- Email: [email protected]
Design of feedback controllers by two-stage methods
Design of feedback controllers by two-stage methods * Magdi S. Malunoud Department
of Electrical
Engineering,
Kuwait
University,
P.O. Box 5969, Kuwait
Yuliu C&en* Department of Electrical Engineering, 204 George Street, Glasgow Cl, UK (Received January 1982)
University
of Strathclyde,
Royal
College Building,
When the eigenvalues of linear, time-invariant discrete systems are widely separated in absolute value, it is shown that the design of state or output feedback controllers can be best approached by two-stage methods. In these methods, the feedback gain matrices are computed by separately placing slow and fast eigenvalues at desired locations. Applications to the problem of independently controlling the speed and torque of petrol engine/dynamometer test rig have demonstrated the potential of the two stage feedback design methods. Key words:
mathematical
Introduction The design of feedback controllers for linear dynamical systems is a fundamental problem in control engineering. The importance of these controllers stem from the wellknown fact’ that feedback control systems have desirable performance characteristics. Using the state space approach there are two basic methods of feedback control design: eigenvalue assignment’ and linear-quadratic theory.3 Both methods are known to be applicable for continuous- and discrete-time systems. With the recent development in microprocessor technology, it becomes important to consider the design of feedback systems using digital controllers. This in turn motivates the study of discrete systems using linear state and output feedback schemes. For a class of discrete systems whose eigenvalues are widely separated in absolute values it is shown4-9 that the system behaviour can be approbated to first order by two lower-order, decoupled subsystems operating on different time-scales. The first subsystem describes the long-term operation and is called the slow subsystem. The second subsystem represents the operation in the short run and is termed the fast subsystem. This means that for discrete two-time-scale systems,4 independent control of the fast and slow modes can be easily achieved by using two separate gain matrices. *On leave from Tsinghua University, Peking, People’s Republic of China 0307-904X/83/0316346/$03.00 0 1983 Butterworth & Co. (Publishers) Ltd.
model,
feedback
controllers,
twostage
methods
The objective of this paper is to apply the two-stage feedback design methods to the problem of independently controlling the speed and torque of an internal combustion engine and dynamometer test rig. Using two different models, it has been shown that using either state or output feedback control schemes the shaft torque and dynamometer speed can be regulated through separate gains. The results obtained on typical data of a petrol engine representative of the type used to power medium-sized passenger cars have demonstrated the potential utility of the twostage design methods.
Analysis A typical feedback control system is shown in Figure 1. We consider that the plant plus the sampler and hold unit is modelled at the kth discrete instant by the difference equations: x(k + 1) = Ax(k) + Bu(k)
x(0) = xa
04
Y(k) = ww (lb) where x(k) is an ndimensional state vector, u(k) is an mdimensional output vector. The matrices A, B and Care constant with appropriate dimensions. It will now be assumed that system (1) is completely controllable and completely observable in the sense that: rank [II, AB, . . . , A”-‘B]
= n
rank[C’, A’C’, . . . , A”-“C’]
Appt. Math. Modelling,
(24
= n
(2b)
1983, Vol. 7, June
163
Design of feedback
controllers
Sampler plus hold
by two-stage
’ Contra’ Input
Physical
$:&es
M. S. Mahmoud
and Y. Chen
Meosurmg ’ devices
plant
l
unit
methods:
Control
I
0
Do=
output
C,(/, - A4)-l Bz
variables Figure
7
Typical
discrete-time
control
Ii is the nj x “j identity matrix, and a fast model of order n, represented by:
system
xf(k + 1) =,4,x/(k) Moreover, we consider that system (1) is asymptotically stable which means that all the eigenvalues of the matrix A are located within the unit disc in the complex plane and on its circumference. In this paper, we are interested in discrete models which possessthe mode-separation property.4 This means that the eigenvalues are divided into a cluster centered around the origin and a cluster distributed near the unit circle. It is known from linear system analysis’ that the response of the first cluster is only important during a short transient period. After that period, the system behaviour is governed by the second cluster of eigenvalues. For this reason we term the eigenvalues of the first cluster the fast modes of system (1) whereas the eigenvalues of the second cluster are termed slow modes. Let the number of slow eigenvalues be ~1 and the number of fast eigenvalues be n2 = II - tll. By rearranging system (l), through permutation and scaling of states,4 it can be put in the form: x,(k + I) = .4,x,(k)
+ A2x2(k) + B+(k) x,(O)
xz(k + 1) = A,xl(k)
+ &x,(k)
=x10
Pa)
+ B+(k)
x2(0) =x20
(3b) (4)
rank[C,] = p1
(5)
The different matrices in equations (3) and (4) have compatible dimensions. The interpretation of (4) is that each row in the output matrix C has one entry which implies that the accessible states are directly and independently measured. It should be noted that the transformation of system (1) into form (3) does not change the eigenvalue distribution. It has been shown using the explicitly invertible linear transformation4” of the discrete quasi-steady state assumptionbm8that systems (3) and (4) are approximated to firstorder by two lower-order models: a slow model of order n I described by:
y,(k)
= Coxdk) =
L&(k)
x,(O) = xl0
(6a)
+ Dous v:&)l’
(6b)
where: A,=A,+A*(I,-A4)-‘A3 B.
