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A Class of Multivariable Controllers and its Application to Power Plant and Power System Control
A CLASS OF MULTIVARIABLE CONTROLLERS AND ITS APPLICATION TO
POWER PLANT AND POWER SYSTEM CONTROL
' ky * H. G. Kwatny * , J. P. HcDonald ** ,K. C. Ka lnlts
A design methodology f or a class of multivariable feedback controllers for industrial processes is developed from the viewpoint of optimal linea r regulator theory. The technique permits s ystematic design of multivariable dynamic compensators to satis f y b o th transient and ultimate state performance requirements and has been applied to loadfollowing control of f ossil-fueled power plants and power s ystem load-frequ ency control.
INTRODUCTION The increasing size and expanse of interconnected power systems, the construction of larger generator stations, the reduced margin of generation capacity over load, and the need to maximize the useful life of individual generating units are among the many factors which have imposed more stringent requirements on both power system and generating station automatic control systems. At both the system and unit levels, as performance requirements have become more difficult to meet, control system designers have introduced a greater degree of coordination between control loops. As the trend towards greater coordination intensifies, it is clear that the practice of 'ad hoc' addition of coordination channels between selected control loops must be replaced by a more formal design procedure. Potentially, the number of parameters to be specified in a fully coordinated control structure equals the number of outputs times the number of control inputs times a factor depending upon the average complexity of the compensating elements. Searching over this many parameters is difficult, and as a practical matter is feasible only with very substa ntial 'a priori' structural constraints, including primary pairing of input-output variables, selection of a limited number of coordination channels and specification of compensation structure. Even with fixed structure the selection of parameters would be greatly facilitated if the search were confined to the domain in the parameter space in which the closed loop system is stable. However, determination of the stability domain is itself a formidable problem. On the other hand, if the design problem is viewed as the selection of a controller which minimizes the quadratic performance index
*
Department of Mechanical Engineering and Mechanics, Drexel University **Philadelphia Electric Company
207
J(u)
~
{y'Qy + u'Ru} dt,
Q
~
0, R > 0
o
where y is a vector of process output deviations and u a vector of control input deviations, and if a realizeable output feedback controller can be obtained for each admissible Q and R, then the control design process becomes a search for suitable parameters Q and R. If Q and R are further restricted to be diagonal, then the parameter space is greatly reduced in dimension. Moreover, the requirements that any open loop unstable modes be controllable, and observable in Q are sufficient to guarantee a stable closed loop system (all in a local context). A further advantage of searching in the Q-R parameter space is that each parameter affects the trajectory in a manner made (more-orless) explicit by the cost functional. There are, however, several often-cited difficulties associated with the application of optimal regulators. These include: the need for state variable feedback, the absence of dynamic compensation, and sensitivity to model accuracy and disturbances. As is now well known, these considerations are all related. In fact, the use of an observer as an alternative to state feedback results in a specific form of dynamic compensation. But there is more to it. Classically, dynamic compensation has been employed in process control system design for two distinct purposes: to achieve either or both transient and ultimate state performance objectives. The former can frequently be stated in terms of gain and/or phase margin, either of which provides some measure of system stability. Ultimate state performance can be described in terms of error coefficients. Optimal regulator design along with dynamic observers can provide a very satisfactory approach to meeting transient objectives. It is the problem of ultimate state performance which limits the application of optimal regulator techniques to process control. The feed forward tracking solutions of early optimal regulator designs are of little use when the plant is nonlinear, models are imprecise, and disturbances are not known beforehand and are frequently unmeasurable. In such cases integral and higher order compensation have been the mainstay of classical single loop methods of design. There is little value in a sophisticated, dynamically integrated power plant control system which cannot locate the desired steady state with reasonable precision or a load-frequency control system which does not produce zero area control error. In recent years there has been considerable interest in developing an understanding of this aspect of the multivariable control design problem. One class of procedures is based on the feedforward of estimated (possibly artificial) disturbances states. Such techniques have been proposed by Kwatny (1), Balchen, et al (2), and Johnson (3). In reference (3) three procedures are suggested. Of interest is the "counteraction mode of disturbance accomodation" which Johnson defines for the situation where the control inputs can be selected to precisely cancel the disturbance terms in the state equations. In essence, the procedure of reference (1) removes this restriction for the constant disturbance case. The procedure presented herein generalizes those results to arbitrary disturbances defined by differential equations. This technique results in the design of dynamic compensation which satisfies ultimate state performance requirements as well as transient objectives.
