LIDS-P- 1327
AUGUST 1983
THE RELATIONSHIP BETWEEN THE ZAMES REPRESENTATION AND LQG COMPENSATORS
by
Michael Athans
Laboratory for Information and Decision Systems
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
ABSTRACT
G (s) = LI+P(s)C(s)]
-CL
-l
P(s)C(s)
Recently Zames E1] has introduced the so-called model
reference transformation which can be used to conveniently parametrize the class of linear time-invariant
multivariable compensators that lead to stable feedbadck
control systems. A very popular design methodology for
The results of Zames (1] state that the closed-loop
The results of Zames 11 state that the closed-loop
designing stable and robust multivariable control sys-
transfer matrix -CL(s) can be expressed in the form
tems is that based on the Linear-Quadratic-Gaussian
design methodology with Loop Transfer Recovery
Thus, it is
'between
'between the
the
pensators.
.1.
= P(s)C(s)[I+P(s)C(s)
(LTR).
natural to inquire upon the relationship
Zames
the LQG/LTR
comZames representation
representation and
and the
LQG/LTR comThis paper summarizes this relationship.
IG
(s) ==- P(s)Q(s)
(3)
(3)
G L(s)
(s)(s)
where the
transfer matrix (s) is defined by the
where
the mxmn
transfer
matrlx
Q(s) is
defined
compensator
C(s)
and the
open-loop
plant
P(s) by
as the
os
follows:
p(s)
The Zames representation [l] has attracted a significant attention in the literature; see also [2].
A recent paper by Gustafson and Desoer (31 presents a
design methodology which has its roots upon the Zames
representation. On the other hand, the most popular
design methodology for linear time-invariant multivariable systems remains the LQG-based design method
with loop-transfer-recovery
(LTR)
(2)
(LQG)
INTRODUCTION
14],[7),
1
C(s) [I+P(s)C(s)|
-
I+C(s)P(s)l
-l
C(s)
It follows that the compensator C(s) is given by
C(s) = Q(s) I-P(s)Q(s)]
-1
= I-(s)P(s)] 2(s)
(4)
(5)
15],[6],[8].
The LQG/LTR method generates nominally stable compensators with superior robustness properties [6] in the
absence of non-minimum phase zeroes. The design of
the output sensitivity matrix S (s) by
-1
SO(s) = 1I+P(s)C(s)]
= I-P(s)Q(s)
(6)
(s)
and the input sensitivity matrix S.i
-1
S.(s) = (I+C(s)P(s)] = I-2(s)P(s)
(7)
LQG/LTR feedback control systems can now be routinely
carried out by loop-shaping methods in the frequency
domain, [5], (6],(8].
Only time will tell whether design methodologies
based upon the Zames representation will approach the
popularity of LQG/LTR methods. In the meantime it is
of interest to relate these methodologies.
In Section 2 we summarize the Zames Representation.
In Section 3 we start with the class of the so-called
Model Based Compensators (MBC) and specialize them to
the class of LQG/LTR compensators; many results not
readily apparent in the literature are stated without
proof. Next, we specify the Zames representation for
,the class of LQG/LTR designs.
-2. THE ZAMES PARAMETRIZATION
;1Cn~
w
umaie
thsseto
h pprmHowever,
In this section we summarize the Q-parametrization
introduced by Zames tl]; see also 12].
Figure 1 illustrates a block diagram of a multivariable linear time-invariant feedback control system.
The mxm transfer matrix P(s) represents the open-loop
;plant, and the mxmn transfer matrix C(s) represents the
dynamic compensator. Elementary algebra leads to the
following closed-loop transfer matrix,
(s), from h
closed-loop transfer matrix, CLS)
following
from the
reference input r(s) to the output y(s), i.e.
X(s)
where
CL( s)r(s)
iwhere~~~~~~~~~~~.
(1)
The recent interest Cl],(2],[3] in this representation
hinges on the fact that the sensitivity matrices S (s)
and Si(s), given by Eqs. (6) and (7), are linearly
related to the matrix i(s). Thus, by shaping Q(s), for
any given P(s), one can obtain "good" sensitivity matrices, and then evaluate the dynamic compensator C(s)
using Eq, (5). The resultant closed loop transfer matrix, given by Eq. (3), is then calculated by
(8)
G (s) = I-S (s)
-o
-CL
Ideally, one would like to have S (jw)=0 for all w.
this is not possible, ai2 approximations always
take place.
take place.
3.
