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Disturbance Error Reduction in Multivariable Optimal Control Systems
Copyright ':C; IFAC Computer Aided Design Indiana. USA 1982
DISTURBANCE ERROR REDUCTION IN MUL TIV ARIABLE OPTIMAL CONTROL SYSTEMS O. A. Solheim and U. Stenhaug Division of Engineering Cybernetics, The Norwegian Institute of Technology, 7034 Trondhez'm - NTH, Norway
Abstract. The paper deals with the design of optimal multivariable controllers, using a modified LQR approach. All controllers discussed contain proportional feedback and, in addition, there may be feedforward, integral action or state estimation. The disturbance reduction properties of the syst e ms arrived at are evaluated and compared, using frequency response curves for the different output variables, together with singular value plots. The ir.vestigations are based on the use of the CAD-systems Keywords. feedback;
1.
DAREK and CYPROS.
Multivariable control systems; optimal control; feedforward; frequency response.
INTRODUCTION
2.
One of the most important tasks of a control system is to reduce the effects of disturbances on the system's outputs. The effectiveness of the controller in reducing disturbance errors is therefore an important design criterion. A complete rejection of the effect of disturbances on the output variables is, in some cases, possible and a suitable controller may be designed via the geometric approach (Wonham, 1979; Takamatsu, Hashimoto and Nakai, 1979; Kummel and Ohrt, 1981). Although this approach may work well in special cases, it also has some rather severe drawbacks as, for example, that the design procedure is not a straightforward procedure and it is difficult to include other design parameters such as optimality and robustness.
state estimation;
PROBLEM STATEMENT
Let
:ic
Ax + Bu + Cv
y
Dx
}
(1)
represent a linear time invariant dynamic system, where x,u,y represent as usual the state, the control input and the output, and v represents process disturbances. In addition there may be sensor noise and disturbances that act directly upon the output y, but these typ e s will not be considered here. A quadratic performance criterion will be used:
Another possibility is to use frequency domain methods, eventually combined with linear quadratic methods (MacFarlan e , 1981; Doyle and Stein, 1981; a number of articles in IEEE TRANS AC-26, February 1981, special issue on linear multivariable control systems). The design of multivariable systems in the frequency domain is, however, rather complicated, at l e ast compared with linear quadratic methods.
~
J [xTQx
+ uTPuldt
(2)
o
where
The purpose of the present paper is to demonstrate the use of a modified LQR (Linear Quadratic Regulator) approach in the design of multivariable c ontrollers, and to show how the disturbance error reduction properties can be visualized using frequency analysis.
and where we have assumed that the reference value Yref = O. The reason for using this performance criterion is to keep the output variables close to zer J under the influence of process disturbances and, at the same time, to economize with the control effort. To evaluate the disturbance reduction properties of the system arrived at, we may look
417
O. A. Solheim and
418
at the transfer matrix Hyv(s) between disturbances v and output y: yes) = H (s)v(s) yv
(3)
u.
Stenhaug H (s) yv
3.
J
a good indication of how one particular disturbance acts upon one particular output variable.
D(sI - A)-l C
(7)
We shall now proceed to a detailed discussion of four possible controller configurations and we start with:
We may for example display the frequency response curves ly.(jw)/v.(jw) l . these giving 1
=
PROPORTIONAL FEEDBACK
See Fig. 1.
It is also possible to have a more general evaluation, for example by using some measure of the "size" of the transfer matrix. The measure we shall use here is the maximum singular value o(s), which is identical to the Euclidian norm, that is:
a( w) =
l A. (H*(j w)H(jw»' max
(4)
: CONTROLLER : '-- _ _ _ _ _ _ _ _ _ _ J
In addition to that o(w) indicates an overall performance, it may also be taken as the "worst case" response, since cr( w) > ly.(j w)/v.(j w)l. -
1
J
(5)
This relation also reveals a ser10US drawback using cr( w) , namely that if, for example, all ly . (j w)/v.(j w) 1 except one, are very 1
J
small, then it will be this larger one that determines o( w). We shall demonstrate this fact below in Section 5. One problem that occurs using the quadratic criterion (2) is the choice of the weighting matrices Qy (output) and P (controls). Let us assume that P is chosen according to the "price" of the different control variables. We may then choose ~ according to which weight we place on keeping the different output variables close to zero. There is, however, the problem that with the chosen Q the system may not have sufficient stabi-
Fig. 1.
Proportional feedback.
We use the control law u
=
(8)
Gx
where G = _P- 1BTR and R is the solution of the usual Riccati equation. The transfer matrix becomes H (s) yv
-1
D[sI - (A+BG) 1 C
=
Thus, in Equation (6): D = D,
A = A+BG,
This problem may be solved using a method developed by Solheim (1972, 1980) whereby it is possible to modify the original Q-matrix so that desired stability properties may be achieved. This is why we stated in the introduction that a modified LQR approach is to be used. In order to simplify the computation of the frequency response curves, we may put the different controller configurations into a general framework. To this end we shall use the following standard form of the transfer matrix H (s) throughout the paper: yv H (s) = D(sI yv
A) -le
e = C
To illustrate the behaviour of the different controller configurations, we shall use the same simple process throughout the paper:
A
°
-1
B
10
y
lity margin, although we know that the system has absolute stability.