208
The procedure described in this paper has been applied in an extensive study of load following control of fossil power plants, McDonald and Kwatny (4) and also to the design of power system load-frequency control, Kwatny (5). Each application finds advantage in different aspects of the technique. DESIGN METHODOLOGY The controller design is based on the model:
+ Ew + Bu,
x
Ax
w
Zw + v,
y
Cx + Fw + Du,
(1)
where x is an n-dimensional state vector, y is a p-dimensional output vector, u is an m-dimensional input vector and w is a q-dimensional random bias vector specifically introduced to characterize external disturbances or model inaccuracies. The bias noise v is a white noise process having zero mean and covariance V c(t). The limiting case as V vanishes is of particular interest. v v The design proceeds in three distinct steps: 1) determination of the nominal (or ultimate state) trajectory, 2) design of the state variable feedback controller, and 3) design of the state and bias variable observer. Each step employs a subset of the output equations as follows. The first step employs the output equations
with Pl= dim (Yl) ~ dim (u)._ Under appropriate conditions the outputs Yl will be driven to desired values Yl in ultimate state. In the second step, a (possibly) different set of outputs is employed
The elements of Y2 represent all of those variables which are of concern during the transient. The outputs selected as elements of Y2 and the weights given them in the cost functional (defined below) shape the character of the transient behavior of the system. The third step, design of the observer-estimator, utilizes a third subset of the output equations
Y3
=
Cx + F w+ Du 3 3 3
(4)
The output included as elements of the P3-dimensional vector Y3 are, of course, only the outputs to be measured. It will be assumed that C and F3 are of full 3 rank. Step 1. Ihe objective is to steer the system so that Yl track~ the desired value Yl while u varies moder~te!y about the nominal value u. With Yl specified, the appropriate values of x, u are obtained by setting v = 0 in (1) to obtain the ultimate state equations: ~
x
. w
Yl
= Ax
+ Ew + Bu,
Zw,
= ex-
+ Fw +
Du.
209
(5)
Solutions of the form x u
=
Xly + X2w
(6)
UlY + U2w
are frequently obtainable and are sought by direct substitution in (5). found that Xl' X , U ' U must satisfy the conditions 2 l 2
It is
(7)
and (8)
Eq. (7) is readily solved using a pseudo-inverse. Note that the ease of solving for the matrix [Xl' :U ']' is due to the fact that each column bi of [Xl': Ul '] ~ l satisfies the equation
(9 )
where ci is the corresponding column of [0': I]'. Thus, the solution is obtained by solving a set of uncoupled (and essentially identical) sub-problems as represented by (9). On the other hand, (8) is now complicated by the appearance of the term [-Z' X2' : 0']'. Although the equations are still linear, a complication arises from the fact that the equations for the columns of[X2':U2']'cannot, in general, be decoupled. Thus, all of the equations must be solved simultaneously and the dimensionality of the problem is multiplied by the dimension of w . However, Z will usually be a sparse matrix and significant advantage can be gained by making use of this property. An algorithm will be given for the special case where the upper triangular part of Z is zero. This special case is important as it includes the situation where the disturbance is characterized as a polynominal in t. Let [A]i denote the ith column of the matrix A. angular part of Z is zero:
Then if the upper tri-
[XZZ]r_l
zr,r-l [X 2 ]r'
[XZZ] r-2
zr,r-2 [X 2 ]r + zr-l,r-2 [X 2 ]r-l'
(10)
etc. Consequently [X 2 Z]r_i depends only on [X 2 ]r-i+l' [X 2 ]r-i+2' ..• , [X 2 ]r and the
210
columns of [x 212 11 U ']' can be computed recursively using the formula:
X2] [ U2 Step 2.
[A . r-l
C
B1 t D
[- E
+ [X 2 Z]
.l
r-'
j
(11)
-F
A quadratic cost functional can now be defined as
+~
(12)
T
where QO' . Q.are non-negative definite and R is positive definite. desired to mlnlmlze
It is
lim
(13)
T-t
If K is the well known state feedback gain matrix minimizing (12) subject to (1) with w = 0, then the optimal controller minimizing (13) is (see reference (1»:
(14)
[K where
(15)
and u * denotes the optimal value of u and xl an estimate of x l l below. Step 3.
as defined
A
The estimate xl is obtained from (16)
.
(17)
~
where the parameters are defined below. The following matrices are introduced
n+q-p H*
°
p
= [~Q! n+p-q H~2! q
J!