=
B1
+ &(I2
Va) -A,)-’
Bz
‘64 Appl. Math. Modelling, 1983, Vol. 7, June
U’b)
+ Bzuf(k)
(84
Xf(O) =x2(0) --xzm
u&> =[ 1 0
xr (k)
C2
= L&(k) &WI’
(8b)
Remarks (I) If it is assumed that system (3) is asymptotically stable, the invertibility of (I2 - A4) is always guaranteed. (2) The quantity E(O) is the initial value of the quasisteady state: Fz(k) = (12 - A,)-%,x,(k)
+ &us(k))
(9)
which has been inrroduced4-’ to account for the separation of modes in discrete systems. (3) The vectors u,(k) and uf(k) are the slow and fast parts of the control input u(k) such that: (10) u(k) = u,(k) + uf(k) In fact, this type of division in the control vector allows for a separate feedback design as we shall see in the next section. (4) It has been established4 that the eigenspectrum of system (3) can be approximated to a first-order perturbation by the disjoint union of the eigenspectra of A0 and A,; this is:? 44
wheres,(k),~~(k) are n,, No state vector components and y,(k),y,(k) are p, ,p2 output vector components. It is assumed in equation (4) that:
x,(k + 1) = /lox,(k) + Bou,(k)
1
=
440)
”
@4)
+
o+w
(11)
where /L is the mode-separation factor defined by: ~1= I max{&14 - (I2 -
A4)-‘A2
)) I/lmin{a(A0))l
(12)
Next we examine the problem of designing feedback controllers for system (3) by utilizing information from the subsystems (6) and (8).
Two-stage feedback
design
In this section, we consider two different schemes for designing feedback controllers using separate gain matrices. The first scheme is based on the assumption that all state variables are accessible for forming feedback signals. The second scheme uses only the output variables in generating the feedback control inputs. State feedback scheme
The control design scheme by state feedback requires the availability of all states for generating feedback signals. In practice, this would require the installation of sensors and measuring devices. By virtue of the fact that the state variables provide a complete picture of the internal of matrix function n(p) of B positive O(pm)if there exist positiveconstantsC and p* t A vector Cpm
for all fi Q Jo*
scalar such
p is said to be that
) n(p)
I <
Design of feedback
dynamics of the system, linear state feedback would then provide the best possible control. Given the slow model (6a) and the fast model (8a), we define: u,(k) = GO-M)
(13a)
U/(k) = Gf-Qk)
(13b)
where Go and G, are feedback gain matrices to be determined subject to design specifications. It has been established’jt7 that the composite control:
u(k) = ~[kG~k--W1&l -GfU,--W1&hW
Go + GfxzW
(14)
when applied to system (3) provides first-order perturbations to the closed-loop trajectories, that is: x 1(k) = x,(k) + 00~)
(Isa)
x,(k) = (I2-A4)-‘(As+B2Go)X,(k) + xf@) + 001)
(15b)
u(k) = u,(k) + q-04 + 001)
(15c)
where I2 is the (n2 x nz) matrix identity. The feedback design procedure is then: (1) Select two distinct sets of arbitrary n, large and n, small eigenvalues (2) compute the gain matrix Gf to place the eigenvalues of (A4 + Bz Gr> at the n2 desired locations (3) compute the gain matrix Go to place the eigenvalues of (A, + B. Go) at the n 1 desired locations (4) implement the feedback control (14). We note that the above design procedure requires separate calculation of gain matrices and yields ‘an order ofp’ approximation to the desired closed-loop eigenvalues.
feedback s-vsrem Here, we consider that only the output variablesy,(k) andy,(k) are accessible for generating the feedback control signals. In order to facilitate implementation of the design in two stages, we assume that:
Ourput
A3 = PC,
(16) which implies that the effect of the slow part on the fast part, which is known to be weak,7 is linearly related to the output of the slow part. This is not a restrictive assumption since there exists a wide classof problems that satisfies (16), an example of which is treated in the next section. In view of (5), the matrix P is computed as: P = A3C;(CI C:)-’
controllers
by two-stage
methods:
M. S. Mahmoud
when applied to systems (6), (8) and’ f3), respectively yield first-order approximations to the closed-Ioopoutput trajectories in the sense that: Y 16) = Y l,(k) + 001) YzW =
GOa)
C,(I,--A,)-‘(P+B,G,)y,,(k)
+uvW
+ 001)
u(k) = u,(k) + q(k)
Wb)
+ 001)
WC)
Here again, G1 and G2 can be separately designed according to the slow and fast subsystems performance specification and implemented as the composite control defined by (19). Using eigenvalue placement techniques,2 the gain matrices G1 and G2 can be designed to place min(nl, pl) and min(n2, p2) eigenvalues arbitrarily close to their desired locations. A block diagram of the control systems designed by two-stage method is shown in Figure 2.