(9)
D
C
o
[~ 13 ,
!J' [~ ~J
P
Example 1 We place equal weight on the two output variables Yl and Y2: Q = [00 Y 0
or 1:0J
Q = DTQ D = [00 Y
°
0 100 0 ,goJ
The eigenvalues of the closed loop system (A+BG) then become -2.92,
-5.62
±
j4.56
(6)
where the tilde ~ indicates modified matrices corresponding to the actual configuration. For the open loop case the transfer matrix is simply
The eigenvalue at -2.92 is too close to the imaginary axis, so we choose to move it to -8. This can be achieved through the modified Q-matrix:
419
Disturbance Error Reduction
Q =
-10.32 101.9 -2.26
55.09 -10.32 [ 12.04
12.04J -2.26 102.6
Q
The feedback matrix becomes G
-1.88 -5.36
= [-5.88
-0.38
14.64 ] -1.43 1004
The feedback matrix becomes G
-1.03.J -0.98
Iy . (j w) / v J. (j w) I
We compute
-5.38 1001 -1.43
54.95 -5.38 [ 14.64
=
i
~
1,2 1,2,3
The result is shown in Fig. 2. In the same figure is also shown the a-plot.
=
-3.62 -16.8
[-5.97 -0.725
-1.48J -5.93
The frequency response curves together with the a-plot are shown in Fig. 3. We note that a considerable improvement has been achieved for most of the curves but very little, for example, for Y2/v3' The general improvement is also indicated by the a-plot.
dB 001
02
10
05
1
20
100 w
50 I
dB 001
10
05
02
20
100 w
50
,
;
i
-10
(J"
Y2 /v 2
- 20
-30
- 10
Y2 Iv)
- -
- r:
Y, Iv]
--- - - -
-
-----............ "---
-----
'/-
....
Y, v]
Y2/ v,
-40
I-------~-
y, Iv,
...
~ , ,
...............
-------\
- 50
- 60
\
\
\ \ \ ~
\
2
- 60 1 - - - - - - - ' -
\
\
i
-..,. y,/v
----.;---<~
,,\ \\
... ~
y, Iv,
\
i
_ r!!v2-
-
-
~
\
-70
,
,
\
\
-70
Fig. 2.
...
K",
Fig. 3.
\
Example 2 Let us start with increasing the weight on the output variables. Let or
Q
=
[~
o 1000
o
We check the eigenvalues :
Example 2.
Next, we try the combination of 4.
y
feedforward or adding integral action. All these measures will be considered in the following.
PROPORTIONAL FEEDBACK AND FEEDFORWARD
Assuming that th e disturbances can be measured, we may introduce a feedforward control matrix G in addition to the feedback con2 trol matrix G , Fig. 4. 1
~--fCl--
~ ~-- -
1
0::: ' l1J '
...J ' ...J .
-2.99,
\
Example 1.
If the performance of this system is not satisfactory, th e re are several possibilities for improvement, as for example increasing the elements of the Q -matrix, introducing
10~0]
\\
y,Iv,
\
,
I
- - - --
- 50
\
I
-30
-40
\\ 1\ \\ \
,
J
Y2/ v2
\~
,
\
1
aY,
- 20
-11.39 ± j5.403
We note that this new Q-matrix does not change very much th e eigenvalue closest to the imaginary axis. We again shift this eigenvalue to -8 through the modified Q-matrix:
0: 0::: '
z>- :'
0 ,
u: Fig. 4.
Proportional feedback and feedforward.
O. A. Solheim and
420
In order to achieve eXdct optimal control, it is necessary to know the future disturbances. If we assume constant disturbances, the situation simplifies much, since the optimal solution in this case is a constant feedforward matrix given by: (10)
u.
Stenhaug
we note that a considerable improvement has been obtained except for Y1/v3 which is better with feedback only. This ~s, however, what we must expect in a multivariable system. The overall si.tuation may improve, as is also indicated by the a-plot in this case, but for some of the variables the situation may deteriorate.
5. where R again is the solution of the usual Riccati equation and G is the same feed1 back matrix as in (8). If the disturbances are constant or slowly varying (slowly relative to the process dynamics) we should expect that the disturbance reduction properties are improved compared to the feedback-only case. For more rapidly varying disturbances the situation could be different.
PROPORTIONAL PLUS INTEGRAL FEEDBACK
In order to improve the steady-state and lowfrequency performance, we may try to include an integral action in the feedback loop, Fig. 6.
We get the transfer matrix H
yv
(s)
The matrices used will then be D = D,
~n
the standard form (6)
A
CONTROLLER
L ________ ________ _
Example 3 We use the same Q-matrix as in Example 1, and thus the feedback matrix will be the same. The feedforward matrix becomes G2
=
-0.754 [ -0.0188
-0.504 -0.671
Proportional plus int e gral feedback.
The control law becomes
~n
this case: ( 1 2)
-0.293J -0.410
The fr e quency response curves and a-plot for this exampl e are shown in Fig. 5. Comparing these curves with those of Fig. 2 (Example 1), dB ° °r'_-,0_.2_ _ -,0.,...5'--- ,_---='--_---=_--','0r-----=2:.::0_ _-=5-=0_...:.:;'00 w
- 10
Fig. 6.
f - - -- - - - +-- - - - -- +-- -- - - - - - I
where by
z
is the integrator output, described
z = Ey
= EDx
( 13)
The E-matrix will singl e out those output variables for which we want to improve the steady-state and low frequency behaviour. Let us establish the augmented syste m
( 14)
-30 c-_~_~ _-_ --_ ,-_~--=~~-e~~--~~~---~ Yl / V 2
- lO
-
I
50~---------~------------_+------~----~
In order that this system be controllable, the system (A,B) must be controllable and in addition must dim z ~ dim u . There can, therefore, be no more int e grators than control variables. Combining (12) and (14) yields the closed loop system
x
=
- 70r_------~~_r-+----------r_~~--;---~
Fig. 5.