211
°
2
Since F3 is of full rank there is no loss of generality in assuming H2 to be nonsingular. Moreover, it is noted that HQ is a right inverse of H and the rows of AO are precisely the n+p-q linearly independent rows of {I - Ha H}. Define the matrix P to be the max imal solution of the Riccati equation. 22
+ (A MA
o
I)
0
(19)
= O.
The matrix H* can now be defined
*
n+p-q 1)-1 where P O P 0 I ( GVV GI + P 0A1 I) HI (F 3Vv F3 ' 0= - 2 22 - 2 . (20)
H
q
Finally, define A
[I
n+p-q
* - HlHl
H* H ] 2 2
Al
Al - BlM.
(21)
Then
r 4 = AA1 0 2 ,
~2
AA H* 1
(22)
The 2n+q eigenvalues of the closed loop system include the q eigenvalues of Z associated with the bias variables w, the n stable eigenvalues corresponding to the closed loop system matrix A= A-BK, and the n+q-p eigenvalues of the matrix AA 0 which are associated with the observer. Moreover, a pre1 2 scribed degree of stability a for the observer eigenvalues is attained by replacing Al in (19) by Al + al. The integral nature of the compensator is most easily exhioited by taking the special case Z = 0, PI = P3 = q. It can be shown, reference (1), that r has precisely PI zero eigenvalues. 4 An alternative form of (17) which has advantages for application is (23) This form of the estimator permits use of the actual applied control variables for estimation of the state deviations. In this way it is possible to partially account for control saturation which is not otherwise considered in the design. APPLICATIONS McDonald and Kwatny (4) report on the investigation of this procedure in the design of fossil power plant control systems. Studies have been made over a wide range of operating conditions via computer simulation using a field validated nonlinear model. Comparisons are made with conventional control systems including the system currently employed as well as a state-of-the-art coordinated control system. Of particular interest are recent investigations concerning variable pressure operation of the plant. In this case attention is
212
focused on the temperature transients in an effort to reduce the damage of low frequency thermal cyclic stress. Temperature is associated with, by far, the slowest of the natural dynamics of the system. As a result the observer degree of stability has a very dramatic effect on the ability to regulate temperature. In the load-frequency control application reported by Kwatny (5), load, which is not directly measurable, is characterized by bias variables w, which may be polynomial in form. It is shown that the response of tie line power flow in a two area interconnection following a change in load in one area can be greatly improved by using greater than a zeroth order polynomial characterization of load. This indicates the importance of predicting load trends and incorporating these predictions in the control of power networks when generating station dynamic limitations dominate bulk power adjustments. CONCLUSIONS The procedures presented herein provide a means for designing dynamic compensation in order to satisfy both ultimate state and transient performance requirements. Applications to the control of a fossil-fueled generating station and to a two area interconnected power system have been briefly noted. The design methodology is based on the theory of optimal linear regulators and minimal order observers. A feature of the design is that although ultimate state tracking is guaranteed for any admissible disturbance, in the absence of disturbances and errors in the initial state estimates, the closed loop response is in fact the optimal state variable feedback response with respect to the original performance index. Once the disturbance model is specified, the order of the compensator is essentially fixed. The ob~ious question is can one get away with less and still meet the control objectives? Assuming that the disturbance model has been minimally selected to satisfy ultimate state requirements, then the question is posed with respect to transient performance objectives. If these objectives are loosely stated then the answer will very likely be yes. Examples abound, with many multivariable industrial processes of obviously high order evidently being suitably stabilized by relatively simple compensators. The efficiency of the use of the proposed technique in this regard is vastly improved by paying careful attention to the process model upon which the design is based. Insuring that only the 'relevant' dynamics of the process are reflected in the model is, indeed, part of the art of control engineering which can also result in considerable payout in many directions. The use of an efficient model will not only result in near minimal order compensation but will also ease the difficulties of design and implementation which can arise, for example, by using a model of excessive bandwidth. REFERENCES 1. Kwatny, H. G., 1972, Proc. 13th Joint Automatic Control Conf., 274. 2. Balchen, J. G., Endressen, T., Field, M., and Olsen, T. 0., 1973, Automatica, 2, 259. 3. Johnson, C. D., 1972, Int. J. Control, 15, 209. 4. McDonald, J. P. and Kwatny, H. G., 1973, IEEE Trans. Auto. Control, 18, 202. 5. Kwatny, H. G., 1973, "Optimal Control of Interconnected Electric Power Systems", Drexel University, U.S.A., Research Report No. 73-0416-1.
