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H∞ Control of System With I/O Delay: A Review of Some
Problem-Oriented Methods
Leonid Mirkin
Faculty of Mechanical Engineering, Technion — IIT, Haifa 32000, Israel
Gilead Tadmor
Electrical & Computer Engineering, Northeastern University, Boston, MA 02115
[Received April 12, 2001]
Systems with input or output delays form the simplest, and yet one of the most
widely applied classes of distributed parameter models. This is a review of some
problem-oriented H∞ methods for that class, with an emphasis on computational
simplicity. Reviewed methods include operator interpolation, game-theoretic
state-space treatments in the time domain, a J-spectral factorization approach,
and methods exploiting various ideas from sampled-data theory. Some interesting
properties of the (sub)optimal solutions are discussed as well.
1. Introduction
This article reviews several H∞ methods for rational systems cascaded with input
or output delays, see Fig. 1. Those are perhaps the simplest distributed parameter
systems, and yet the most widely used as models in industrial applications.
Hence the interest in methods developed specifically for that class. In view of the
voluminous literature on robust distributed systems design, articles cited here are
intended merely as samples, and by no means provide an exhaustive, or even a
representative list.
Historical perspectives. Solutions of H∞ problems in distributed systems trace
back to the mid 1980’s. Some early work [21, 17, 18, 64, 82] appealed to operator
interpolation/commutant lifting theorem methods [61, 63]. One major outgrowth,
-
w
u
z
-
P
Kτ
? −τ s
e
y
Fig. 1. General LFT setup with single delay
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L. Mirkin & G. Tadmor
with later impact on nonlinear robust control, was the development of the SkewToeplitz methodology [20, 4, 36, 52, 51, 53, 49, 19]. Yet another approach followed
early on, was the use of reduction to a finite dimensional (rational) problem [10,
33, 24, 25, 56, 83, 29]. The introduction of time domain/game theoretic state space
methods [66, 59, 80, 81, 68, 3] was immediately followed by extensions to a general
framework of distributed parameter systems [67, 9, 70, 76, 32, 37]. A feature that
facilitates design in systems with input (or output) delays is that their distributed
component has very simple dynamics (essentially, an FIR). Indeed, in “classical”
and some robust stabilization problems, this feature allows a reduction to a purely
rational problem [35, 1, 71], using a Smith predictor.
Scope and organization of the paper. The common thread in the several reviewed
methods is the utilization of the special structure and time domain representations
of systems with I/O delay. The review does not cover the use of tools of broader
scope, such as low order approximations, or general factorization techniques, as well
as a variety of interesting ideas leading to simplified transform-domain solutions
(such as elegant reproducing kernels methods [16], or the simplified computation of
Hankel singular values [50], which requires an inner/outer factorization).
Some early work meet the set criterion in the context of operator interpolation
methods and will be discussed in Section 2. Here, time domain analysis is used to
characterize the interpolation conditions for one-block problems. Sections 3 and 4
are devoted to time- and frequency-domain treatments of the “standard problem”
in Fig. 1. §3.1 reviews the use of a “lifting” trick, borrowed from work in sampled
data control. It transforms the delay problem to a discrete time problem with
distributed I/O operators, but with no delays. The game theoretic solution leads
to a periodic, time varying compensator. Realizing that periodicity is clearly a
technical artifact of “lifting”, immediate followups - reviewed in §3.2 - removed the
resulting time variation. Here one considers the interplay between abstract evolution
models over the product space M2 = Rn ×L2 [−τ, 0] and the original delay equation,
in the associated differential games. A J-spectral factorization based approach is
reviewed in §4.1. Here the Smith predictor trick is used to reduce the problem
to an equivalent rational problem. The extraction of dead-time controllers from
delay-free parameterizations is discussed in §4.2. This approach also has its roots in
the sampled-data theory and leads to a simple and intuitively appealing solution.
Finally, the controller structure and the achievable cost in the H∞ delay problems
are discussed in Section 5.
Preliminaries and notation. Given an LTI system G(s) = C(sI − A)−1 B, its
0
conjugate is defined as G∼ = −B
− A0 )−1 C 0 . The completion of e−τ s G to [0, τ ]
(sI
.
−τ s
−Aτ
is defined
as
follows:
π
e
G
=
C(e
− e−τ s I)(sI − A)−1 B. It can be seen
τ
−τ s that π τ e
G is an entire function whose impulse response has support on [0, τ ]
(FIR system).
H∞ control of system with i/o delay: review
3
2. Time-domain methods in operator interpolation
Arguably the most significant early development in robust H∞ control was the
connection made with operator interpolation/commutant lifting problems [61, 63],
involving the following general setting. Given are w, m ∈ H∞ , with m(s) inner.