Example 3.
A;{' Here
p
+ Cv
dim z , n
(15 ) dim x.
421
Disturbance Error Reduction From (14) we not e that the eigenva1ues of the open-loop system consist of the eigenva1ues of the process A and as many eigenva1ues at the origin as there are integrators. The effectiveness of the integral action will now depend upon how far to the left we shift these last eigenva1ues. Compared with a conventional PI-controller, a large integral time corresponds to a small shift of the eigenva1ues. The matrices in the standard transfer matrix (6) are for this mu1tivariab 1e PI-controller: [D
D where A
m
mxp 1
dim y
C
and
0
and
0.2
dB 001
5
0.5
50
100 w
- 20
--- 30 Y1/ v,
- .0
y,/v,
,. - 50
,-
Example 4 We choose E = [0 1 1 and start with the same Q as in Example 1. We need an augmen-
20
-10
p = dim z .
are as given 1.n ( 15)
10
,-
/
, ,-
- 60
y
-70
ted Q-matrix :
Q
[:
=
0 0
0 100 0 0
0 0 100 0
~l
Fig. 7.
With this 'Q-matrix we get the following eigenvalues 0,
-2.92,
-5 .62 ± j4.56
We choose to move the eigenva1ue at -2.9 2 to - 8, and, starting with a rather weak integral action, we move the eigenva1ue at the origin to -0.5. The corresponding 'Q-rnatrix becomes
[551
_ Q-
-10. 23 12.1 0.51
~
-10.23 103.1 -1.47 6.45
12.1 -1. 47 103.1 4.19
Example 4.
We also note that the a-plot in Fig. 7 does not exhibit the shape that we perhaps should expect when using a PI-type controller, al though we note some low-frequency improvement comparing with the a-plot in Fig. 2. The reason why a does not have a typical integral shape (+ - 00 dB as w + 0) is, of course, that there is still a large element in the Hyv(S) matrix when w + 0, namely Yl/v3' This is a weakness in a-plots and shows the usefulness of individual y./v . plots. 1. J Example 5
0.51] 6.45 4.19 34.28
We start in the same way as in Example 4 and move the eigenvalue a - 2.92 at -8 as before, but we will now try a stronger integral action and move the eigenva lue at the origin to -2. The result is
which yie ld the f ee dback matrices [-5. 9 -0.414
G
l
G
2
=
-2.07 -5.84
-1.15J -1. 3
Q
[-1.0 2 J - 2.58
-8.8 121.3 10.35 10 3. 1
-5.9 4 [ -0.5 3
We should now expect that all !Y2(j w)/v (jw) ! j
(- 00 dB) as w + O. This is also th e case but, since the frequence range for the plots starts at w = 0.1, this is not quite clear from the frequency response curves, Fig. 7. We see, however, the trend. These curves are to be compared with those in Fig. 2 (Example 1) for the corresponding proportional feedback case. We note that the integral action is effective below w ~ 2. We note also that for Yl/v3 the situation has deteriorated.
55 . 2 -8.8 [ 13.04 8. 17
-2.64 -7.3
13.04 10.35 110.8 67.06
8.17] 103.1 67.06 548.4
-1 .53 ] -2.25
+
Remembering that Yl is not input to an inte grator, this is what can be expected .
G2
lO • 13 =t,-4.09J
The frequency response curves together with the a-plot are shown i Fig. 8. Compared with the previous example, Fig. 7, we have now improved the low frequency performance for all curves, except for Yl/v3'
422 dB 001
O. A. Solheim and U. Stenhaug 0.2
10
0.5 I
20
100 w
50
I
-10
- 20
-30
- .0
- 50
mator (Kalman-filter) or a modal type, based on eigenvalue placement. The design of an optimal estimator is based on a stochastic description of process disturbance and sensor noise. As we are considering more deterministic signals here, such a design is not directly applicable. It is, however, possible to modify the design by combining an optimal and a modal estimator. Let us assume that we have some knowledge about the sensor noise, so that we can specify the noise covariance matrix W. If we now in addition specify the eigenvalues of the estimator closed-loop matrix (A-KD), we may use the same technique as for designing optimal feedback systems with specified stability, since these are dual problems (Solheim, 1972). The result is not directly an optimal estimator but an estimator with prescribed stability that weighs the measurements according to their noise contents. The matrices in the standard transfer matrix (6) are in this case:
'i5 Fig. 8. 6.
=
[D
I
0
Example 5. A
PROPORTIONAL FEEDBACK WITH STATE ESTIMATION
and
C
mxn
1
as given in (16).
Example 6
See Fig. 9.
We use the same feedback controller as in Example 1. We specify the sensor noise covariance matrix as W =
. . ...... ............ . . .~.
--~ -y-~
:
[~ ~J
which means that both measurements contain the same amount of noise. Let us specify the eigenvalues of the estimator (A-KD) as -4, -6, -7.
Comparing these with the eigenvalues of the feedback system CONTROLLER
-8, - 5 . 62 ± j 4 . 56
Fig. 9.
Proportional feedback with state estimation.
we note that they all lie in the same area. The specified eigenvalues of the estimator may be obtained by using the process disturbance covariance matrix
The total system is described by
v
[;~:~
20.8
'Ai+cv where x is the estimated state and augmented state.
