- Email: [email protected]
Integrated Control of Distributed Parameter Systems
Cl lp \rig lll ([) 11-' .\( <:1111 11"11 1 01 l> i"d r i hll l t' d P"Lt tll t'l t'l" S \ " l t' III' . 1.11' .\IH.!, t'k, . ( ..ditol"lIi.1. IqX"
INTEGRATED CONTROL OF DISTRIBUTED PARAMETER SYSTEMS G. Capitani*, M. Tibaldi*, G. Bertoni* and A. Yousuff** "'/) I'!JIlrl llll' lIl or t ." ln lm llirs , SySlflll; alld Cumpllln Srinlrp, L' llil 'Prsi(l' of B ologlla. "iait' Risurgimnl lu 2, I -·J()] 36 Bulogll{/ , Ilaly """ /) I'!JIlrlll/f' lIl 0( ,\l ('rha ll i({ll E llg ill Pnillg {/ lId A[pr/wlli rs, Du'xl' l L' lI il'l'/'si ly. Ph i lad(,lphia. P A 19 /0 -1 . USA
Abstract. This work develops a new method for designing low-order dynami c compensators. This design procedure integrates the LOG-based reduced - order dynamic compensator synthesis with the low-authority algebraic controller synthesis. A comparison with a well-known method which exploits similar concepts is also presented. Keywords Reduced-order compensators; large scale systems; control system synthesis; controllers; multivariable control systems.
reasons. In the scientific literature many efforts are devoted to the design of reduced-order control systems (e.g. [Aubrun and others (1979)), [Verriest (1981)), [Jonckheere and Sllverman (1983)), [Yousuff and Skelton (1984a and 1984b], [Capltanl, Penati and Tibaldi (I984a and 1984b)1, [Ly and others (1985)]). This work aims to develop a new synthesis method for designing low-order compensators of large scale systems (LSS) suggested by Capitani , Tibaldi and Bertonl (1985). This method is in some sense Similar to the one presented by Aubrun and others (1979 and 1982) and Gupta and others (1981)
INTRODUCTION 1. The dynamic behavior of distributed parameter systems (oPSs) is represented by means of mathematical models formed by partial differential equations. However, in order to exploit the relevant results of the modern control theory about compensator synthesis, mathematical models formed by ordinary differential equations must be used. This kind of mathematical models (state space models) describe the dynamic behavior of a DPS with some accuracy only if the number of differential equations is very large. Let us summarize some terms quite fami 1iar to control system engineers (e.g. [Gupta and others ( 1981)), [Bertoni and others (1983))) and which we are going to use extensively in this work. The physical system to be controlled (in general a DPS) will be referred to as plant and the mathematical model (formed by ordinary differentia l equations) which is believed to represent the plant with the best accuracy (concerning the goal we are interested in) will be referred to as evaluation model (EM); finally, we will name
REDUCED-ORDER COMPENSATOR DESI GN 2. In general, we can outline a reduced-order compensator design method by the scheme shown in Fig. 1. In this methodological frame some steps can be disregarded In order to build a particular design method. For instance, step 2 can be obviously neglected when the EM has Riccat/-so/vab/e dimenSions, whl1e step 3 can be omitted if we want to design the highest order LOG compensator. Moreover note that, for the sake of generality, the scheme of Fig. 1 shows steps 2 and 3 separate ly. In fact, some model reduction methods (e.g. [Jonckheere and Silverman (1983)]) can be applied In step 3, but not in step 2. Finally, step 5 can be neglected when the dynamic order of the CSM is suffiCiently low.
control synthesis model (CSM) a mathematical model (of order lower than the order of the EM, in genera\) used for the design of (or a part of) a control system for the plant. In general , as said before, the dynamic order of the EM of a DPS is very high while a control system must have low dynamic order for implementability
269
~7t)
C. Ca pitalli 1'1 Il /.
Whatever is the method used to design the reduced-order compensator, it is well-known that the behavior of the overall system (plant+compensat or) may turn out to be quite poor (the overall system can be unstable even). Once we have designed a compensator via a synthesis method belonging to the scheme of Fig. 1, we can try to improve its performance by optimizing the parameters of the compensator itself (e.g. [Ly and others (1985)]) Another approach, aiming to design a satisfactory low-order compensator, consists on integrating the design method outlined in Fig. 1, based mainly on the LOG synthesis, with the design method of (algebraic) controllers based on Jacobi 's formula [Jacobi (1846)] concerning the eigenvalue perturbation theory. This kind of controllers was suggested by Aubrun ( 1980). In order to present the new method of reduced-order compensator syntheSiS, section 3 recalls Jacobl's formula for estimating the variations of the elgenvalues of a matrix and presents some comments on its use.
elgenvectors, Nk « r k holds for each k = 1, 2, ... , n), eqs. (3.3) can be approximated by eqs. (3.4). That is:
ll.\ :::: 8\
=
4T N
4
T
rk
, k = 1,2, ... , n. (3.5)
rk
As said before, let us rearrange the basic eq. (3.5), known as Jacobi 's formula, in a form suitable for the LDWB controller design. Given the linear, time-invariant model: x=Fx+6u,
y = H X,
(3 .6)
where x(nx 1) is the state vector, u(rx I) is the Input vector, y(mx 1) Is the output vector, and F, 6, H are matrices of suitable dimensions, let us assume that the input vector u is given by an algebraic feedback of the output vector through the matrix C(rxm):
u = C y.
