimalEFRFDB controllers stems from the speci"cation of the admissible set of RFDB controllers fromwhich the optimal controller is determined. It is shown that any "nite-step RFDB controlsequence can be obtained as a weighted linear combination of time-shifted responses of the



minimum-step RFDB controller and the optimization is conditioned by a parametrization ofa competitive set of RFDB controllers, de"ned on a "nite-dimensional space of polynomials. Thecriteria are expressed in a "nite-dimensional parametric space of polynomials, their minimizationis transformed to a classic problem of linear programming and leads to the optimal EFRFDBcontrollers with the chosen step position change, settling time and degree of freedom. Severalexamples illustrate the general properties of the algorithm and the behaviour of the controlledplant. Further, new interesting possibilities are shown for the usage of the elaborated algorithmfor the discrete time-optimal control with a control e!ort limitation. The obtained results con"rmthe feasibility of the algorithm and hold for the whole class of stable linear systems. A slightmodi"cation of the algorithm seems to be promising for the solution of the above-mentionedproblem concerning the minimum e!ort &braking' control of a servomechanism, moving alonga certain time interval with a constant velocity.

Finally, it is remarked that the suggested procedure was elaborated in the reaction to theproblems solved mainly in References [9}12], where a linear position servomechanism wasconsidered. Thus, the considered problem su!ers from some restrictions, which require somecommentary. Dealing only with the optimal RFDB control of the step response does not seem tobe restrictive, if the internal model principle is taken into account. The extension to other inputsignals, however, requires that the continuous plant should contain the speci"c reference signalmodel. If this condition is not satis"ed by the original plant, an appropriate precompensatormustbe used; however, the zero-order hold will always deteriorate good tracking properties. Thereappears also the possibility of using higher-order holds for input signals of a polynomial type. Theassumption of the stability of the plant polynomial is not necessary for the design of optimalEFRFDB controllers; however, the minimum-step RFDB controller for an unstable system mustbe found in the "rst instance. The direct extension of the presented design procedure to MIMOsystems appears to be complicate, whilst the state approach seems to be more appropriate.

ACKNOWLEDGEMENT

This work was supported by the Ministry of Education of the Czech Republic*Project No: MSM 2352 00004.

REFERENCES

1. Bergen AR, Ragazzini CA. Sampled-data processing techniques for feedback control systems. Transactions of AIEE(Industry and Applications), 1954; 73:236}247.

2. Jury EI, Tsypkin YZ. Output theory of discrete systems. Automatica, 1971; 7:89}107.3. Za"riou E, Morari M. Digital controllers for SISO systems: a review and a new algorithm. International Journal of

Control, 1985; 42:855}876.4. Urikura S, Nagata A. Ripple-free deadbeat control for sampled-data systems. IEEE Transations on Automatic Control,

1987; AC-32:474}482.5. Sirisena HR. Ripple-free deadbeat control of SISO discrete systems. IEEE Transactions on Automatic Control, 1985;

AC-30:168}170.6. Jetto L. Deadbeat ripple-free tracking with disturbance rejection: a polynomial approach. IEEE Transactions on

Automatic Control, 1994; AC-39:1459}1764.7. Isermann R. Digital Control Systems. Springer: Berlin, 1981.8. Janiszowsky K. A linear digital controller for single loop control systems. International Journal of Control, 1983;

37:159.9. Barbargires CA, Karybakas CA. Optimum dead-beat controller design for SISO systems using time-weighted

performance indices. International Journal of Control, 1994; 60:641}648.10. Barbargires CA, Karybakas CA. Dead-beat response of SISO systems to parabolic inputs with optimum step and

ramp responses. Kybernetika, 1994; 30:689}699.



11. Barbargires CA, Karybakas CA. Ripple-free dead-beat control of dc servo motors. Proceedings of the second IEEEMediterranean Symposium on New Directions in Control and Automation, Chania, Crete, 1994; 469}476.

12. Barbargires CA, Karybakas CA. Optimal control of "rst-order plants by dead-beat techniques. Optimal ControlApplications & Methods, 1997; 18:355}362.

13. Z[ ampa P. Importance of directional causation in system theory. The 9th IFAC World Congress, China, 1999.14. Z[ ampa P, Mos\ na J, Prautsch P. New approach to optimal control theory. The Second IFAC Workshop on New Trends

in Desgin of Control Systems, Smolenice, Slovak Republic, 1998.



minimum effort dead-beat control of linear servomechanisms with ripple-free response

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