=
16.6 16.7
30.1
30.1
74.3
20.8]
(16) x
the
Fortunately, the design problem can be divided into two independent parts, since we can design the feedback system (A+BC) independently of the estimator (A-KD). For the estimator design there are several possibilities, as for example an optimal esti-
We may consider this matrix as a description of some sort of fictional process disturbances. The estimator gain matrix becomes 2.15 K =
1.8
[ 2.9
2.54J 2.9 7.2
The frequency response curves of the trans-
423
Disturbance Error Reduction fer matrix
H (s) are shown yv gether with the a-plot. dB 0 0.1
0.5
0.2
10
~n
Fig. 10 to-
20
10
20
50
100 w
100 w
50
er
- 10
~====:Y;2/~V:,~:;:::;:;;;;:::~~t~-----J Yl Iv?
.J._
- 20 ~_-_. _-_-_-_~~+ , _~14--~~---~~----~ --r;---_~
y] / v)
Yl /v3
_
" ,
,
-30r------~----~'~\~+-~~~~-~ \
- .0
~--------+_------~_.--~--~
-5 0r-------~----------~r_~~--~r_4
-60r_----------~----------_4r_--~~r---~
Fig. 11.
\ \
Example 7.
\\
- 70~-------_1------_+--~~~~~
Fig. 10.
Example 6.
Comparing with Example 1, Fig. 2, which is the same system without estimator, we note that the performance has deteriorated. We will there for try a new estimator that is somewhat faster than the feedback system itself. Example 7 We specify the eigenvalues of the estimator as
In order to compare the different cases more easily, all the a-plots are shown in Fig. 12, all the Y2/vl-plots in Fig. 13 and all the Y2/v3-plots in Fig. 14.
The design technique is fairly straightforward and can be adapted to different controller configurations. The frequency analysis is made easier by putting all the different configurations into a standard framework. dB 001
Comparing again with the eigenvalues of the feedback system ±
we note that the estimator is now somewhat faster than the feedback system.
02
05
--- ~itho
Ex. 6
---'-
, tx .1 _. -
-- -
We get the covariance matrix - 30
V
= [6773 1228 398
1228 295 151
10
20
50
: I
-10
j4.56
In addition we have
also shown the case without control in these figures.
-10, -12, -15
-8, -5.62
CONCLUDING REMARKS
7.
"
~
~
~t
control
~
'-.:
'\
398] 151 164
and estimator gain matrix - 50
K
=
[68.7 17 . 0 9.5
22.8] 9.5 12 . 0
The frequency response curves and a-plot are shown in Fig. 11. We note that the performance has now improved from the previous example and is not very much inferior to the case without estimator, Fig. 2.
100 w
I
,
- 60 ;
-70
Fig. 12.
a-plots for all examples.
O. A. Solheim and U. Stenhaug
424
dB 0 or 1'____-=o-= .2'_____o;..:.5_-i-_....::..._ _T-'____lTo~__'2::;0~_ _.: .5;:: O-.:.:::100 w
With out contr ol
· 10
I-......""""""""'--k~~~~+---------j Ex. 6 ....£--
- 2 0 r-----------+---~~~--4-----------~
Ex . 7 ~-- 3 0 r---------~r------2~~-----------
Ex.1 -.0 r-----~~~r-----~~-4r---------,-
.-
.- "' Ex . ..... ,../ Ex.S - 50 ~'----~.-'----1_---~._-~-4._-----~
,
,,
" - 60r-"L---'----'----~--~~----~~~~----~
/
- 70 r-------~--+-------~--~~~~----~
Fig. 13. dB 00 1
Y2/ v1-pl o ts for all examples. 05
02
5
10
20
50
100 w
!
-10 Ex. 7
r-..L.
.).x 1
_~Ih
ut control
, ,
-~
::
- 20
Ex . 2~
E~i.-" .
.-
-30
'I Ex ."
.- /
- .0
Ex . 3
, .-
'Ex. "
, ,.-
~
~~
.-
i
~
5
i - 50
, !
- 60
, I
:
, I
I
-70 Fig . 14.
Y2/ v3-plots for all examples.
The a-plots give a useful evaluation of the performance of the system, although there are some weaknesses in using such overall descriptions. A better evaluation can therefore be obtained by also inc luding curves that show how the indiv idual disturbance variables influence the different output variables. In producing the above results we have used the computer-aided sy st ems DAREK (Pedersen, P~hner and Solheim, 1972) and CYPROS (Tyss~, 1980) at the Division of Engine ering Cybernetic s, The Norwegian Institute of Technology.
REFERENCES Doyle, J.C. and G. Stein (1981). Multivariable feedback design: Concepts for a classical/modern synthesis. IEEE Trans. Autom. Control, AC-26, 4-16 Klirnmel, M. and E. Oh~981). Practical evaluation of geometric control. 8 IFAC Congress, Kyoto, Aug. 24-28,1981. Mac Farlane, A.G.J. (1981). Characteristic and principal gains and phase s and their use as multivariable control design tools. AGARD Lecture Se ri e s No. 117, Paper No. 2, 1-34. Pedersen, J.O., F. P~hn e r and O.A. Solheim (1972). Computer aided de sign of multivariable control s ys t e ms. 5 . IFAC Congr e ss, Paris, June 197 2 . Solheim, O.A. (1972). Des i gn of optimal control systems with pr e scribed eigenvalues. Int. J. Contr., 15, 143-160. Solheim, O.A. (1980). On the-Us e of loworder Riccati equations in the design of a class of f ee dback contro ll e rs and estimators. MIC, 1,231-245. Ta kamat su, T.~ Hashimo t o and Y. Nakai (1979). A ge ometri c a pproac h t o multivariable control sy stem de s ign of a distillation c olumn. Automa ti ca ,~, 38 7-402. Ty ss~, A. (1980) . CYPROS - c yb e rnetic pr ogram packages . MIC, 1,215-229 . Wonham, W. M. (1979).--Ceome tri c state-space theory in linear multiva riable control: A status report. Automati c a,~, 5-13.