(3 .7)
From eqs. (3.6) and (}.7) we have:
x = (F
LDWB CONTROLLERS
3. First of all this section presents the basic formulas concerning the eigenvalue perturbation theory. Then these equations are rearranged for the LDWB (L ow-Oamping Wide-Bandwidth ) controller design. Let F be a (nxn) matrix and \ ' k = 1, 2,., n, its eigenvalues. Let us denote by r k the right eigen-
+ 6 C H) x .
(3.8)
Because the state matrix variation is:
N
(3.9)
== 6CH,
eqs. (3.5) turn out to be:
ll.\::::8\ =
4T6CHrk
vector and by -'-r. the left eigenvector, both related
k = 1, 2, ... , n.
to \ . As it is well-known, they are defined by:
F rk = \ rk
1./ F =
\
,
1./,
= 1,2, ... , n,
(3 . 1)
EQs. (}. IO) give approximated estimates 8\ of
k = 1, 2, ..., n.
(3.2)
the act(Jal variations
k
Both each elgenvalue \ and each eigenvector r k and
-'-r. change if the matrix F changes into F+N. Let us denote such variations by ll.\ ,N k and ll.4. respect ive ly. Using eqs (3. 1) and (3.2) and similar equations for F+N, we can easily derive the following expressions:
4T N ( rk + N k ) 4 T ( r k + Nk ) k
= 1, 2,
ll.\ of the poles of the
model (}.8), due to the algebraic output feedback through the matrix C. These estimates are reliable if the variation M , defined by eq. (3.9), can be considered small in the sense stated before. Let us rewrite eqs. (}. IO) in a form more suitable for the LDWB design. To this aim let us define:
-'-r. == 6 T -'-r., r k == H r k ,
k = 1,2, ... , n. (3 .11)
Usually, -'-r. and r k are referred to as general.. , n.
(3 .3)
Defining:
IZed elgenyeclorsCleft and right respectively). Subst itut ing eqs (3. 1 1) into (}. 10) we have
8\ =
4T N r k 4T r k
(3. 10)
k = I, 2, ... , n,
(3.4)
we can note that, assum ing that pert(Jrbat ions are small (1. e. supposing that, with normalized
4T C
~
k
= 1, 2, ... , n.
(3. 12)
Moreover, let us rearrange the elements of the feedback matrix C selecting them by columns:
~I l
Illt t'l,( r a te d C Olltrol
q
==
[C 11 C21 ... CrI
C 12 C22 ... Cr2 C
(3.21)
C ... C ] T. (3. 13) rm 1m 2m
where cij is the generic element of the matrix C.
holds, then the function (3. 1g) is strictly convex and the necessary condition 0.20) is sufficient too. In this case eQ. (3 .20) has the unique solution
The number of elements of q is P =' mr. EQs. (3 12) can be rewritten in the form: k
= 1.2• ... • n.
(314)
where: SkT-
J../rk
and r kj , j = 1,2"
m, is the j-th element of the
°
generalized elgenvector r k . Defining (315)
eQs. (3. 14) can be written in a compact form 6), = S q = ( SR + j SJ ) q ,
(3. 16)
where: (317) The matrices SR e SJ are the real part and the Imaginary part of 5 respectively. Now we are able to outline a possible way for designing a LDWB controller. Fix a real vector T, representing the desired variations, and a weight-matrix W(nxn); then, using the equation: (3 .18) where
6~
variations function:
Is the vector of the real parts of the pole given
J = (6~ -
by
eQ. (3. 16),
T) T W (6~
minimize
- T) + qT q,
the
(3 .1g)
which depends only on q because of eQ. (3 .18) The design of the LDWBC consists in finding output feedback gains sufficiently small (to this aim the function (3 19) includes the term qT q) which give variations of the real parts of the poles of the system (38) sufficiently near to the desired values T.
The necessary condition for a minimum of (319) Is: d.)(q)/d(q) = 0, tl'lat Is: (SRT W ~ + Jp)
If:
q
The weight-matrix W Is usually chosen positive Oeflnlte and then eQ. 0 .21l hOlOs. The term qTq In eQ. (3. 19) penalizes large values of the feedback gains and then It alms to give small perturbatlons. Furthermore the term qTq guarantees the uniqueness of the solution of the minimum problem when W Is positive Oeflnlte. Finally, for a flxeO weight - matrix W>O, missing the term qTq causes a worse conditioning of the minimum problem, even if SR T W SR > holOs.