Applications A variety of investigations on internal combustion engines have been carried out using engine/dynamometer test rigs.“.” Recent concern with pollution has caused further studies on petrol and diesel engines to become necessary with a view to reducing their contribution to pollution of the atmosphere. For this purpose, it is desirable to be able to control the torque and speed developed by such engines independently so that the engines can be subjected to a test cycle which will simulate typical road driving conditions. The purpose of this section is to study the design of control schemes for such a system to see how useful two-stage design methods are in providing independent control of torque and speed. An existing analogue model of an engine/dynamometer test rig” is used throughout this study, and two different models of the system are considered” having order 3 and 5. The engine considered is a petrol engine representative of the type used to power medium-sized passenger cars. The energy developed by the engine is controlled by the input voltage to the throttle servo system used on the test rig. The dynamometer, which acts as a load for the engine, consumes energy at a rate determined by the input voltage to the dynamometer field-current controller. The engine/ dynamometer configuration is shown schematically in Figure 3. By linearizing the whole system model about a typical operating point, an electrical analogue model” is shown in Figure 4 with experimental values for the various elements. Here, the engine is represented as a constant velocity source VE with internal resistance RE. The inertia of the engine is represented by the capacitance CE. The dynamometer is
(17)
Suppose now that: u,(k) = Glydk)
(18a)
u#)
(18b)
= Gy&)
- Control
(19)
pSFuTp’er
+ slgnol
are the output feedback controls for the subsystems (6) and (8) where the gains G1 and Gz contain the design parameters. By following parallel development to that of the previous section it has been shown9 that the output feedback controls (18a), (18b) and:
u(k)= { [I,,,- GzWI, - A# &I GI -G,C,(I,--A,)-‘P)y,(k) + Gzydk)
and Y. Chen
hold
Control Input
’
System model
Slw gain matrix
P
Fast
modes
Slow
modes
I*
1 l
Fast goln matrix Figure
2
Two-stage
feedback
control
system
Appl. Math. Modelling, 1983, Vol. 7, June 165
Design of feedback
controllers
by two-stage
methods:
Field current
Throttle setting
1
1
Engine
Coupling
shaft
Dynamometer
1Torque Figure
3
M. S. Mahmoud
Engine/dynamometer
1
Speed
schematic
and Y. Chen
where: dynamometer rotor speed x = engine speed [ shaft torque
1
throttle servo voltage II= r Ldynamometer source currentA
The eigenvalues of this model are 0.8275,0.283 +iO.3030 which shows that the open-loop separation ratio has the value 0.5. It is obvious that n, = 1 and n, = 2. The slow subsystem is described by: A0 = 0.7885 B. = [0.1970
co=[l.::21
l]
-0.241
Do=
[0.0:84
-0.:59,1
The fast subsystem is given by: 0.4704
Ad= B2 = RE
= l.lQR
RD=
Rs
q
c,
= 0.447F
Figure
455mR
4
Electrical
analogue
3R
RW
= 154pR
LS = 0.532mH
Lw
=
C,
Cw
= 7.57kF
= 1.38F
1.2mH
model
represented by the constant current source ID with internal resistance RD. The shaft connecting the engine to the dynamometer is modelled by the combination of the dissipator R, and the inductance L,, where the shaft is essentially considered as a rotational spring. The dynamics of the dynamometer clamping gear are represented by the dissipator R,, the capacitance C,, and the inductance, L,. The control studies to be described here were carried out on two versions of this basic model by using a pertinent sampler plus hold units. Third-order model One model of the test rig considered is for the case where the dynamometer caseing was firmly clamped to the reference frame. This mode of operation allows the design control systems with a marginally greater bandwidth,” and results in a simple third-order model, where the parallel combination of R,, C,, f., are replaced by a direct connection as in Figure 3. Using a sampling period of 0.18 s with zero-order hold, the discrete model of the type (3) and (4) is given by the following matrices: 0.0866
A=
0.4704 0.1731
B =
[
0.1637
-0.2056
0.2010
-0.2155
0.0169
0.0152
0.0733 -0.4206 0.2027 I
1
Appl.
Math. Modelling,
1983, Vol. 7, June
0.2010 [ 0.0169
0.2027 I -0.2155
0.0152 I
c, = [O 1.O] Since the dynamometer rotor speed (output of fast subsystem) are directly measurable, hence we can apply the output feedback design scheme. Note that condition (5) is satisfied. To calculate the gains Gr and Gz of (18), we place the slow eigenvalue at 0.88 and the fast eigenvalue at 0.2 kjO.3 yield:
[ 1 [ 1 [ 1 -2.0726
Cl =
-2.0726 -9.9615
C2=
-6.8717
From (17) we obtain: P=
0.43 12
-0.3262
and using (19) we obtain the output feedback control: u(k) =
-4.652
-9.961
-3.852
-6.872
1y(k)
This control law results in the closed-loop eigenvalues 0.8978,0.1258 kjO.3407 which are close to the desired eigenvalues. We note that the inherent oscillation has not been eliminated and this is mainly due to the nature of output feedback design.’ If we can install a device to measure the engine speed, we can then apply the state feedback design scheme. Assuming that this job can be done, we choose the desired eigenvalues to be 0.8,0.3,0.2. The use of dyadic-type eigenvalue assignment’ gives the gain matrices Gr and G2 of (13) as: Go=
Gf’
166
[ 0.1731
-0.4206
[
-0.2608 -0.2608 -0.7531 -0.7531
1
-5.773 -5.773
1
Design of feedback
controllers
From (14) the closed-loop state feedback control law is given by: u(k) =
-0.7077
-0.7855
-5.8005
1x(k)
The corresponding closed-loop eigenvalues are 0.7885, 0.3012,0.1801, which again are ‘order of’ approximations to the desired eigenvalues. The results of both design schemes validate the analysis of the section on two-stage feedback design. Fifth-order model This model is an extension of the third-order model by including the dynamics of an existing dynamometer fieldcurrent amplifier. The resulting model is now of order 5 and has the state variables as the dynamometer rotor speed, shaft torque, engine speed and current amplifier states. Using a sampling period of 0.1928 s and zero-order hold, which is selected to ensure matching the response of the continuous model, the discrete model of type (3) and (4) is given by: 0.8070 A=