213
POWER PLANT AND POWER SYSTEM CONTROL
' ky * H. G. Kwatny * , J. P. HcDonald ** ,K. C. Ka lnlts
A design methodology f or a class of multivariable feedback controllers for industrial processes is developed from the viewpoint of optimal linea r regulator theory. The technique permits s ystematic design of multivariable dynamic compensators to satis f y b o th transient and ultimate state performance requirements and has been applied to loadfollowing control of f ossil-fueled power plants and power s ystem load-frequ ency control.
INTRODUCTION The increasing size and expanse of interconnected power systems, the construction of larger generator stations, the reduced margin of generation capacity over load, and the need to maximize the useful life of individual generating units are among the many factors which have imposed more stringent requirements on both power system and generating station automatic control systems. At both the system and unit levels, as performance requirements have become more difficult to meet, control system designers have introduced a greater degree of coordination between control loops. As the trend towards greater coordination intensifies, it is clear that the practice of 'ad hoc' addition of coordination channels between selected control loops must be replaced by a more formal design procedure. Potentially, the number of parameters to be specified in a fully coordinated control structure equals the number of outputs times the number of control inputs times a factor depending upon the average complexity of the compensating elements. Searching over this many parameters is difficult, and as a practical matter is feasible only with very substa ntial 'a priori' structural constraints, including primary pairing of input-output variables, selection of a limited number of coordination channels and specification of compensation structure. Even with fixed structure the selection of parameters would be greatly facilitated if the search were confined to the domain in the parameter space in which the closed loop system is stable. However, determination of the stability domain is itself a formidable problem. On the other hand, if the design problem is viewed as the selection of a controller which minimizes the quadratic performance index
*
Department of Mechanical Engineering and Mechanics, Drexel University **Philadelphia Electric Company
207
J(u)
~
{y'Qy + u'Ru} dt,
Q
~
0, R > 0
o
where y is a vector of process output deviations and u a vector of control input deviations, and if a realizeable output feedback controller can be obtained for each admissible Q and R, then the control design process becomes a search for suitable parameters Q and R. If Q and R are further restricted to be diagonal, then the parameter space is greatly reduced in dimension. Moreover, the requirements that any open loop unstable modes be controllable, and observable in Q are sufficient to guarantee a stable closed loop system (all in a local context). A further advantage of searching in the Q-R parameter space is that each parameter affects the trajectory in a manner made (more-orless) explicit by the cost functional. There are, however, several often-cited difficulties associated with the application of optimal regulators. These include: the need for state variable feedback, the absence of dynamic compensation, and sensitivity to model accuracy and disturbances. As is now well known, these considerations are all related. In fact, the use of an observer as an alternative to state feedback results in a specific form of dynamic compensation. But there is more to it. Classically, dynamic compensation has been employed in process control system design for two distinct purposes: to achieve either or both transient and ultimate state performance objectives. The former can frequently be stated in terms of gain and/or phase margin, either of which provides some measure of system stability. Ultimate state performance can be described in terms of error coefficients. Optimal regulator design along with dynamic observers can provide a very satisfactory approach to meeting transient objectives. It is the problem of ultimate state performance which limits the application of optimal regulator techniques to process control. The feed forward tracking solutions of early optimal regulator designs are of little use when the plant is nonlinear, models are imprecise, and disturbances are not known beforehand and are frequently unmeasurable. In such cases integral and higher order compensation have been the mainstay of classical single loop methods of design. There is little value in a sophisticated, dynamically integrated power plant control system which cannot locate the desired steady state with reasonable precision or a load-frequency control system which does not produce zero area control error. In recent years there has been considerable interest in developing an understanding of this aspect of the multivariable control design problem. One class of procedures is based on the feedforward of estimated (possibly artificial) disturbances states. Such techniques have been proposed by Kwatny (1), Balchen, et al (2), and Johnson (3). In reference (3) three procedures are suggested. Of interest is the "counteraction mode of disturbance accomodation" which Johnson defines for the situation where the control inputs can be selected to precisely cancel the disturbance terms in the state equations. In essence, the procedure of reference (1) removes this restriction for the constant disturbance case. The procedure presented herein generalizes those results to arbitrary disturbances defined by differential equations. This technique results in the design of dynamic compensation which satisfies ultimate state performance requirements as well as transient objectives.