One defines a subspace K := H2 m H2 , the orthogonal projection π : H2 7→ K,
and an operator T : K 7→ K, via T f = π(wf ), f ∈ K. A function w0 ∈ H∞ is
said to interpolate the operator T if it satisfies: (i) T f = π(w0 f ), f ∈ K, and
(ii) kT k = kw0 k∞ = kΛw0 k, where Λw0 : H2 7→ H2 is the multiplication operator
associated with w0 . In the context of control design, “m” is usually the inner part of
the open loop plant, whereas “w” is a frequency shaping weight function [14, 22]. In
the interest of simplicity, w can be selected rational, yet non-rational, non-minimum
phase elements in a distributed plant, such as delays, cannot be removed from m.
The following is a key result Lemma 1 ([61, Proposition 5.1]) Suppose that f ∈ K is a maximal function
for the operator T . Then w0 = T f /f ∈ H∞ is an all-pass function, and is the
unique solution of the interpolation problem.
The main solution steps are thus the characterization of K, π, and T . Early case
studies of distributed systems [21, 17, 18, 82, 25] concerned the example of scalar
rational systems with a pure input delay. The distributed non-minimum phase part
is already inner, m(s) = e−τ s , leading to a particularly simple characterization
of K, π and T : assuming an outer rational part, K = H2 e−τ s H2 is isometric
L2 [0, τ ], π : L2 [0, ∞) 7→ L2 [0, τ ] is the truncation and T is the restriction to [0, τ ] of
convolution with the inverse Laplace transform of w. Thus, T is realized by a causal
ordinary differential equation (ODE), its adjoint has an anti causal ODE realization,
and Schmidt pairs of T are solutions of a Hamiltonian, two point boundary value
problem. Once that observation is made, the step leading to a solution based on
transcendental equation characterization of Schmidt pairs, is immediate.
The pleasant feature of an inner distributed component fails in the next simplest
example of a scalar system with several, commensurate input delays, and neither an
inner/outer plant factorization, nor a characterization of K in terms of the plant’s
inner part, would then be as simple. The solution outlined in [65, 64] utilizes the fact
that the plants outer part has no effect on the definition of K, and on Fredholm’s
alternative, whereby K = H2 gH2 = N (Λ0g ), with g being the plant’s transfer
function. The main point is that the adjoint multiplication operator, Λ0g , has a
system realization and can be characterized in terms of that realization. In a plant
with multiple commensurate input delays and an outer rational part, it suffices
to consider g as the delay polynomial. Then K is the stable component of the
solution space of a difference equation and is isomorphic to initial L2 [0, lτ ] portions
of trajectories, A new challenge is to define the right Hilbert space structure in
K, to make this isomorphism an isometry, as is needed in order to compute the
orthogonal projection π, and adjoint of operators over K, as follows.
Let S be the lτ -time units shift along trajectories in K. Identifying trajectories
l
i
f ∈ L2 [0, ∞) ↔ {f i }∞
i=0 ∈ `2 (L2 ([0, τ ], R )), where f (s) = [f (ilτ + s), f ((il +
T
1)τ + s), . . . , f (((i + 1)l − 1)τ + s)] , s ∈ [0, τ ], i = 0, 1, . . . , the shift S admits a
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L. Mirkin & G. Tadmor
representation by an l × l matrix E with eigenvalues magnitude smaller than one.
Let Q > 0 solve Q − E 0 QE = I. Then, inner products between trajectories in K
satisfy
hx, yiL2 [0,∞) =
∞
X
hxk , y k iL2 [0,τ ] =
k=0
∞
X
hE k x0 , E k y 0 iL2 [0,τ ] = hx0 , Qy 0 iL2 [0,τ ]
k=0
The Q - weighted L2 ([0, τ ], Rl ) inner product thus defines the isometric
representation of K by [0, τ ] trajectories. The projection π : L2 [0, ∞) 7→ K is
P∞
k
then [πf ]i = E i Q−1 k=0 E 0 f k . The operator T = πΛw |K is the compression of
a convolution operator; here too, it admits a causal ODE realization over [0, τ ],
albeit subject to certain non-trivial (integral) boundary conditions, its adjoint T 0
has an anti causal ODE realization and the equation (σ 2 I − T 0 T )f = 0 amounts
to a two point boundary value problem. Thus singular values are again solutions of
a transcendental equation and Schmidt pairs are derived from algebraic conditions
on admissible boundary values.
Stressed in closing is the key step of utilizing the time domain structure of the
delay operator to provide a time domain realization of the interpolation space K
as the solution space of a difference equation. This leads to a problem reduction to
a solution of a two point boundary value problem.