0
0.0092
-0.0267
0.5527
0.0171 -0.0002
-0.1998
5.9560
Oil599
-0.0018
0
-0.0381
-5.0795
0
_ 0.0243
-6.8493
0 B=
0
0.0012 -0.2576 0
0.0003 -0.3805
0.8511
0.0766
-0.0106
0.7019
-0.0832
0
22.3995
0.1418 0
0
0 0 1.0
B,,=
0.7621
1 1
0.6885
0
1.0492
0.0899
-0.0179 I
c, = [ -0.2213 Do=
0
-0.0294
0 0 0.7648
C,=[O
M. S. Mahmoud
and Y. Chen
;;;;:;I 0
1 .O]
Through the use of auxiliary devices, we can consider that all the state variables become available for generating feedback signals, and proceed to apply the state feedback design scheme. The desired eigenvalues are to be placed at 0.8,0.7, 0.0999, -0.2026, -0.2173 to eliminate system oscillation. The gain matrix Cr is computed as: Go=
0.0076
-0.0913
0.0076
-0.0913 I
whereas the fast gain Gf takes the form: -0.2861
0.0011
-0.0787
G,= [ -0.2861
0.0011
-0.0787
1
From (14), the composite control law is given by: u(k) =
0.0541
0.0301
-0.2877
0.0012
-0.0784
1x(k)
This state-feedback contol yields the closed-loop eigenvalues as 0.8,0.7001,0.0998, -0.2179, -0.2021 which are very close to the desired ones. In the case of an output feedback design scheme, we n place the desired slow eigenvalue at 0.88 and obtain the gain matrix: 3.7690
[ 1 0.5276
For the desired fast eigenvalues to be positioned at -0.16 * j0.16, we obtain the gain matrix:
The open-loop eigenvalues are 0.7487,0.7476, -0.2083 + jO.2274,0.0213, which shows that the static separation ratio has the value of 0.4125. It is readily seen that this model has two slow states (nr = 2) and three fast states (nz = 3). Direct calculation gives the slow subsystem matrices as: A,,=
methods:
B2 = [ ;;L;;
Cl =
0.0257
0 1.0 0 0 c= [ 0
0.2311
0
by two-stage
1.o 8.191 I 0 -0.1439
1
and the fast subsystem matrices as:
-0.038 1 -0.0018 0.0003
0 -0.2576 -0.3805
1
From (19), the output feedback control law takes the form: u(k) =
0.6644 -0.2697
-0.2614 0.2932
1y(k)
and yields the closed-loop eigenvalues 0.9069,0.9038, -0.1864 kjO.1356, -0.0076. It is interesting to note that these eigenvalues are close to the desired ones. From the above results, we can draw the following conclusions: (1) The two-stage feedback design schemes, based on the multiple-time-scale analysis,4-9 provide effective methods for separate eigenvalue assignment. (2) Since the output variables of the engine system model have different time responses, independent control actions can easily be obtained which enables the design engineer to improve the performance of the engine system. (3) In general, state feedback control provides more robust results than output control scheme on the expense of using more measuring devices. (4) On comparing the results obtained in this work with those of others, ‘i we note that the elements of the feedback gain matrix are smaller than those obtained by applying state-variable decoupling approach.” On the other hand, application of the inverse Nyquist array design method”
Appl.
Math. Modelling,
1983, Vol. 7, June
167
Design of feedback
controllers
by two-stage
methods:
M. S. Mahmoud
requires the use of compensators in addition to unity output feedback. It is thus clear that the present two-stage methods are simpler and yield more manageable control schemes.
Conclusions This paper has been concerned with the design of digital controllers for linear, time-invariant systems using state and output feedback schemes. It is shown that when the eigenvalues of the discretized model are widely separated in absolute value, then the model behaviour can be approximated to first order by two lower-order submodel operating on two different scales. These submodels are the low submodels which represents long-term studies and the fast submodel which represents short-term studies. It is further shown that the feedback control design, whether using state or output variables, can be implemented in two-stages by utilizing two independent gain matrices. Application of the two-stage design methods to the problem of independently controlling the speed and torque of a petrol engine/dynamometer test rig system is undertaken. By using two different models of the engine system, one of order three and the other of order five, the results of feedback control design have clearly showed the effectiveness and potential utility of the two-stage methods.
168
Appl.
Math.
Modelling,
1983,
Vol.
7, June
and Y. Chen
References Kailath, T. ‘Linear systems’, Prentice-Hall, Inc., N.J., 1980 IEE 1979, 126,549 Kwakernaak, H. and Sivan, R. ‘Linear optimal control systems’, Wiley Interscience, N.Y., 1972 Mahmoud, M. S., Chen, Y. and Singh, M. G. ‘Discrete twotime-scale systems’, Control systems centre report No. 497, UMIST, England, 1980 (submitted for publication) 5 Mahmoud, M. S. and Chen, Y. ‘Feedback design of discrete two-time-scale systems’, Control Systems Centre Report No. 5 12, UMIST, UK, 1981 (submitted for publication) 6 Mahmoud, M. S. ‘Order reduction and control of discrete systems’, Proc. European Con/. Circuit Theory and Design, 25-28 August, 1981, The Netherlands, pp. 619-625 I Mahmoud, M. S., Chen, Y. and Singh, M. G. ‘On the eigenvalue assignment in discrete systems with fast and slow modes’, Control Systems Centre Report No. 499, UK, England, 1981 (submitted for publication) 8 Mahmoud, M. S. ‘Design of observer-based controllers for a class of discrete systems’, Automatica (to appear) 9 Mahmoud, M. S., Chen, Y. and Singh, M. G. ‘A two-stage outout feedback design’, Control Systems Centre Report No. 5 16, UMIST, UK, 1981 (submitted for publication) 10 Monk, J. and Comfort, J. ‘Mathematical model of an internal combustion engine and dynamometer test rig’,Measurement Control 1970.3, T93-TlOO 11 Munro, N. and Hirbod, S. N. ‘Multivariable control of an engine/dynamometer test rig’, Proc. Seventh IFAC Congress, Helsinki, 1978, pp. 369-376 12. Falb, P. L. and Wolovich, W. A. ‘Decoupling in the design and synthesis of multivariable control systems’, IEEE Trans. on Automat. Contr. 1967, AC-12,651
MUNO,N. ‘Pole assignment’,Proc.