0.20)
THE NEW DESIGN METHOD HDNBC+LDWBC
4. We will name HDNBC (High-Damping NarrowBandwidth Control/er) a dynamic compensator designed by the methodology outlined in Fig. !. Such a compensator is able to provide (at least theoretically) an arbitrary shifting of the real parts of a limited number of poles We will name LDWBC (Low-Damping WldeBandwidth Controller) an algebraiC controller which Is able to provide a limited shifting of the real parts of a large number of poles (le in a frequency range wider tl'lan that In which the HDNBC works). The HDNBC+LDWBC design method can be summarlzeO as follows: a- Form a linear, time-Invariant EM In the state space (step 1 of Fig. 1). IJ - ChOose one among the several syntheSIS methOOs representeO by the scheme of FIg. 1. e - Design a reduced-order compensator via the methOO selecteO. This Is the HDNBC d- Check the stability of the overall system EM+HDNBC If you are satisfied then stop. e- Design an algebraic controller to improve the stabi I ity of the overall system EM+HDNBC. This controller is the LDWBC. It exploits both the outputs of the EM and the states of the HDNBC. f - Check the stab i I i ty of the overall system EM+HDNBC+LDWBC If you are satisfied then stop. g - Choose one of the following steps: ;, / - Exploiting the results of step f change the requirements about the LDWBC and go to step e. g.2 - Choose a higher order f or the HDNBC and go to step c ;,3- Choose a different design method for the HDNBC and go to step c
272
(; . Ca pi ta lli 1'1 al.
The HDNBC+LDWBC design algorithm can be detailed as follows: 1) form the Evaluation Model :
Xp = Fp xp + Gp Y
=
U + Dp W,
cov(w) = Rw '
Hp xp + v ,
cov(v) = Ry ;
2) design the HDNBC via a suitable method (Fig. I):
XC
= Fc Xc + Gy Y + Gu u,
U =
Uti+ul
is the actual goal of the LDWBC design itself. Moreover the HDNB+LDWB control system seems to be useful for a wide class of dynamic systems (not necessari ly flexible structures): i.e. every time the system EM+HDNBC has poles with small real parts. The performance improvement achievable by a LDWBC depends heavi lyon the HDNBC and, therefore, on the method selected (Fig. 1) for its synthesis. The choice of a suitable HDNBC synthesis method is an attractive research item.
,
uH = Hc xc'
EXAMPLES
3) using the EM+HDNBC model:
6.1 Let us resume an example introduced by Capi-
11 :: I+ I:: I
tani , Penati and Tibaldi (1984b). The EM is:
Ul '
I:J = I:p ; 11 :: I'
1 1
Fp =
design the LDWBC
2
o .5
o o
4
REMARKS 5. As said before, the HDNBC+LDWBC synthesis method is in some sense similar to the LAC+HAC synthesis method developed by Aubrun and others (1979). Both methods exploit the eigenvalue perturbation theory in order to improve the stability of a plant via an algebraic controller. The main difference consists on the choice of the system which the algebraic controller is designed for. In fact, the LAC design is based on the EM alone, or on a reduced-order CSM: it does not take into account the presence of the (reduced-order) dynamic compensator (the HAC>. The only practical way for implementing a LAC ([Aubrun and others (1982)], [Aubrun and others (1984)] seems to consIst on the so-called c%cated feed/Jack and therefore on Implementing a c%cated sensors/ actuators confIguration. ObvIously thIs solution Is always expensIve and, sometimes, unlmplementable. Furthermore It seems to be restricted to flexIble structure control. The LDWBC design eXPlICitly takes Into account the presence of the (reduced-order) dynamiC compensator HDNBC. This fact makes the LDWBC design more dIfficult because it Is based upon a model whose dynamic order is certainly larger then the order of the model used in the LAC design. Furthermore a new HDNBC design implies a new LDWBC design. However, a successful LDWBC design does not request any further check'ebout the stability of the overall system (EM+HDNBC+LDWBC) because this
o
0 -2 0
0-3
Hp= [ 1 0 0 0
Finally, we get the overall system outlined in Fig. 2.
o o
3
J.
Rw =Ry=1.
Let us design a HDNBC of order 2 via the scheme of Fig. 1 in which i) steps 2 and 3 are neglected, ii) step 5 uses the model reduction via Moore's balancing [Moore (1981 »). The overall system (EM+ HDNBC) po I es are:
.0293±j .6754, -.7901±j2.453, -2.925, -6.229. Choosing: T
= [ -1 -1 0 0 0 0] T
and
W = 1as 16 '
we obtain: ).
),+6),
.0293±J .6754 - .7901±J2.453 -2.925 -6.229
- .9408±J.7311 - .7862±J4.824 -2.923 -6.227
),+6),
- .2886± J .5903 -1.438±J4 .191 -2.923 -6.227
where). are the overall system poles, ),+6), are the estimated poles and ),+6), are the actual poles. Designing a HDNBC of order 1 via the same method, we obtain the following overall system poles:
.6218±j2.151, - .9266, -2.952, -8.039. Choosing: T
= [ -1 -1
0 0 0 ]T
and
we obtain: ).
.6218±j2.151 - .9266 -2.952 -8.039
),+6),
- .3101±j4.080 - .6522 -2.955 -8.035
),+6),
- .2469± j 3.552 -.7794 -2.955 -8.035
Illl q~ rat e d
6.2 Let us consider an hypothetical flexible structure, whose 8- order EM is (nonzero elements): f p(1,5) = -10, f p(2,6) = -121, f pO,7) = - 40, f p(4,8) = -4.84, fp(5, 1) = f p(6,2) = f P (7,3) P = f (84) = 10 ' . , gp( 1, 1) = gp(2, 1) = gp(3, 1) = gp(4, 1) = 1.0, hp( 1,5) = hp(1 ,6) = 10, hP (1 ,7) P = h (1 8) = -1 0 ' . Dp = Gp, Rw = ' \ = 10 .