DISTURBANCE ERROR REDUCTION IN MUL TIV ARIABLE OPTIMAL CONTROL SYSTEMS O. A. Solheim and U. Stenhaug Division of Engineering Cybernetics, The Norwegian Institute of Technology, 7034 Trondhez'm - NTH, Norway
Abstract. The paper deals with the design of optimal multivariable controllers, using a modified LQR approach. All controllers discussed contain proportional feedback and, in addition, there may be feedforward, integral action or state estimation. The disturbance reduction properties of the syst e ms arrived at are evaluated and compared, using frequency response curves for the different output variables, together with singular value plots. The ir.vestigations are based on the use of the CAD-systems Keywords. feedback;
1.
DAREK and CYPROS.
Multivariable control systems; optimal control; feedforward; frequency response.
INTRODUCTION
2.
One of the most important tasks of a control system is to reduce the effects of disturbances on the system's outputs. The effectiveness of the controller in reducing disturbance errors is therefore an important design criterion. A complete rejection of the effect of disturbances on the output variables is, in some cases, possible and a suitable controller may be designed via the geometric approach (Wonham, 1979; Takamatsu, Hashimoto and Nakai, 1979; Kummel and Ohrt, 1981). Although this approach may work well in special cases, it also has some rather severe drawbacks as, for example, that the design procedure is not a straightforward procedure and it is difficult to include other design parameters such as optimality and robustness.
state estimation;
PROBLEM STATEMENT
Let
:ic
Ax + Bu + Cv
y
Dx
}
(1)
represent a linear time invariant dynamic system, where x,u,y represent as usual the state, the control input and the output, and v represents process disturbances. In addition there may be sensor noise and disturbances that act directly upon the output y, but these typ e s will not be considered here. A quadratic performance criterion will be used:
Another possibility is to use frequency domain methods, eventually combined with linear quadratic methods (MacFarlan e , 1981; Doyle and Stein, 1981; a number of articles in IEEE TRANS AC-26, February 1981, special issue on linear multivariable control systems). The design of multivariable systems in the frequency domain is, however, rather complicated, at l e ast compared with linear quadratic methods.
~
J [xTQx
+ uTPuldt
(2)
o
where
The purpose of the present paper is to demonstrate the use of a modified LQR (Linear Quadratic Regulator) approach in the design of multivariable c ontrollers, and to show how the disturbance error reduction properties can be visualized using frequency analysis.
and where we have assumed that the reference value Yref = O. The reason for using this performance criterion is to keep the output variables close to zer J under the influence of process disturbances and, at the same time, to economize with the control effort. To evaluate the disturbance reduction properties of the system arrived at, we may look
417
O. A. Solheim and
418
at the transfer matrix Hyv(s) between disturbances v and output y: yes) = H (s)v(s) yv
(3)
u.
Stenhaug H (s) yv
3.
J
a good indication of how one particular disturbance acts upon one particular output variable.
D(sI - A)-l C
(7)
We shall now proceed to a detailed discussion of four possible controller configurations and we start with:
We may for example display the frequency response curves ly.(jw)/v.(jw) l . these giving 1
=
PROPORTIONAL FEEDBACK
See Fig. 1.
It is also possible to have a more general evaluation, for example by using some measure of the "size" of the transfer matrix. The measure we shall use here is the maximum singular value o(s), which is identical to the Euclidian norm, that is:
a( w) =
l A. (H*(j w)H(jw»' max
(4)
: CONTROLLER : '-- _ _ _ _ _ _ _ _ _ _ J
In addition to that o(w) indicates an overall performance, it may also be taken as the "worst case" response, since cr( w) > ly.(j w)/v.(j w)l. -
1
J
(5)
This relation also reveals a ser10US drawback using cr( w) , namely that if, for example, all ly . (j w)/v.(j w) 1 except one, are very 1
J
small, then it will be this larger one that determines o( w). We shall demonstrate this fact below in Section 5. One problem that occurs using the quadratic criterion (2) is the choice of the weighting matrices Qy (output) and P (controls). Let us assume that P is chosen according to the "price" of the different control variables. We may then choose ~ according to which weight we place on keeping the different output variables close to zero. There is, however, the problem that with the chosen Q the system may not have sufficient stabi-
Fig. 1.
Proportional feedback.
We use the control law u
=
(8)
Gx
where G = _P- 1BTR and R is the solution of the usual Riccati equation. The transfer matrix becomes H (s) yv
-1
D[sI - (A+BG) 1 C
=
Thus, in Equation (6): D = D,
A = A+BG,
This problem may be solved using a method developed by Solheim (1972, 1980) whereby it is possible to modify the original Q-matrix so that desired stability properties may be achieved. This is why we stated in the introduction that a modified LQR approach is to be used. In order to simplify the computation of the frequency response curves, we may put the different controller configurations into a general framework. To this end we shall use the following standard form of the transfer matrix H (s) throughout the paper: yv H (s) = D(sI yv
A) -le
e = C
To illustrate the behaviour of the different controller configurations, we shall use the same simple process throughout the paper:
A
°
-1
B
10
y
lity margin, although we know that the system has absolute stability.