of Electrical
Engineering,
Kuwait
University,
P.O. Box 5969, Kuwait
Yuliu C&en* Department of Electrical Engineering, 204 George Street, Glasgow Cl, UK (Received January 1982)
University
of Strathclyde,
Royal
College Building,
When the eigenvalues of linear, time-invariant discrete systems are widely separated in absolute value, it is shown that the design of state or output feedback controllers can be best approached by two-stage methods. In these methods, the feedback gain matrices are computed by separately placing slow and fast eigenvalues at desired locations. Applications to the problem of independently controlling the speed and torque of petrol engine/dynamometer test rig have demonstrated the potential of the two stage feedback design methods. Key words:
mathematical
Introduction The design of feedback controllers for linear dynamical systems is a fundamental problem in control engineering. The importance of these controllers stem from the wellknown fact’ that feedback control systems have desirable performance characteristics. Using the state space approach there are two basic methods of feedback control design: eigenvalue assignment’ and linear-quadratic theory.3 Both methods are known to be applicable for continuous- and discrete-time systems. With the recent development in microprocessor technology, it becomes important to consider the design of feedback systems using digital controllers. This in turn motivates the study of discrete systems using linear state and output feedback schemes. For a class of discrete systems whose eigenvalues are widely separated in absolute values it is shown4-9 that the system behaviour can be approbated to first order by two lower-order, decoupled subsystems operating on different time-scales. The first subsystem describes the long-term operation and is called the slow subsystem. The second subsystem represents the operation in the short run and is termed the fast subsystem. This means that for discrete two-time-scale systems,4 independent control of the fast and slow modes can be easily achieved by using two separate gain matrices. *On leave from Tsinghua University, Peking, People’s Republic of China 0307-904X/83/0316346/$03.00 0 1983 Butterworth & Co. (Publishers) Ltd.
model,
feedback
controllers,
twostage
methods
The objective of this paper is to apply the two-stage feedback design methods to the problem of independently controlling the speed and torque of an internal combustion engine and dynamometer test rig. Using two different models, it has been shown that using either state or output feedback control schemes the shaft torque and dynamometer speed can be regulated through separate gains. The results obtained on typical data of a petrol engine representative of the type used to power medium-sized passenger cars have demonstrated the potential utility of the twostage design methods.
Analysis A typical feedback control system is shown in Figure 1. We consider that the plant plus the sampler and hold unit is modelled at the kth discrete instant by the difference equations: x(k + 1) = Ax(k) + Bu(k)
x(0) = xa
04
Y(k) = ww (lb) where x(k) is an ndimensional state vector, u(k) is an mdimensional output vector. The matrices A, B and Care constant with appropriate dimensions. It will now be assumed that system (1) is completely controllable and completely observable in the sense that: rank [II, AB, . . . , A”-‘B]
= n
rank[C’, A’C’, . . . , A”-“C’]
Appt. Math. Modelling,
(24
= n
(2b)
1983, Vol. 7, June
163
Design of feedback
controllers
Sampler plus hold
by two-stage
’ Contra’ Input
Physical
$:&es
M. S. Mahmoud
and Y. Chen
Meosurmg ’ devices
plant
l
unit
methods:
Control
I
0
Do=
output
C,(/, - A4)-l Bz
variables Figure
7
Typical
discrete-time
control
Ii is the nj x “j identity matrix, and a fast model of order n, represented by:
system
xf(k + 1) =,4,x/(k) Moreover, we consider that system (1) is asymptotically stable which means that all the eigenvalues of the matrix A are located within the unit disc in the complex plane and on its circumference. In this paper, we are interested in discrete models which possessthe mode-separation property.4 This means that the eigenvalues are divided into a cluster centered around the origin and a cluster distributed near the unit circle. It is known from linear system analysis’ that the response of the first cluster is only important during a short transient period. After that period, the system behaviour is governed by the second cluster of eigenvalues. For this reason we term the eigenvalues of the first cluster the fast modes of system (1) whereas the eigenvalues of the second cluster are termed slow modes. Let the number of slow eigenvalues be ~1 and the number of fast eigenvalues be n2 = II - tll. By rearranging system (l), through permutation and scaling of states,4 it can be put in the form: x,(k + I) = .4,x,(k)
+ A2x2(k) + B+(k) x,(O)
xz(k + 1) = A,xl(k)
+ &x,(k)
=x10
Pa)
+ B+(k)
x2(0) =x20
(3b) (4)
rank[C,] = p1
(5)
The different matrices in equations (3) and (4) have compatible dimensions. The interpretation of (4) is that each row in the output matrix C has one entry which implies that the accessible states are directly and independently measured. It should be noted that the transformation of system (1) into form (3) does not change the eigenvalue distribution. It has been shown using the explicitly invertible linear transformation4” of the discrete quasi-steady state assumptionbm8that systems (3) and (4) are approximated to firstorder by two lower-order models: a slow model of order n I described by:
y,(k)
= Coxdk) =
L&(k)
x,(O) = xl0
(6a)
+ Dous v:&)l’
(6b)
where: A,=A,+A*(I,-A4)-‘A3 B.