J
Note that no velocIty sensor Is avallable. Let us desIgn a HDNBC of order 2 vIa the scheme of FIg. I In whIch 1) step 2 Is neglected, 11) step 3 gIves a CSM of order 2 vIa the modal truncatIon, 111) step 5 Is neglected. The overall system poles are: .1623±j 1.881, .0066±j2. 133, - .0056±j 1.053 - .5624±j.5346, - .7321±j1.716
Co ntro l
control system However, as said before, the HDNBC+LDWBC method is slight ly more complex.
REFERENCES Aubrun J.N., NK Gupta, M.G. Lyons, and G. Margulies (1979). Large space structures control: an integrated approach AIAA Guidance and Control Conf, Boulder, CO, August 1979. Aubrun J.N. (1980). Theory of the control of structures by low-authority controllers. J
Guidance and Control, l. 5. Aubrun J.N., JA Breakwell, NK Gupta, M.G. Lyons and G. Margulles (1982). ACOSS Five, Phase la. Tech. Report RADC-TR-82-21, Rome Air Development Center, Griffiss AIr Force Base, New York.
Choosing: ,. = [-1 -1 - .5 - .5 - .5 - .5 0 0 .2 .2 )T,
Guidance , L 5.
W = .75 110' we obtain: ~
.1623±j 1.881 .0066±j2.133 - .0056±j 1.053 - .5624±j .5346
Aubrun J.N., M.J. Ratner and M.G. Lyons (1984). Structural control for a circular plate. J
~+6~
-
~+t.~
.0254±j 1.879 .0050±j2. 161 .0065±j 1.054 .5439± j .2602
- .0180±j 1.944 - .0263±j2.171 -.0056±j 1.055 -.2782 - .7852 - .7321±J 1.716 - .5190±J 1633 - .5181±J 1.579 These results were obtained on a VM 111750 computer by means of the CACSD package CTRL -C, version 3.0, produced by Systems Control Technology, Palo Alto (August 1985). Let us note that the choice of T and W is not straightforward and sometimes a satisfying result may be obtained only by trials. This drawback can be overcome via a different formulation of the LDWBC design. This item is the subject of a forthcoming paper.
CONCLUSION
Bertoni G., G. Capitani, M.E. Penati and M. Tibaldi modellng and control : an (1983). LFS investigation (In itallan). Tech. Report TC-2/83 , Dept. of ElectronIcs, Systems and Computer SCience, University of Bologna, Italy. Capltanl G., M.E. Penatl , and M. Tlbaldl (1984a). Numerical synthesis of optImum reduced-order dynamic regulators. 1st European WorkShop
on the ''Real Time Control of Large Scale Systems '; Patras, Greece. Capitanl G., M.E. Penatl , and M. Tibaldl (1984b). Initial configuration and ill - conditioning problems In optimum reduced-order dynamic regulators synthesis. 3rd Int. Conference on Systems Engineering, Day ton, OhIo, USA. Capitanl G., M. Tlbaldi and G. Bertonl (1985). A new integrated approach to large flexible structure Symposium on "Identification, control.
Control and Optimization of Dynamic Systems'; Villa Olmo, Como, Italy, June 3-5, 1985.
7. This work develops a new design method for the synthesis of low-order controllers for LSSs, i.e. for systems which are practically represented by means of very large dimensional EMs, like some DPSs. The resulting compensator, named HDNBC+ LDWBC, gives better stabi I ity performances of the overall system if compared with a similar design method, known as HAC+LAC. Furthermore, since the LDWB controller exploits also the HDNBC states as measurements, we can argue that a HDNB+LDWB control system, although exploiting fewer sensors, can give overall performances similar to a HA+LA
Gupta N.K., M.G. Lyons, J.N. Aubrun, and G. Margulles and system (1981). Modeling, control IdentifIcation methods for flexible structures.
AGARDograph 2QQ. Jacobi c.G.J. (1846). Crelle's Journal fur die Reine und Angewand Mathematik. ~ de Gruyter, Berlin. Jonckheere EA and L.M. Si Iverman ( 1(83). A new set of invariants for linear systems - applications to reduced order compensator design. IEEE
c . Capitalli Transact ions on AC, 2L 5. Ly U.L., AE. Bryson and RH. Cannon (1985). Design of low-order compensators using parameter optimization Automatica, 2.L 3. Moore B.L (1981). Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Transactions on A( 2§.,
1.
step
operetion
1'1 Ill.
Verriest E.!. (1981). Suboptimal LOG-Design via Balanced Realizations. 20tIJ IEEE Conference on Oeci sion and Control, San Diego, CA Yousuff A. and RE. Skelton (1984a). Controller reduction by component cost analysis. IEEE
Transactions on AC, 22., 6. Yousuff A and RE. Skelton (I 984b). Controller reduction by covariance equivalent realizations. American Control Conference, San Diego, CA
resul t
ldentificetion end modellng 2
model reduct lOn
3
mode I reduc ti on
4
LQG synthesis
5
compensetor reduction compensetor
Fig . 1
Reduced-order compensetor (HDNBC) design .