(9)
D
C
o
[~ 13 ,
!J' [~ ~J
P
Example 1 We place equal weight on the two output variables Yl and Y2: Q = [00 Y 0
or 1:0J
Q = DTQ D = [00 Y
°
0 100 0 ,goJ
The eigenvalues of the closed loop system (A+BG) then become -2.92,
-5.62
±
j4.56
(6)
where the tilde ~ indicates modified matrices corresponding to the actual configuration. For the open loop case the transfer matrix is simply
The eigenvalue at -2.92 is too close to the imaginary axis, so we choose to move it to -8. This can be achieved through the modified Q-matrix:
419
Disturbance Error Reduction
Q =
-10.32 101.9 -2.26
55.09 -10.32 [ 12.04
12.04J -2.26 102.6
Q
The feedback matrix becomes G
-1.88 -5.36
= [-5.88
-0.38
14.64 ] -1.43 1004
The feedback matrix becomes G
-1.03.J -0.98
Iy . (j w) / v J. (j w) I
We compute
-5.38 1001 -1.43
54.95 -5.38 [ 14.64
=
i
~
1,2 1,2,3
The result is shown in Fig. 2. In the same figure is also shown the a-plot.
=
-3.62 -16.8
[-5.97 -0.725
-1.48J -5.93
The frequency response curves together with the a-plot are shown in Fig. 3. We note that a considerable improvement has been achieved for most of the curves but very little, for example, for Y2/v3' The general improvement is also indicated by the a-plot.
dB 001
02
10
05
1
20
100 w
50 I
dB 001
10
05
02
20
100 w
50
,
;
i
-10
(J"
Y2 /v 2
- 20
-30
- 10
Y2 Iv)
- -
- r:
Y, Iv]
--- - - -
-
-----............ "---
-----
'/-
....
Y, v]
Y2/ v,
-40
I-------~-
y, Iv,
...
~ , ,
...............
-------\
- 50
- 60
\
\
\ \ \ ~
\
2
- 60 1 - - - - - - - ' -
\
\
i
-..,. y,/v
----.;---<~
,,\ \\
... ~
y, Iv,
\
i
_ r!!v2-
-
-
~
\
-70
,
,
\
\
-70
Fig. 2.
...
K",
Fig. 3.
\
Example 2 Let us start with increasing the weight on the output variables. Let or
Q
=
[~
o 1000
o
We check the eigenvalues :
Example 2.
Next, we try the combination of 4.
y
feedforward or adding integral action. All these measures will be considered in the following.
PROPORTIONAL FEEDBACK AND FEEDFORWARD
Assuming that th e disturbances can be measured, we may introduce a feedforward control matrix G in addition to the feedback con2 trol matrix G , Fig. 4. 1
~--fCl--
~ ~-- -
1
0::: ' l1J '
...J ' ...J .
-2.99,
\
Example 1.
If the performance of this system is not satisfactory, th e re are several possibilities for improvement, as for example increasing the elements of the Q -matrix, introducing
10~0]
\\
y,Iv,
\
,
I
- - - --
- 50
\
I
-30
-40
\\ 1\ \\ \
,
J
Y2/ v2
\~
,
\
1
aY,
- 20
-11.39 ± j5.403
We note that this new Q-matrix does not change very much th e eigenvalue closest to the imaginary axis. We again shift this eigenvalue to -8 through the modified Q-matrix:
0: 0::: '
z>- :'
0 ,
u: Fig. 4.
Proportional feedback and feedforward.
O. A. Solheim and
420
In order to achieve eXdct optimal control, it is necessary to know the future disturbances. If we assume constant disturbances, the situation simplifies much, since the optimal solution in this case is a constant feedforward matrix given by: (10)
u.
Stenhaug
we note that a considerable improvement has been obtained except for Y1/v3 which is better with feedback only. This ~s, however, what we must expect in a multivariable system. The overall si.tuation may improve, as is also indicated by the a-plot in this case, but for some of the variables the situation may deteriorate.
5. where R again is the solution of the usual Riccati equation and G is the same feed1 back matrix as in (8). If the disturbances are constant or slowly varying (slowly relative to the process dynamics) we should expect that the disturbance reduction properties are improved compared to the feedback-only case. For more rapidly varying disturbances the situation could be different.
PROPORTIONAL PLUS INTEGRAL FEEDBACK
In order to improve the steady-state and lowfrequency performance, we may try to include an integral action in the feedback loop, Fig. 6.
We get the transfer matrix H
yv
(s)
The matrices used will then be D = D,
~n
the standard form (6)
A
CONTROLLER
L ________ ________ _
Example 3 We use the same Q-matrix as in Example 1, and thus the feedback matrix will be the same. The feedforward matrix becomes G2
=
-0.754 [ -0.0188
-0.504 -0.671
Proportional plus int e gral feedback.
The control law becomes
~n
this case: ( 1 2)
-0.293J -0.410
The fr e quency response curves and a-plot for this exampl e are shown in Fig. 5. Comparing these curves with those of Fig. 2 (Example 1), dB ° °r'_-,0_.2_ _ -,0.,...5'--- ,_---='--_---=_--','0r-----=2:.::0_ _-=5-=0_...:.:;'00 w
- 10
Fig. 6.
f - - -- - - - +-- - - - -- +-- -- - - - - - I
where by
z
is the integrator output, described
z = Ey
= EDx
( 13)
The E-matrix will singl e out those output variables for which we want to improve the steady-state and low frequency behaviour. Let us establish the augmented syste m
( 14)
-30 c-_~_~ _-_ --_ ,-_~--=~~-e~~--~~~---~ Yl / V 2
- lO
-
I
50~---------~------------_+------~----~
In order that this system be controllable, the system (A,B) must be controllable and in addition must dim z ~ dim u . There can, therefore, be no more int e grators than control variables. Combining (12) and (14) yields the closed loop system
x
=
- 70r_------~~_r-+----------r_~~--;---~
Fig. 5.
Example 3.
A;{' Here
p
+ Cv
dim z , n
(15 ) dim x.