THE CLASS OF LQG COMPENSATORS
In this section we examine the class of LinearQuadratic-Gaussian (LQG) compensators with Loop-Transfer-Recovery (LTR) [4],[5], [6]. It is quite helpful
to first define the class of Model-Based-Compensators
(M.BC); the MBC nonmeclature is non-standard.
3.1
The Open-Loop Plant
heOe
The time domain description of the open-loop plant
* Research supported by NASA Ames and Langley Research Centers under grant NGL-22-009-124 and by the Office of
Naval Research under grant ONR/N00014-82-K-0582 (NR 606-003).
*
Proc. 22nd IEEE Conference on Decision and Control, San Antonio, Texas, December 1983.
P(s) is as follows:
P-s)- is as follows:no
x(t) = A x(t) + B u(t)
with LTR [5], [8].
Suppose that the open-loop plant has
right-half plane (non-minimumn phase) transmission
zeros. Furthermore, suppose that the con-trol gain matrix G in Fig. 2 and in Eq. (12) is computed as follows:
(9)
f(t) = C x(t)
=
m R efnetGh
e
(t)e R . Define the nxm
m
with x(t)e R , u(t)e R ,
matrix ¢(s) by
-10)
_(s) = (sI-A)
0 = -K A-A'K - $C'C + - K B B'K
-P
In particular in the LTR procedure we let B-o.
-1
the results in Refs.
P(s) = C(sI-A)
B = C ;(s)B
Model Based Compensators (MBC)
(s)= G[sI-A+B G+H C
-1
(s) - [C _(s)B]
C(s) = C
-LQG
Thus, the LQG/LTR compensator
[7].
and
(s) in Eq.
(12)
all parameters in C CM(s)
Note that
(12) aBC
in E.
Note that all parameters in
Fig. 2 are fixed, except for the mxn filter gain matrix
H, and the nxm control gain matrix G.
If [A,B) is
stabilizable and [A,C] is detectable, then it is wellknown (from non-minimal observer theory and pole-placement theory) that there exist gain matrices H and G
such that the closed loop system of Fig. I is stable
using the NBC of Eq. (12).
Next we present the closed-loop transfer function
MBC
BC (s), Y(s) = GBC(s)r(s) associated with the feed-CL
-CL
back control system of Fig. 1 using the MBC of Fig. 2.
:Tedious algebra and the use of the matrix inversion
lemma lead to
~~~~~~~~~~~~~MBC
G b(s)H[I+C I(s)H]
-
1
_(s)H]
(s)B-
(18)
(s) performs an au-
-
LQG(s) -_ Pl(s)C _(s)H[I+C
-
1
-(s)]
(20)
(21)
~~~~LQG
The interpretation of LQG (s) is that it corresponds to
the closed-loop transfer matrix of the system shown in
P-(s).
Fig. 3 followed by the plant inverse
In the LQG/LTR design procedure the filter gain matrix H is calculated by suitable solution of a Kalman
1
(22)
meet the posed performance and robustness specifications
Indeed, at low frecuencies,
in the frequency domain.
it is possible to select H so that
?(jw)H]
C
-(j)H[I+C
resulting in
LQG(jW):
p
(jw)
(23)
(24)
and output sensitivity which is almost zero -see Eq.(6)-
LQG/LTR Compensators
S (jw)n-0
-0
The structure of an LQG compensator is identical to
that of an MBC (see Figure 2). What distinguishes an
LQG compensator from an MBC is the specific way that
one calculates the gain matrices H and G via the appropriate algebraic
equations for
propriate
algebraic Riccati
Riccati equations
for the
the LQG
LQG opoptimization problem [4),[7]. These will not be reviewed,
since an arbitrary LQG compensator does not provide any
more insight than that contained in Eq. (14).
___l_~~~~~~
edacklos.POP--Hwvr,
· =1~~~- i~~i~~rT( · e
G
(s) l
QLQG(s) = C
(s)[I+P(s)C
- Q
-QG
we deduce from Eqs. (18) and (20) that
As $-,
-
shape Q(s).
.A greater insight can be obtained
C _(s)H
which is the closed-loop transfer of the feedback system shown in Figure 3.
in Figure 3.Zames parametrization 2(s)-=2LQG (s)
ten The
shown
corresponding
can now be readily calculated. From Eq. (4)
It is the selection of the constant filter gain matrix
H and of the constant control gain matrix G that would
3.3
(s)-
there are no right-half plane pole-zero cancellations.