208
The procedure described in this paper has been applied in an extensive study of load following control of fossil power plants, McDonald and Kwatny (4) and also to the design of power system load-frequency control, Kwatny (5). Each application finds advantage in different aspects of the technique. DESIGN METHODOLOGY The controller design is based on the model:
+ Ew + Bu,
x
Ax
w
Zw + v,
y
Cx + Fw + Du,
(1)
where x is an n-dimensional state vector, y is a p-dimensional output vector, u is an m-dimensional input vector and w is a q-dimensional random bias vector specifically introduced to characterize external disturbances or model inaccuracies. The bias noise v is a white noise process having zero mean and covariance V c(t). The limiting case as V vanishes is of particular interest. v v The design proceeds in three distinct steps: 1) determination of the nominal (or ultimate state) trajectory, 2) design of the state variable feedback controller, and 3) design of the state and bias variable observer. Each step employs a subset of the output equations as follows. The first step employs the output equations
with Pl= dim (Yl) ~ dim (u)._ Under appropriate conditions the outputs Yl will be driven to desired values Yl in ultimate state. In the second step, a (possibly) different set of outputs is employed
The elements of Y2 represent all of those variables which are of concern during the transient. The outputs selected as elements of Y2 and the weights given them in the cost functional (defined below) shape the character of the transient behavior of the system. The third step, design of the observer-estimator, utilizes a third subset of the output equations
Y3
=
Cx + F w+ Du 3 3 3
(4)
The output included as elements of the P3-dimensional vector Y3 are, of course, only the outputs to be measured. It will be assumed that C and F3 are of full 3 rank. Step 1. Ihe objective is to steer the system so that Yl track~ the desired value Yl while u varies moder~te!y about the nominal value u. With Yl specified, the appropriate values of x, u are obtained by setting v = 0 in (1) to obtain the ultimate state equations: ~
x
. w
Yl
= Ax
+ Ew + Bu,
Zw,
= ex-
+ Fw +
Du.
209
(5)
Solutions of the form x u
=
Xly + X2w
(6)
UlY + U2w
are frequently obtainable and are sought by direct substitution in (5). found that Xl' X , U ' U must satisfy the conditions 2 l 2
It is
(7)
and (8)
Eq. (7) is readily solved using a pseudo-inverse. Note that the ease of solving for the matrix [Xl' :U ']' is due to the fact that each column bi of [Xl': Ul '] ~ l satisfies the equation
(9 )
where ci is the corresponding column of [0': I]'. Thus, the solution is obtained by solving a set of uncoupled (and essentially identical) sub-problems as represented by (9). On the other hand, (8) is now complicated by the appearance of the term [-Z' X2' : 0']'. Although the equations are still linear, a complication arises from the fact that the equations for the columns of[X2':U2']'cannot, in general, be decoupled. Thus, all of the equations must be solved simultaneously and the dimensionality of the problem is multiplied by the dimension of w . However, Z will usually be a sparse matrix and significant advantage can be gained by making use of this property. An algorithm will be given for the special case where the upper triangular part of Z is zero. This special case is important as it includes the situation where the disturbance is characterized as a polynominal in t. Let [A]i denote the ith column of the matrix A. angular part of Z is zero:
Then if the upper tri-
[XZZ]r_l
zr,r-l [X 2 ]r'
[XZZ] r-2
zr,r-2 [X 2 ]r + zr-l,r-2 [X 2 ]r-l'
(10)
etc. Consequently [X 2 Z]r_i depends only on [X 2 ]r-i+l' [X 2 ]r-i+2' ..• , [X 2 ]r and the
210
columns of [x 212 11 U ']' can be computed recursively using the formula:
X2] [ U2 Step 2.
[A . r-l
C
B1 t D
[- E
+ [X 2 Z]
.l
r-'
j
(11)
-F
A quadratic cost functional can now be defined as
+~
(12)
T
where QO' . Q.are non-negative definite and R is positive definite. desired to mlnlmlze
It is
lim
(13)
T-t
If K is the well known state feedback gain matrix minimizing (12) subject to (1) with w = 0, then the optimal controller minimizing (13) is (see reference (1»:
(14)
[K where
(15)
and u * denotes the optimal value of u and xl an estimate of x l l below. Step 3.
as defined
A
The estimate xl is obtained from (16)
.
(17)
~
where the parameters are defined below. The following matrices are introduced
n+q-p H*
°
p
= [~Q! n+p-q H~2! q
J!