=
B1
+ &(I2
Va) -A,)-’
Bz
‘64 Appl. Math. Modelling, 1983, Vol. 7, June
U’b)
+ Bzuf(k)
(84
Xf(O) =x2(0) --xzm
u&> =[ 1 0
xr (k)
C2
= L&(k) &WI’
(8b)
Remarks (I) If it is assumed that system (3) is asymptotically stable, the invertibility of (I2 - A4) is always guaranteed. (2) The quantity E(O) is the initial value of the quasisteady state: Fz(k) = (12 - A,)-%,x,(k)
+ &us(k))
(9)
which has been inrroduced4-’ to account for the separation of modes in discrete systems. (3) The vectors u,(k) and uf(k) are the slow and fast parts of the control input u(k) such that: (10) u(k) = u,(k) + uf(k) In fact, this type of division in the control vector allows for a separate feedback design as we shall see in the next section. (4) It has been established4 that the eigenspectrum of system (3) can be approximated to a first-order perturbation by the disjoint union of the eigenspectra of A0 and A,; this is:? 44
wheres,(k),~~(k) are n,, No state vector components and y,(k),y,(k) are p, ,p2 output vector components. It is assumed in equation (4) that:
x,(k + 1) = /lox,(k) + Bou,(k)
1
=
440)
”
@4)
+
o+w
(11)
where /L is the mode-separation factor defined by: ~1= I max{&14 - (I2 -
A4)-‘A2
)) I/lmin{a(A0))l
(12)
Next we examine the problem of designing feedback controllers for system (3) by utilizing information from the subsystems (6) and (8).
Two-stage feedback
design
In this section, we consider two different schemes for designing feedback controllers using separate gain matrices. The first scheme is based on the assumption that all state variables are accessible for forming feedback signals. The second scheme uses only the output variables in generating the feedback control inputs. State feedback scheme
The control design scheme by state feedback requires the availability of all states for generating feedback signals. In practice, this would require the installation of sensors and measuring devices. By virtue of the fact that the state variables provide a complete picture of the internal of matrix function n(p) of B positive O(pm)if there exist positiveconstantsC and p* t A vector Cpm
for all fi Q Jo*
scalar such
p is said to be that
) n(p)
I <
Design of feedback
dynamics of the system, linear state feedback would then provide the best possible control. Given the slow model (6a) and the fast model (8a), we define: u,(k) = GO-M)
(13a)
U/(k) = Gf-Qk)
(13b)
where Go and G, are feedback gain matrices to be determined subject to design specifications. It has been established’jt7 that the composite control:
u(k) = ~[kG~k--W1&l -GfU,--W1&hW
Go + GfxzW
(14)
when applied to system (3) provides first-order perturbations to the closed-loop trajectories, that is: x 1(k) = x,(k) + 00~)
(Isa)
x,(k) = (I2-A4)-‘(As+B2Go)X,(k) + xf@) + 001)
(15b)
u(k) = u,(k) + q-04 + 001)
(15c)
where I2 is the (n2 x nz) matrix identity. The feedback design procedure is then: (1) Select two distinct sets of arbitrary n, large and n, small eigenvalues (2) compute the gain matrix Gf to place the eigenvalues of (A4 + Bz Gr> at the n2 desired locations (3) compute the gain matrix Go to place the eigenvalues of (A, + B. Go) at the n 1 desired locations (4) implement the feedback control (14). We note that the above design procedure requires separate calculation of gain matrices and yields ‘an order ofp’ approximation to the desired closed-loop eigenvalues.
feedback s-vsrem Here, we consider that only the output variablesy,(k) andy,(k) are accessible for generating the feedback control signals. In order to facilitate implementation of the design in two stages, we assume that:
Ourput
A3 = PC,
(16) which implies that the effect of the slow part on the fast part, which is known to be weak,7 is linearly related to the output of the slow part. This is not a restrictive assumption since there exists a wide classof problems that satisfies (16), an example of which is treated in the next section. In view of (5), the matrix P is computed as: P = A3C;(CI C:)-’
controllers
by two-stage
methods:
M. S. Mahmoud
when applied to systems (6), (8) and’ f3), respectively yield first-order approximations to the closed-Ioopoutput trajectories in the sense that: Y 16) = Y l,(k) + 001) YzW =
GOa)
C,(I,--A,)-‘(P+B,G,)y,,(k)
+uvW
+ 001)
u(k) = u,(k) + q(k)
Wb)
+ 001)
WC)
Here again, G1 and G2 can be separately designed according to the slow and fast subsystems performance specification and implemented as the composite control defined by (19). Using eigenvalue placement techniques,2 the gain matrices G1 and G2 can be designed to place min(nl, pl) and min(n2, p2) eigenvalues arbitrarily close to their desired locations. A block diagram of the control systems designed by two-stage method is shown in Figure 2.