421
Disturbance Error Reduction From (14) we not e that the eigenva1ues of the open-loop system consist of the eigenva1ues of the process A and as many eigenva1ues at the origin as there are integrators. The effectiveness of the integral action will now depend upon how far to the left we shift these last eigenva1ues. Compared with a conventional PI-controller, a large integral time corresponds to a small shift of the eigenva1ues. The matrices in the standard transfer matrix (6) are for this mu1tivariab 1e PI-controller: [D
D where A
m
mxp 1
dim y
C
and
0
and
0.2
dB 001
5
0.5
50
100 w
- 20
--- 30 Y1/ v,
- .0
y,/v,
,. - 50
,-
Example 4 We choose E = [0 1 1 and start with the same Q as in Example 1. We need an augmen-
20
-10
p = dim z .
are as given 1.n ( 15)
10
,-
/
, ,-
- 60
y
-70
ted Q-matrix :
Q
[:
=
0 0
0 100 0 0
0 0 100 0
~l
Fig. 7.
With this 'Q-matrix we get the following eigenvalues 0,
-2.92,
-5 .62 ± j4.56
We choose to move the eigenva1ue at -2.9 2 to - 8, and, starting with a rather weak integral action, we move the eigenva1ue at the origin to -0.5. The corresponding 'Q-rnatrix becomes
[551
_ Q-
-10. 23 12.1 0.51
~
-10.23 103.1 -1.47 6.45
12.1 -1. 47 103.1 4.19
Example 4.
We also note that the a-plot in Fig. 7 does not exhibit the shape that we perhaps should expect when using a PI-type controller, al though we note some low-frequency improvement comparing with the a-plot in Fig. 2. The reason why a does not have a typical integral shape (+ - 00 dB as w + 0) is, of course, that there is still a large element in the Hyv(S) matrix when w + 0, namely Yl/v3' This is a weakness in a-plots and shows the usefulness of individual y./v . plots. 1. J Example 5
0.51] 6.45 4.19 34.28
We start in the same way as in Example 4 and move the eigenvalue a - 2.92 at -8 as before, but we will now try a stronger integral action and move the eigenva lue at the origin to -2. The result is
which yie ld the f ee dback matrices [-5. 9 -0.414
G
l
G
2
=
-2.07 -5.84
-1.15J -1. 3
Q
[-1.0 2 J - 2.58
-8.8 121.3 10.35 10 3. 1
-5.9 4 [ -0.5 3
We should now expect that all !Y2(j w)/v (jw) ! j
(- 00 dB) as w + O. This is also th e case but, since the frequence range for the plots starts at w = 0.1, this is not quite clear from the frequency response curves, Fig. 7. We see, however, the trend. These curves are to be compared with those in Fig. 2 (Example 1) for the corresponding proportional feedback case. We note that the integral action is effective below w ~ 2. We note also that for Yl/v3 the situation has deteriorated.
55 . 2 -8.8 [ 13.04 8. 17
-2.64 -7.3
13.04 10.35 110.8 67.06
8.17] 103.1 67.06 548.4
-1 .53 ] -2.25
+
Remembering that Yl is not input to an inte grator, this is what can be expected .
G2
lO • 13 =t,-4.09J
The frequency response curves together with the a-plot are shown i Fig. 8. Compared with the previous example, Fig. 7, we have now improved the low frequency performance for all curves, except for Yl/v3'
422 dB 001
O. A. Solheim and U. Stenhaug 0.2
10
0.5 I
20
100 w
50
I
-10
- 20
-30
- .0
- 50
mator (Kalman-filter) or a modal type, based on eigenvalue placement. The design of an optimal estimator is based on a stochastic description of process disturbance and sensor noise. As we are considering more deterministic signals here, such a design is not directly applicable. It is, however, possible to modify the design by combining an optimal and a modal estimator. Let us assume that we have some knowledge about the sensor noise, so that we can specify the noise covariance matrix W. If we now in addition specify the eigenvalues of the estimator closed-loop matrix (A-KD), we may use the same technique as for designing optimal feedback systems with specified stability, since these are dual problems (Solheim, 1972). The result is not directly an optimal estimator but an estimator with prescribed stability that weighs the measurements according to their noise contents. The matrices in the standard transfer matrix (6) are in this case:
'i5 Fig. 8. 6.
=
[D
I
0
Example 5. A
PROPORTIONAL FEEDBACK WITH STATE ESTIMATION
and
C
mxn
1
as given in (16).
Example 6
See Fig. 9.
We use the same feedback controller as in Example 1. We specify the sensor noise covariance matrix as W =
. . ...... ............ . . .~.
--~ -y-~
:
[~ ~J
which means that both measurements contain the same amount of noise. Let us specify the eigenvalues of the estimator (A-KD) as -4, -6, -7.
Comparing these with the eigenvalues of the feedback system CONTROLLER
-8, - 5 . 62 ± j 4 . 56
Fig. 9.
Proportional feedback with state estimation.
we note that they all lie in the same area. The specified eigenvalues of the estimator may be obtained by using the process disturbance covariance matrix
The total system is described by
v
[;~:~
20.8
'Ai+cv where x is the estimated state and augmented state.
=
16.6 16.7
30.1
30.1
74.3
20.8]
(16) x
the
Fortunately, the design problem can be divided into two independent parts, since we can design the feedback system (A+BC) independently of the estimator (A-KD). For the estimator design there are several possibilities, as for example an optimal esti-
We may consider this matrix as a description of some sort of fictional process disturbances. The estimator gain matrix becomes 2.15 K =
1.8
[ 2.9
2.54J 2.9 7.2
The frequency response curves of the trans-
423
Disturbance Error Reduction fer matrix
H (s) are shown yv gether with the a-plot. dB 0 0.1
0.5
0.2
10
~n
Fig. 10 to-
20
10
20
50
100 w
100 w
50
er
- 10
~====:Y;2/~V:,~:;:::;:;;;;:::~~t~-----J Yl Iv?