,- - - - - - - - - - - - - - - - - I
Fi Iter
+
U
L
r------------, I
C I
lOWBC
I
, _____________ J
Fig 2. The overell system.
\
INTEGRATED CONTROL OF DISTRIBUTED PARAMETER SYSTEMS G. Capitani*, M. Tibaldi*, G. Bertoni* and A. Yousuff** "'/) I'!JIlrl llll' lIl or t ." ln lm llirs , SySlflll; alld Cumpllln Srinlrp, L' llil 'Prsi(l' of B ologlla. "iait' Risurgimnl lu 2, I -·J()] 36 Bulogll{/ , Ilaly """ /) I'!JIlrlll/f' lIl 0( ,\l ('rha ll i({ll E llg ill Pnillg {/ lId A[pr/wlli rs, Du'xl' l L' lI il'l'/'si ly. Ph i lad(,lphia. P A 19 /0 -1 . USA
Abstract. This work develops a new method for designing low-order dynami c compensators. This design procedure integrates the LOG-based reduced - order dynamic compensator synthesis with the low-authority algebraic controller synthesis. A comparison with a well-known method which exploits similar concepts is also presented. Keywords Reduced-order compensators; large scale systems; control system synthesis; controllers; multivariable control systems.
reasons. In the scientific literature many efforts are devoted to the design of reduced-order control systems (e.g. [Aubrun and others (1979)), [Verriest (1981)), [Jonckheere and Sllverman (1983)), [Yousuff and Skelton (1984a and 1984b], [Capltanl, Penati and Tibaldi (I984a and 1984b)1, [Ly and others (1985)]). This work aims to develop a new synthesis method for designing low-order compensators of large scale systems (LSS) suggested by Capitani , Tibaldi and Bertonl (1985). This method is in some sense Similar to the one presented by Aubrun and others (1979 and 1982) and Gupta and others (1981)
INTRODUCTION 1. The dynamic behavior of distributed parameter systems (oPSs) is represented by means of mathematical models formed by partial differential equations. However, in order to exploit the relevant results of the modern control theory about compensator synthesis, mathematical models formed by ordinary differential equations must be used. This kind of mathematical models (state space models) describe the dynamic behavior of a DPS with some accuracy only if the number of differential equations is very large. Let us summarize some terms quite fami 1iar to control system engineers (e.g. [Gupta and others ( 1981)), [Bertoni and others (1983))) and which we are going to use extensively in this work. The physical system to be controlled (in general a DPS) will be referred to as plant and the mathematical model (formed by ordinary differentia l equations) which is believed to represent the plant with the best accuracy (concerning the goal we are interested in) will be referred to as evaluation model (EM); finally, we will name
REDUCED-ORDER COMPENSATOR DESI GN 2. In general, we can outline a reduced-order compensator design method by the scheme shown in Fig. 1. In this methodological frame some steps can be disregarded In order to build a particular design method. For instance, step 2 can be obviously neglected when the EM has Riccat/-so/vab/e dimenSions, whl1e step 3 can be omitted if we want to design the highest order LOG compensator. Moreover note that, for the sake of generality, the scheme of Fig. 1 shows steps 2 and 3 separate ly. In fact, some model reduction methods (e.g. [Jonckheere and Silverman (1983)]) can be applied In step 3, but not in step 2. Finally, step 5 can be neglected when the dynamic order of the CSM is suffiCiently low.
control synthesis model (CSM) a mathematical model (of order lower than the order of the EM, in genera\) used for the design of (or a part of) a control system for the plant. In general , as said before, the dynamic order of the EM of a DPS is very high while a control system must have low dynamic order for implementability
269
~7t)
C. Ca pitalli 1'1 Il /.
Whatever is the method used to design the reduced-order compensator, it is well-known that the behavior of the overall system (plant+compensat or) may turn out to be quite poor (the overall system can be unstable even). Once we have designed a compensator via a synthesis method belonging to the scheme of Fig. 1, we can try to improve its performance by optimizing the parameters of the compensator itself (e.g. [Ly and others (1985)]) Another approach, aiming to design a satisfactory low-order compensator, consists on integrating the design method outlined in Fig. 1, based mainly on the LOG synthesis, with the design method of (algebraic) controllers based on Jacobi 's formula [Jacobi (1846)] concerning the eigenvalue perturbation theory. This kind of controllers was suggested by Aubrun ( 1980). In order to present the new method of reduced-order compensator syntheSiS, section 3 recalls Jacobl's formula for estimating the variations of the elgenvalues of a matrix and presents some comments on its use.
elgenvectors, Nk « r k holds for each k = 1, 2, ... , n), eqs. (3.3) can be approximated by eqs. (3.4). That is:
ll.\ :::: 8\
=
4T N
4
T
rk
, k = 1,2, ... , n. (3.5)
rk
As said before, let us rearrange the basic eq. (3.5), known as Jacobi 's formula, in a form suitable for the LDWB controller design. Given the linear, time-invariant model: x=Fx+6u,
y = H X,
(3 .6)
where x(nx 1) is the state vector, u(rx I) is the Input vector, y(mx 1) Is the output vector, and F, 6, H are matrices of suitable dimensions, let us assume that the input vector u is given by an algebraic feedback of the output vector through the matrix C(rxm):
u = C y.