The resulting closed-loop transfer matrix
LQG(s) tends to
L
(19)
LQG s)-C f(s)HtI+C 4(s)H]
C D(s)H[I+C 4(s)H]
(14)
G _(s)H[I+C 4'(s)H]
(17)
(s);
he assmtion
proximate inversion of the lant
that P(s) has no non-minimum phase zeros guarantees that
(13)
By comparing Eqs. (3) and (13) we can readily conclude
that the q(s) matrix in the Zames representation [1)
is as follows for MBC derived control systems
(s)=2MBC(s)=[I+G
the
Filter design problem so that the loop transfer-atrix
C 4(s)H in Fig. 3 and the closed-loop transfer function
-1
-1
Then,
S-~,
(s)C '(s)H
= P
(12)
(s) is shown in Fig. 2. The
A block diagram of C
MBCblock
in
such
diagram
wa
is drso
so that
that the
block diagram is drawn
in such
aa wak
the rela
relationship of the MBC to the LQG compensator is apparent;
if we set r(s) = d (s)=0 in Fig. 1, then v(s) would represent the innova-ions vector, and x(s) the state estimate. Also, one can think of the MBC in Fig. 2 as
been obtained by cascading a non-minimal stable
Luenberger observer with a suitable control gain, G,
that may be obtained from a pole-placement algorithm
-1G _(s)H[I+C
indicate that as
and that the LQG compensator transfer matrix C
see Eq. (12)- approaches
H
-1
= G[y l(s)+B G+H Cl
H
MBC
GCL (s)=C t(s)B[I+G .(s)B]
[5],[8]
(16)
1
control
gain matrix G has the property that
orthonormal matrix
W
G - W C ;
()
In reference to the feedback system in Fig. 1 we
define a model based compensator (MBC) with the transfer matrix
_(s)B
= P(s)[I+G
(15)
1
(10)
Evidently, the open-loop plant transfer matrix P(s) is
given by
C(s) =-C
p>0
where the matrix K is the solution of the algebraic
Riccati equation
~()A
3.2
K;
-B
for LQG
designs
this--
~
(25)
matrix for the class
We have calculated the Za
of LQG/LTR designs. In the opinion of the author, the
Zames decomposition offers no particular insights into
the LQG/LTR problem, except that it confirms (in a
round-about way) that in the absence of non-minimum
A greater
hase insight
zeros
can
be
the
obtained
LQG/LTR
forL designs
procedure can result in excellent
However, this has been known ans-;ay.
eedback loops.
has
been k1wn an.:-ay.
i~-·-
5.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
G. Zames, "Feedback and Optimal Sensitivity:
Model Reference Transformations, Multiplicative
Seminorms, and Approximate Inverses,n IEEE Trans.
on Automatic Control, Vol. AC-26, No. 2, April
1981, pp. 301-320.
C.A. Desoer and M.J. Chen, "Design of Multivariable Feedback Systems with Known Plant," IEEE
Trans. on Auto. Control, Vol. AC-26, No. 2,
April 1981, pp. 408-415.
C.L. Gustafson and C.A. Desoer, "Controller
(31~ C rFigure
Gsfoa C"Cto
Design for Linear Multivariable Feedback Systems
with Stable Plants, Using Optimization with
Inequality Contraints," Int. J. Control, Vol.
37, No. 5, 1983, pp. 881-907.
M. Athans, "The Role and Use of the Stochastic
LQG Problem in Control Systems Design," IEEE
Trans. on Automatic Control, Vol. AC-16, Dec.
1971, pp. 529-552.
J.C. Doyle and G. Stein, "Multivariable Feedback
Design Concepts for a Classical/Modern Synthesis,"
IEEE Trans. on Auto. Control, Vol. AC-26, No. 1,
Feb. 1981, pp. 4-16.
N.A. Lehtomaki, N.R. Sandell, Jr., and M. Athans,
"Robustness Results in LQG-Based Multivariable
Control Designs," IEEE Trans. on Auto. Control,
Vol. AC-26, No. 1, Feb. 1981,' pp. 75-92.
H. Kwackernaak and R. Sivan, Linear Optimal
Control Systems, NY, J. Wiley and Sons, 1972.
G. Stein, "LQG-Based Multivariable Design:
Frequency Domain Interpretation," AGARD Lecture
Series No. 117, NATO, September 1981.
FILTER
GAIN
CONTROL
GAIN
Figure 2: Block diagram of HBC and LQG compensator, driven by the error
signal e(s) and generating the control u(s) in Figure 1.
d o(s)
(o(S)
Figure 3:
Block diagram of the limiting feedback control system obtained
by the LQG/LTR procedure. The constant matrix H is the
Kalman Filter gain matrix.
Id(s)
r()
e(s)
e(
(s)
s
(s)
P(s)
PLANT
1: Block diagram of multivariable control system.