Applications A variety of investigations on internal combustion engines have been carried out using engine/dynamometer test rigs.“.” Recent concern with pollution has caused further studies on petrol and diesel engines to become necessary with a view to reducing their contribution to pollution of the atmosphere. For this purpose, it is desirable to be able to control the torque and speed developed by such engines independently so that the engines can be subjected to a test cycle which will simulate typical road driving conditions. The purpose of this section is to study the design of control schemes for such a system to see how useful two-stage design methods are in providing independent control of torque and speed. An existing analogue model of an engine/dynamometer test rig” is used throughout this study, and two different models of the system are considered” having order 3 and 5. The engine considered is a petrol engine representative of the type used to power medium-sized passenger cars. The energy developed by the engine is controlled by the input voltage to the throttle servo system used on the test rig. The dynamometer, which acts as a load for the engine, consumes energy at a rate determined by the input voltage to the dynamometer field-current controller. The engine/ dynamometer configuration is shown schematically in Figure 3. By linearizing the whole system model about a typical operating point, an electrical analogue model” is shown in Figure 4 with experimental values for the various elements. Here, the engine is represented as a constant velocity source VE with internal resistance RE. The inertia of the engine is represented by the capacitance CE. The dynamometer is
(17)
Suppose now that: u,(k) = Glydk)
(18a)
u#)
(18b)
= Gy&)
- Control
(19)
pSFuTp’er
+ slgnol
are the output feedback controls for the subsystems (6) and (8) where the gains G1 and Gz contain the design parameters. By following parallel development to that of the previous section it has been shown9 that the output feedback controls (18a), (18b) and:
u(k)= { [I,,,- GzWI, - A# &I GI -G,C,(I,--A,)-‘P)y,(k) + Gzydk)
and Y. Chen
hold
Control Input
’
System model
Slw gain matrix
P
Fast
modes
Slow
modes
I*
1 l
Fast goln matrix Figure
2
Two-stage
feedback
control
system
Appl. Math. Modelling, 1983, Vol. 7, June 165
Design of feedback
controllers
by two-stage
methods:
Field current
Throttle setting
1
1
Engine
Coupling
shaft
Dynamometer
1Torque Figure
3
M. S. Mahmoud
Engine/dynamometer
1
Speed
schematic
and Y. Chen
where: dynamometer rotor speed x = engine speed [ shaft torque
1
throttle servo voltage II= r Ldynamometer source currentA
The eigenvalues of this model are 0.8275,0.283 +iO.3030 which shows that the open-loop separation ratio has the value 0.5. It is obvious that n, = 1 and n, = 2. The slow subsystem is described by: A0 = 0.7885 B. = [0.1970
co=[l.::21
l]
-0.241
Do=
[0.0:84
-0.:59,1
The fast subsystem is given by: 0.4704
Ad= B2 = RE
= l.lQR
RD=
Rs
q
c,
= 0.447F
Figure
455mR
4
Electrical
analogue
3R
RW
= 154pR
LS = 0.532mH
Lw
=
C,
Cw
= 7.57kF
= 1.38F
1.2mH
model
represented by the constant current source ID with internal resistance RD. The shaft connecting the engine to the dynamometer is modelled by the combination of the dissipator R, and the inductance L,, where the shaft is essentially considered as a rotational spring. The dynamics of the dynamometer clamping gear are represented by the dissipator R,, the capacitance C,, and the inductance, L,. The control studies to be described here were carried out on two versions of this basic model by using a pertinent sampler plus hold units. Third-order model One model of the test rig considered is for the case where the dynamometer caseing was firmly clamped to the reference frame. This mode of operation allows the design control systems with a marginally greater bandwidth,” and results in a simple third-order model, where the parallel combination of R,, C,, f., are replaced by a direct connection as in Figure 3. Using a sampling period of 0.18 s with zero-order hold, the discrete model of the type (3) and (4) is given by the following matrices: 0.0866
A=
0.4704 0.1731
B =
[
0.1637
-0.2056
0.2010
-0.2155
0.0169
0.0152
0.0733 -0.4206 0.2027 I
1
Appl.
Math. Modelling,
1983, Vol. 7, June
0.2010 [ 0.0169
0.2027 I -0.2155
0.0152 I
c, = [O 1.O] Since the dynamometer rotor speed (output of fast subsystem) are directly measurable, hence we can apply the output feedback design scheme. Note that condition (5) is satisfied. To calculate the gains Gr and Gz of (18), we place the slow eigenvalue at 0.88 and the fast eigenvalue at 0.2 kjO.3 yield:
[ 1 [ 1 [ 1 -2.0726
Cl =
-2.0726 -9.9615
C2=
-6.8717
From (17) we obtain: P=
0.43 12
-0.3262
and using (19) we obtain the output feedback control: u(k) =
-4.652
-9.961
-3.852
-6.872
1y(k)
This control law results in the closed-loop eigenvalues 0.8978,0.1258 kjO.3407 which are close to the desired eigenvalues. We note that the inherent oscillation has not been eliminated and this is mainly due to the nature of output feedback design.’ If we can install a device to measure the engine speed, we can then apply the state feedback design scheme. Assuming that this job can be done, we choose the desired eigenvalues to be 0.8,0.3,0.2. The use of dyadic-type eigenvalue assignment’ gives the gain matrices Gr and G2 of (13) as: Go=
Gf’
166
[ 0.1731
-0.4206
[
-0.2608 -0.2608 -0.7531 -0.7531
1
-5.773 -5.773
1
Design of feedback
controllers
From (14) the closed-loop state feedback control law is given by: u(k) =
-0.7077
-0.7855
-5.8005
1x(k)
The corresponding closed-loop eigenvalues are 0.7885, 0.3012,0.1801, which again are ‘order of’ approximations to the desired eigenvalues. The results of both design schemes validate the analysis of the section on two-stage feedback design. Fifth-order model This model is an extension of the third-order model by including the dynamics of an existing dynamometer fieldcurrent amplifier. The resulting model is now of order 5 and has the state variables as the dynamometer rotor speed, shaft torque, engine speed and current amplifier states. Using a sampling period of 0.1928 s and zero-order hold, which is selected to ensure matching the response of the continuous model, the discrete model of type (3) and (4) is given by: 0.8070 A=