(3 .7)
From eqs. (3.6) and (}.7) we have:
x = (F
LDWB CONTROLLERS
3. First of all this section presents the basic formulas concerning the eigenvalue perturbation theory. Then these equations are rearranged for the LDWB (L ow-Oamping Wide-Bandwidth ) controller design. Let F be a (nxn) matrix and \ ' k = 1, 2,., n, its eigenvalues. Let us denote by r k the right eigen-
+ 6 C H) x .
(3.8)
Because the state matrix variation is:
N
(3.9)
== 6CH,
eqs. (3.5) turn out to be:
ll.\::::8\ =
4T6CHrk
vector and by -'-r. the left eigenvector, both related
k = 1, 2, ... , n.
to \ . As it is well-known, they are defined by:
F rk = \ rk
1./ F =
\
,
1./,
= 1,2, ... , n,
(3 . 1)
EQs. (}. IO) give approximated estimates 8\ of
k = 1, 2, ..., n.
(3.2)
the act(Jal variations
k
Both each elgenvalue \ and each eigenvector r k and
-'-r. change if the matrix F changes into F+N. Let us denote such variations by ll.\ ,N k and ll.4. respect ive ly. Using eqs (3. 1) and (3.2) and similar equations for F+N, we can easily derive the following expressions:
4T N ( rk + N k ) 4 T ( r k + Nk ) k
= 1, 2,
ll.\ of the poles of the
model (}.8), due to the algebraic output feedback through the matrix C. These estimates are reliable if the variation M , defined by eq. (3.9), can be considered small in the sense stated before. Let us rewrite eqs. (}. IO) in a form more suitable for the LDWB design. To this aim let us define:
-'-r. == 6 T -'-r., r k == H r k ,
k = 1,2, ... , n. (3 .11)
Usually, -'-r. and r k are referred to as general.. , n.
(3 .3)
Defining:
IZed elgenyeclorsCleft and right respectively). Subst itut ing eqs (3. 1 1) into (}. 10) we have
8\ =
4T N r k 4T r k
(3. 10)
k = I, 2, ... , n,
(3.4)
we can note that, assum ing that pert(Jrbat ions are small (1. e. supposing that, with normalized
4T C
~
k
= 1, 2, ... , n.
(3. 12)
Moreover, let us rearrange the elements of the feedback matrix C selecting them by columns:
~I l
Illt t'l,( r a te d C Olltrol
q
==
[C 11 C21 ... CrI
C 12 C22 ... Cr2 C
(3.21)
C ... C ] T. (3. 13) rm 1m 2m
where cij is the generic element of the matrix C.
holds, then the function (3. 1g) is strictly convex and the necessary condition 0.20) is sufficient too. In this case eQ. (3 .20) has the unique solution
The number of elements of q is P =' mr. EQs. (3 12) can be rewritten in the form: k
= 1.2• ... • n.
(314)
where: SkT-
J../rk
and r kj , j = 1,2"
m, is the j-th element of the
°
generalized elgenvector r k . Defining (315)
eQs. (3. 14) can be written in a compact form 6), = S q = ( SR + j SJ ) q ,
(3. 16)
where: (317) The matrices SR e SJ are the real part and the Imaginary part of 5 respectively. Now we are able to outline a possible way for designing a LDWB controller. Fix a real vector T, representing the desired variations, and a weight-matrix W(nxn); then, using the equation: (3 .18) where
6~
variations function:
Is the vector of the real parts of the pole given
J = (6~ -
by
eQ. (3. 16),
T) T W (6~
minimize
- T) + qT q,
the
(3 .1g)
which depends only on q because of eQ. (3 .18) The design of the LDWBC consists in finding output feedback gains sufficiently small (to this aim the function (3 19) includes the term qT q) which give variations of the real parts of the poles of the system (38) sufficiently near to the desired values T.
The necessary condition for a minimum of (319) Is: d.)(q)/d(q) = 0, tl'lat Is: (SRT W ~ + Jp)
If:
q
The weight-matrix W Is usually chosen positive Oeflnlte and then eQ. 0 .21l hOlOs. The term qTq In eQ. (3. 19) penalizes large values of the feedback gains and then It alms to give small perturbatlons. Furthermore the term qTq guarantees the uniqueness of the solution of the minimum problem when W Is positive Oeflnlte. Finally, for a flxeO weight - matrix W>O, missing the term qTq causes a worse conditioning of the minimum problem, even if SR T W SR > holOs.
0.20)
THE NEW DESIGN METHOD HDNBC+LDWBC
4. We will name HDNBC (High-Damping NarrowBandwidth Control/er) a dynamic compensator designed by the methodology outlined in Fig. !. Such a compensator is able to provide (at least theoretically) an arbitrary shifting of the real parts of a limited number of poles We will name LDWBC (Low-Damping WldeBandwidth Controller) an algebraiC controller which Is able to provide a limited shifting of the real parts of a large number of poles (le in a frequency range wider tl'lan that In which the HDNBC works). The HDNBC+LDWBC design method can be summarlzeO as follows: a- Form a linear, time-Invariant EM In the state space (step 1 of Fig. 1). IJ - ChOose one among the several syntheSIS methOOs representeO by the scheme of FIg. 1. e - Design a reduced-order compensator via the methOO selecteO. This Is the HDNBC d- Check the stability of the overall system EM+HDNBC If you are satisfied then stop. e- Design an algebraic controller to improve the stabi I ity of the overall system EM+HDNBC. This controller is the LDWBC. It exploits both the outputs of the EM and the states of the HDNBC. f - Check the stab i I i ty of the overall system EM+HDNBC+LDWBC If you are satisfied then stop. g - Choose one of the following steps: ;, / - Exploiting the results of step f change the requirements about the LDWBC and go to step e. g.2 - Choose a higher order f or the HDNBC and go to step c ;,3- Choose a different design method for the HDNBC and go to step c
272
(; . Ca pi ta lli 1'1 al.