211
°
2
Since F3 is of full rank there is no loss of generality in assuming H2 to be nonsingular. Moreover, it is noted that HQ is a right inverse of H and the rows of AO are precisely the n+p-q linearly independent rows of {I - Ha H}. Define the matrix P to be the max imal solution of the Riccati equation. 22
+ (A MA
o
I)
0
(19)
= O.
The matrix H* can now be defined
*
n+p-q 1)-1 where P O P 0 I ( GVV GI + P 0A1 I) HI (F 3Vv F3 ' 0= - 2 22 - 2 . (20)
H
q
Finally, define A
[I
n+p-q
* - HlHl
H* H ] 2 2
Al
Al - BlM.
(21)
Then
r 4 = AA1 0 2 ,
~2
AA H* 1
(22)
The 2n+q eigenvalues of the closed loop system include the q eigenvalues of Z associated with the bias variables w, the n stable eigenvalues corresponding to the closed loop system matrix A= A-BK, and the n+q-p eigenvalues of the matrix AA 0 which are associated with the observer. Moreover, a pre1 2 scribed degree of stability a for the observer eigenvalues is attained by replacing Al in (19) by Al + al. The integral nature of the compensator is most easily exhioited by taking the special case Z = 0, PI = P3 = q. It can be shown, reference (1), that r has precisely PI zero eigenvalues. 4 An alternative form of (17) which has advantages for application is (23) This form of the estimator permits use of the actual applied control variables for estimation of the state deviations. In this way it is possible to partially account for control saturation which is not otherwise considered in the design. APPLICATIONS McDonald and Kwatny (4) report on the investigation of this procedure in the design of fossil power plant control systems. Studies have been made over a wide range of operating conditions via computer simulation using a field validated nonlinear model. Comparisons are made with conventional control systems including the system currently employed as well as a state-of-the-art coordinated control system. Of particular interest are recent investigations concerning variable pressure operation of the plant. In this case attention is
212
focused on the temperature transients in an effort to reduce the damage of low frequency thermal cyclic stress. Temperature is associated with, by far, the slowest of the natural dynamics of the system. As a result the observer degree of stability has a very dramatic effect on the ability to regulate temperature. In the load-frequency control application reported by Kwatny (5), load, which is not directly measurable, is characterized by bias variables w, which may be polynomial in form. It is shown that the response of tie line power flow in a two area interconnection following a change in load in one area can be greatly improved by using greater than a zeroth order polynomial characterization of load. This indicates the importance of predicting load trends and incorporating these predictions in the control of power networks when generating station dynamic limitations dominate bulk power adjustments. CONCLUSIONS The procedures presented herein provide a means for designing dynamic compensation in order to satisfy both ultimate state and transient performance requirements. Applications to the control of a fossil-fueled generating station and to a two area interconnected power system have been briefly noted. The design methodology is based on the theory of optimal linear regulators and minimal order observers. A feature of the design is that although ultimate state tracking is guaranteed for any admissible disturbance, in the absence of disturbances and errors in the initial state estimates, the closed loop response is in fact the optimal state variable feedback response with respect to the original performance index. Once the disturbance model is specified, the order of the compensator is essentially fixed. The ob~ious question is can one get away with less and still meet the control objectives? Assuming that the disturbance model has been minimally selected to satisfy ultimate state requirements, then the question is posed with respect to transient performance objectives. If these objectives are loosely stated then the answer will very likely be yes. Examples abound, with many multivariable industrial processes of obviously high order evidently being suitably stabilized by relatively simple compensators. The efficiency of the use of the proposed technique in this regard is vastly improved by paying careful attention to the process model upon which the design is based. Insuring that only the 'relevant' dynamics of the process are reflected in the model is, indeed, part of the art of control engineering which can also result in considerable payout in many directions. The use of an efficient model will not only result in near minimal order compensation but will also ease the difficulties of design and implementation which can arise, for example, by using a model of excessive bandwidth. REFERENCES 1. Kwatny, H. G., 1972, Proc. 13th Joint Automatic Control Conf., 274. 2. Balchen, J. G., Endressen, T., Field, M., and Olsen, T. 0., 1973, Automatica, 2, 259. 3. Johnson, C. D., 1972, Int. J. Control, 15, 209. 4. McDonald, J. P. and Kwatny, H. G., 1973, IEEE Trans. Auto. Control, 18, 202. 5. Kwatny, H. G., 1973, "Optimal Control of Interconnected Electric Power Systems", Drexel University, U.S.A., Research Report No. 73-0416-1.
213