0
0.0092
-0.0267
0.5527
0.0171 -0.0002
-0.1998
5.9560
Oil599
-0.0018
0
-0.0381
-5.0795
0
_ 0.0243
-6.8493
0 B=
0
0.0012 -0.2576 0
0.0003 -0.3805
0.8511
0.0766
-0.0106
0.7019
-0.0832
0
22.3995
0.1418 0
0
0 0 1.0
B,,=
0.7621
1 1
0.6885
0
1.0492
0.0899
-0.0179 I
c, = [ -0.2213 Do=
0
-0.0294
0 0 0.7648
C,=[O
M. S. Mahmoud
and Y. Chen
;;;;:;I 0
1 .O]
Through the use of auxiliary devices, we can consider that all the state variables become available for generating feedback signals, and proceed to apply the state feedback design scheme. The desired eigenvalues are to be placed at 0.8,0.7, 0.0999, -0.2026, -0.2173 to eliminate system oscillation. The gain matrix Cr is computed as: Go=
0.0076
-0.0913
0.0076
-0.0913 I
whereas the fast gain Gf takes the form: -0.2861
0.0011
-0.0787
G,= [ -0.2861
0.0011
-0.0787
1
From (14), the composite control law is given by: u(k) =
0.0541
0.0301
-0.2877
0.0012
-0.0784
1x(k)
This state-feedback contol yields the closed-loop eigenvalues as 0.8,0.7001,0.0998, -0.2179, -0.2021 which are very close to the desired ones. In the case of an output feedback design scheme, we n place the desired slow eigenvalue at 0.88 and obtain the gain matrix: 3.7690
[ 1 0.5276
For the desired fast eigenvalues to be positioned at -0.16 * j0.16, we obtain the gain matrix:
The open-loop eigenvalues are 0.7487,0.7476, -0.2083 + jO.2274,0.0213, which shows that the static separation ratio has the value of 0.4125. It is readily seen that this model has two slow states (nr = 2) and three fast states (nz = 3). Direct calculation gives the slow subsystem matrices as: A,,=
methods:
B2 = [ ;;L;;
Cl =
0.0257
0 1.0 0 0 c= [ 0
0.2311
0
by two-stage
1.o 8.191 I 0 -0.1439
1
and the fast subsystem matrices as:
-0.038 1 -0.0018 0.0003
0 -0.2576 -0.3805
1
From (19), the output feedback control law takes the form: u(k) =
0.6644 -0.2697
-0.2614 0.2932
1y(k)
and yields the closed-loop eigenvalues 0.9069,0.9038, -0.1864 kjO.1356, -0.0076. It is interesting to note that these eigenvalues are close to the desired ones. From the above results, we can draw the following conclusions: (1) The two-stage feedback design schemes, based on the multiple-time-scale analysis,4-9 provide effective methods for separate eigenvalue assignment. (2) Since the output variables of the engine system model have different time responses, independent control actions can easily be obtained which enables the design engineer to improve the performance of the engine system. (3) In general, state feedback control provides more robust results than output control scheme on the expense of using more measuring devices. (4) On comparing the results obtained in this work with those of others, ‘i we note that the elements of the feedback gain matrix are smaller than those obtained by applying state-variable decoupling approach.” On the other hand, application of the inverse Nyquist array design method”
Appl.
Math. Modelling,
1983, Vol. 7, June
167
Design of feedback
controllers
by two-stage
methods:
M. S. Mahmoud
requires the use of compensators in addition to unity output feedback. It is thus clear that the present two-stage methods are simpler and yield more manageable control schemes.
Conclusions This paper has been concerned with the design of digital controllers for linear, time-invariant systems using state and output feedback schemes. It is shown that when the eigenvalues of the discretized model are widely separated in absolute value, then the model behaviour can be approximated to first order by two lower-order submodel operating on two different scales. These submodels are the low submodels which represents long-term studies and the fast submodel which represents short-term studies. It is further shown that the feedback control design, whether using state or output variables, can be implemented in two-stages by utilizing two independent gain matrices. Application of the two-stage design methods to the problem of independently controlling the speed and torque of a petrol engine/dynamometer test rig system is undertaken. By using two different models of the engine system, one of order three and the other of order five, the results of feedback control design have clearly showed the effectiveness and potential utility of the two-stage methods.
168
Appl.
Math.
Modelling,
1983,
Vol.
7, June
and Y. Chen
References Kailath, T. ‘Linear systems’, Prentice-Hall, Inc., N.J., 1980 IEE 1979, 126,549 Kwakernaak, H. and Sivan, R. ‘Linear optimal control systems’, Wiley Interscience, N.Y., 1972 Mahmoud, M. S., Chen, Y. and Singh, M. G. ‘Discrete twotime-scale systems’, Control systems centre report No. 497, UMIST, England, 1980 (submitted for publication) 5 Mahmoud, M. S. and Chen, Y. ‘Feedback design of discrete two-time-scale systems’, Control Systems Centre Report No. 5 12, UMIST, UK, 1981 (submitted for publication) 6 Mahmoud, M. S. ‘Order reduction and control of discrete systems’, Proc. European Con/. Circuit Theory and Design, 25-28 August, 1981, The Netherlands, pp. 619-625 I Mahmoud, M. S., Chen, Y. and Singh, M. G. ‘On the eigenvalue assignment in discrete systems with fast and slow modes’, Control Systems Centre Report No. 499, UK, England, 1981 (submitted for publication) 8 Mahmoud, M. S. ‘Design of observer-based controllers for a class of discrete systems’, Automatica (to appear) 9 Mahmoud, M. S., Chen, Y. and Singh, M. G. ‘A two-stage outout feedback design’, Control Systems Centre Report No. 5 16, UMIST, UK, 1981 (submitted for publication) 10 Monk, J. and Comfort, J. ‘Mathematical model of an internal combustion engine and dynamometer test rig’,Measurement Control 1970.3, T93-TlOO 11 Munro, N. and Hirbod, S. N. ‘Multivariable control of an engine/dynamometer test rig’, Proc. Seventh IFAC Congress, Helsinki, 1978, pp. 369-376 12. Falb, P. L. and Wolovich, W. A. ‘Decoupling in the design and synthesis of multivariable control systems’, IEEE Trans. on Automat. Contr. 1967, AC-12,651
MUNO,N. ‘Pole assignment’,Proc.