.J._
- 20 ~_-_. _-_-_-_~~+ , _~14--~~---~~----~ --r;---_~
y] / v)
Yl /v3
_
" ,
,
-30r------~----~'~\~+-~~~~-~ \
- .0
~--------+_------~_.--~--~
-5 0r-------~----------~r_~~--~r_4
-60r_----------~----------_4r_--~~r---~
Fig. 11.
\ \
Example 7.
\\
- 70~-------_1------_+--~~~~~
Fig. 10.
Example 6.
Comparing with Example 1, Fig. 2, which is the same system without estimator, we note that the performance has deteriorated. We will there for try a new estimator that is somewhat faster than the feedback system itself. Example 7 We specify the eigenvalues of the estimator as
In order to compare the different cases more easily, all the a-plots are shown in Fig. 12, all the Y2/vl-plots in Fig. 13 and all the Y2/v3-plots in Fig. 14.
The design technique is fairly straightforward and can be adapted to different controller configurations. The frequency analysis is made easier by putting all the different configurations into a standard framework. dB 001
Comparing again with the eigenvalues of the feedback system ±
we note that the estimator is now somewhat faster than the feedback system.
02
05
--- ~itho
Ex. 6
---'-
, tx .1 _. -
-- -
We get the covariance matrix - 30
V
= [6773 1228 398
1228 295 151
10
20
50
: I
-10
j4.56
In addition we have
also shown the case without control in these figures.
-10, -12, -15
-8, -5.62
CONCLUDING REMARKS
7.
"
~
~
~t
control
~
'-.:
'\
398] 151 164
and estimator gain matrix - 50
K
=
[68.7 17 . 0 9.5
22.8] 9.5 12 . 0
The frequency response curves and a-plot are shown in Fig. 11. We note that the performance has now improved from the previous example and is not very much inferior to the case without estimator, Fig. 2.
100 w
I
,
- 60 ;
-70
Fig. 12.
a-plots for all examples.
O. A. Solheim and U. Stenhaug
424
dB 0 or 1'____-=o-= .2'_____o;..:.5_-i-_....::..._ _T-'____lTo~__'2::;0~_ _.: .5;:: O-.:.:::100 w
With out contr ol
· 10
I-......""""""""'--k~~~~+---------j Ex. 6 ....£--
- 2 0 r-----------+---~~~--4-----------~
Ex . 7 ~-- 3 0 r---------~r------2~~-----------
Ex.1 -.0 r-----~~~r-----~~-4r---------,-
.-
.- "' Ex . ..... ,../ Ex.S - 50 ~'----~.-'----1_---~._-~-4._-----~
,
,,
" - 60r-"L---'----'----~--~~----~~~~----~
/
- 70 r-------~--+-------~--~~~~----~
Fig. 13. dB 00 1
Y2/ v1-pl o ts for all examples. 05
02
5
10
20
50
100 w
!
-10 Ex. 7
r-..L.
.).x 1
_~Ih
ut control
, ,
-~
::
- 20
Ex . 2~
E~i.-" .
.-
-30
'I Ex ."
.- /
- .0
Ex . 3
, .-
'Ex. "
, ,.-
~
~~
.-
i
~
5
i - 50
, !
- 60
, I
:
, I
I
-70 Fig . 14.
Y2/ v3-plots for all examples.
The a-plots give a useful evaluation of the performance of the system, although there are some weaknesses in using such overall descriptions. A better evaluation can therefore be obtained by also inc luding curves that show how the indiv idual disturbance variables influence the different output variables. In producing the above results we have used the computer-aided sy st ems DAREK (Pedersen, P~hner and Solheim, 1972) and CYPROS (Tyss~, 1980) at the Division of Engine ering Cybernetic s, The Norwegian Institute of Technology.
REFERENCES Doyle, J.C. and G. Stein (1981). Multivariable feedback design: Concepts for a classical/modern synthesis. IEEE Trans. Autom. Control, AC-26, 4-16 Klirnmel, M. and E. Oh~981). Practical evaluation of geometric control. 8 IFAC Congress, Kyoto, Aug. 24-28,1981. Mac Farlane, A.G.J. (1981). Characteristic and principal gains and phase s and their use as multivariable control design tools. AGARD Lecture Se ri e s No. 117, Paper No. 2, 1-34. Pedersen, J.O., F. P~hn e r and O.A. Solheim (1972). Computer aided de sign of multivariable control s ys t e ms. 5 . IFAC Congr e ss, Paris, June 197 2 . Solheim, O.A. (1972). Des i gn of optimal control systems with pr e scribed eigenvalues. Int. J. Contr., 15, 143-160. Solheim, O.A. (1980). On the-Us e of loworder Riccati equations in the design of a class of f ee dback contro ll e rs and estimators. MIC, 1,231-245. Ta kamat su, T.~ Hashimo t o and Y. Nakai (1979). A ge ometri c a pproac h t o multivariable control sy stem de s ign of a distillation c olumn. Automa ti ca ,~, 38 7-402. Ty ss~, A. (1980) . CYPROS - c yb e rnetic pr ogram packages . MIC, 1,215-229 . Wonham, W. M. (1979).--Ceome tri c state-space theory in linear multiva riable control: A status report. Automati c a,~, 5-13.