The HDNBC+LDWBC design algorithm can be detailed as follows: 1) form the Evaluation Model :
Xp = Fp xp + Gp Y
=
U + Dp W,
cov(w) = Rw '
Hp xp + v ,
cov(v) = Ry ;
2) design the HDNBC via a suitable method (Fig. I):
XC
= Fc Xc + Gy Y + Gu u,
U =
Uti+ul
is the actual goal of the LDWBC design itself. Moreover the HDNB+LDWB control system seems to be useful for a wide class of dynamic systems (not necessari ly flexible structures): i.e. every time the system EM+HDNBC has poles with small real parts. The performance improvement achievable by a LDWBC depends heavi lyon the HDNBC and, therefore, on the method selected (Fig. 1) for its synthesis. The choice of a suitable HDNBC synthesis method is an attractive research item.
,
uH = Hc xc'
EXAMPLES
3) using the EM+HDNBC model:
6.1 Let us resume an example introduced by Capi-
11 :: I+ I:: I
tani , Penati and Tibaldi (1984b). The EM is:
Ul '
I:J = I:p ; 11 :: I'
1 1
Fp =
design the LDWBC
2
o .5
o o
4
REMARKS 5. As said before, the HDNBC+LDWBC synthesis method is in some sense similar to the LAC+HAC synthesis method developed by Aubrun and others (1979). Both methods exploit the eigenvalue perturbation theory in order to improve the stability of a plant via an algebraic controller. The main difference consists on the choice of the system which the algebraic controller is designed for. In fact, the LAC design is based on the EM alone, or on a reduced-order CSM: it does not take into account the presence of the (reduced-order) dynamic compensator (the HAC>. The only practical way for implementing a LAC ([Aubrun and others (1982)], [Aubrun and others (1984)] seems to consIst on the so-called c%cated feed/Jack and therefore on Implementing a c%cated sensors/ actuators confIguration. ObvIously thIs solution Is always expensIve and, sometimes, unlmplementable. Furthermore It seems to be restricted to flexIble structure control. The LDWBC design eXPlICitly takes Into account the presence of the (reduced-order) dynamiC compensator HDNBC. This fact makes the LDWBC design more dIfficult because it Is based upon a model whose dynamic order is certainly larger then the order of the model used in the LAC design. Furthermore a new HDNBC design implies a new LDWBC design. However, a successful LDWBC design does not request any further check'ebout the stability of the overall system (EM+HDNBC+LDWBC) because this
o
0 -2 0
0-3
Hp= [ 1 0 0 0
Finally, we get the overall system outlined in Fig. 2.
o o
3
J.
Rw =Ry=1.
Let us design a HDNBC of order 2 via the scheme of Fig. 1 in which i) steps 2 and 3 are neglected, ii) step 5 uses the model reduction via Moore's balancing [Moore (1981 »). The overall system (EM+ HDNBC) po I es are:
.0293±j .6754, -.7901±j2.453, -2.925, -6.229. Choosing: T
= [ -1 -1 0 0 0 0] T
and
W = 1as 16 '
we obtain: ).
),+6),
.0293±J .6754 - .7901±J2.453 -2.925 -6.229
- .9408±J.7311 - .7862±J4.824 -2.923 -6.227
),+6),
- .2886± J .5903 -1.438±J4 .191 -2.923 -6.227
where). are the overall system poles, ),+6), are the estimated poles and ),+6), are the actual poles. Designing a HDNBC of order 1 via the same method, we obtain the following overall system poles:
.6218±j2.151, - .9266, -2.952, -8.039. Choosing: T
= [ -1 -1
0 0 0 ]T
and
we obtain: ).
.6218±j2.151 - .9266 -2.952 -8.039
),+6),
- .3101±j4.080 - .6522 -2.955 -8.035
),+6),
- .2469± j 3.552 -.7794 -2.955 -8.035
Illl q~ rat e d
6.2 Let us consider an hypothetical flexible structure, whose 8- order EM is (nonzero elements): f p(1,5) = -10, f p(2,6) = -121, f pO,7) = - 40, f p(4,8) = -4.84, fp(5, 1) = f p(6,2) = f P (7,3) P = f (84) = 10 ' . , gp( 1, 1) = gp(2, 1) = gp(3, 1) = gp(4, 1) = 1.0, hp( 1,5) = hp(1 ,6) = 10, hP (1 ,7) P = h (1 8) = -1 0 ' . Dp = Gp, Rw = ' \ = 10 .
J
Note that no velocIty sensor Is avallable. Let us desIgn a HDNBC of order 2 vIa the scheme of FIg. I In whIch 1) step 2 Is neglected, 11) step 3 gIves a CSM of order 2 vIa the modal truncatIon, 111) step 5 Is neglected. The overall system poles are: .1623±j 1.881, .0066±j2. 133, - .0056±j 1.053 - .5624±j.5346, - .7321±j1.716
Co ntro l
control system However, as said before, the HDNBC+LDWBC method is slight ly more complex.
REFERENCES Aubrun J.N., NK Gupta, M.G. Lyons, and G. Margulies (1979). Large space structures control: an integrated approach AIAA Guidance and Control Conf, Boulder, CO, August 1979. Aubrun J.N. (1980). Theory of the control of structures by low-authority controllers. J
Guidance and Control, l. 5. Aubrun J.N., JA Breakwell, NK Gupta, M.G. Lyons and G. Margulles (1982). ACOSS Five, Phase la. Tech. Report RADC-TR-82-21, Rome Air Development Center, Griffiss AIr Force Base, New York.
Choosing: ,. = [-1 -1 - .5 - .5 - .5 - .5 0 0 .2 .2 )T,
Guidance , L 5.
W = .75 110' we obtain: ~
.1623±j 1.881 .0066±j2.133 - .0056±j 1.053 - .5624±j .5346
Aubrun J.N., M.J. Ratner and M.G. Lyons (1984). Structural control for a circular plate. J
~+6~
-
~+t.~
.0254±j 1.879 .0050±j2. 161 .0065±j 1.054 .5439± j .2602
- .0180±j 1.944 - .0263±j2.171 -.0056±j 1.055 -.2782 - .7852 - .7321±J 1.716 - .5190±J 1633 - .5181±J 1.579 These results were obtained on a VM 111750 computer by means of the CACSD package CTRL -C, version 3.0, produced by Systems Control Technology, Palo Alto (August 1985). Let us note that the choice of T and W is not straightforward and sometimes a satisfying result may be obtained only by trials. This drawback can be overcome via a different formulation of the LDWBC design. This item is the subject of a forthcoming paper.
CONCLUSION
Bertoni G., G. Capitani, M.E. Penati and M. Tibaldi modellng and control : an (1983). LFS investigation (In itallan). Tech. Report TC-2/83 , Dept. of ElectronIcs, Systems and Computer SCience, University of Bologna, Italy. Capltanl G., M.E. Penatl , and M. Tlbaldl (1984a). Numerical synthesis of optImum reduced-order dynamic regulators. 1st European WorkShop
on the ''Real Time Control of Large Scale Systems '; Patras, Greece. Capitanl G., M.E. Penatl , and M. Tibaldl (1984b). Initial configuration and ill - conditioning problems In optimum reduced-order dynamic regulators synthesis. 3rd Int. Conference on Systems Engineering, Day ton, OhIo, USA. Capitanl G., M. Tlbaldi and G. Bertonl (1985). A new integrated approach to large flexible structure Symposium on "Identification, control.
Control and Optimization of Dynamic Systems'; Villa Olmo, Como, Italy, June 3-5, 1985.
7. This work develops a new design method for the synthesis of low-order controllers for LSSs, i.e. for systems which are practically represented by means of very large dimensional EMs, like some DPSs. The resulting compensator, named HDNBC+ LDWBC, gives better stabi I ity performances of the overall system if compared with a similar design method, known as HAC+LAC. Furthermore, since the LDWB controller exploits also the HDNBC states as measurements, we can argue that a HDNB+LDWB control system, although exploiting fewer sensors, can give overall performances similar to a HA+LA
Gupta N.K., M.G. Lyons, J.N. Aubrun, and G. Margulles and system (1981). Modeling, control IdentifIcation methods for flexible structures.
AGARDograph 2QQ. Jacobi c.G.J. (1846). Crelle's Journal fur die Reine und Angewand Mathematik. ~ de Gruyter, Berlin. Jonckheere EA and L.M. Si Iverman ( 1(83). A new set of invariants for linear systems - applications to reduced order compensator design. IEEE
c . Capitalli Transact ions on AC, 2L 5. Ly U.L., AE. Bryson and RH. Cannon (1985). Design of low-order compensators using parameter optimization Automatica, 2.L 3. Moore B.L (1981). Principal component analysis in linear systems: controllability, observability and model reduction. IEEE Transactions on A( 2§.,
1.
step
operetion
1'1 Ill.
Verriest E.!. (1981). Suboptimal LOG-Design via Balanced Realizations. 20tIJ IEEE Conference on Oeci sion and Control, San Diego, CA Yousuff A. and RE. Skelton (1984a). Controller reduction by component cost analysis. IEEE
Transactions on AC, 22., 6. Yousuff A and RE. Skelton (I 984b). Controller reduction by covariance equivalent realizations. American Control Conference, San Diego, CA
resul t
ldentificetion end modellng 2
model reduct lOn
3
mode I reduc ti on
4
LQG synthesis
5
compensetor reduction compensetor
Fig . 1
Reduced-order compensetor (HDNBC) design .
,- - - - - - - - - - - - - - - - - I
Fi Iter
+
U
L
r------------, I
C I
lOWBC
I
, _____________ J
Fig 2. The overell system.
\