3. Time-domain methods
By the late 1980’s, state space / Riccati equations methods became dominant,
beginning with the finite dimensional LTI case [2, 15], continuing with time varying
(LTV) / finite time systems, [67, 70, 3] and eventually allowing extensions to
distributed systems. Practical impediments in the latter include the need to solve
infinite dimensional operator Riccati equations, and compensator realizations in
terms of abstract evolution models. Here we review methods utilizing the structure
of systems with a pure input lag, to address both issues: operator Riccati equations
are reduced to fixed size matrix equations and explicit delay system compensator
realizations are provided. Early developments are reported in [3]. The focus of this
review is on the subsequent results in [46, 47], followed by [34, 72, 74, 73, 71].
3.1
A sampled data method: matrix Riccati eqs. and “periodic” solutions
Several control and observation problems are considered in [47], and reduced (using
standard unitary loop transformations) to a single representative, akin to the
“standard problem” in the context of Figure 1. Roughly, the idea here is to break
the basic problem into two parts, one of which is just a delay-free estimation of z
over [τ, ∞) and another one can be reduced to a Nehari-like problem defined over
the finite horizon [0, τ ]. Nagpal and Ravi [47] observed an opportunity in adapting
the so-called “lifting” technique similar to that used for some sampled-data control
problems [8]. In one guise or another, the various approaches to sampled data design
built on the “lifting” trick (already used in in §2) where, in I/O trajectories one
identifies f ∈ L2 [0, ∞) ↔ {f k } ∈ `2 (L2 [0, τ ]), with f k (s) = f (kτ + s). This reduces
H∞ control of system with i/o delay: review
5
the original (analog) system to a discrete time system with the same state space, say
Rn and distributed I/O signals. Two algebraic Riccati equations (AREs) determine
a discrete time design, which leads to periodic actuation between samples. Since the
system acts in open loop between successive decisions, an added differential Riccati
equation (DRE) is also needed, in some variants, to evaluate the I/O gain during
these uncontrolled periods.
The adaptation of these ideas to the context of the basic delay problem required
sophisticated and elaborate computations, and we shall be content with highlights
of some important characteristics of [46, 47]. First, as in sampled data control,
simple solvability conditions are derived, comprising the AREs for the delay-free
problem and a DRE over [0, τ ]. The latter quantifies the extra cost for tolerating
delay. In contrast, the controller structure is cumbersome, even in the simplest case.
It is derived in terms of the “lifted” representation, and, as mentioned earlier, its
“unlifted” form is periodic, time varying. The challenge of “peeling-off” a timeinvariant representation is addressed next.
3.2
A semigroup based reduction: matrix Riccati equations and LTI solutions
Inspired by [46, 47], the studies [34, 72, 74, 73, 71] searched for an alternative
reduction that would lead to a similar computational component (i.e., the combining
AREs and DRE), but that avoids compensator periodicity. Following are the guiding
principles, underlying these developments.
The first is the use abstract evolution models for the delay system [30, 11, 5,
13, 77, 6, 7, 60, 12, 75]. A natural state in the current setting would be f (t) =
(x(t), ut ) ∈ M2 := Rn × L2 [−τ, 0], where ut (s) = u(t + s), s ∈ [−τ, 0], is the relevant
control history at the time t. Abstract state space/operator Riccati equations
solutions are well known for the linear quadratic (LQ) optimal control problem
[78, 31, 57, 58] and have been established also for H∞ problems [67, 9, 70, 76, 32, 37].
As in the non-distributed case, the solution of the operator Riccati equation (ORE)
is completely characterized by the quadratic form for the optimal cost in terms of
the initial state. In the H∞ case, that is the cost of an associated differential game.
Second, as in [46, 47] the explicit solution of the said game is computed in the
original setting, based on fixed size matrix AREs/DRE. A positive semidefinite
quadratic form for the game’s optimal value is then derived as a function of the
initial data f (0) = (x(0), u0 ) ∈ M2 . Returning to the abstract model, that quadratic
form provides the desired explicit expression for the solution of the ORE.
Third, in complete analogy to finite dimensional results, suboptimal compensators are parameterized by state space equations, in an abstract evolution
model. These state space equations are based on perturbations of the infinitesimal
generator and of input and output operators from the original model. As it
turns out, these perturbed generators are, by themselves, abstract models of delay
differential equations. The tools used here are provided by the detailed analysis of
the interplay between functional differential equations and their abstract models,
in [6, 75].
In summary, the family of solutions described in this section comprise an
analysis along two parallel tracks. Along one track, a formal solution of the H∞
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L. Mirkin & G. Tadmor
problem, including a parameterization of suboptimal compensators, is provided in
terms of abstract evolution model counterparts of familiar state space solutions.
Along a second track, explicit solution of operator Riccati equations is reduced
to matrix computations, via a solution of a differential game in the original delay
equation setting. Finally, the abstract evolution model realization of suboptimal
compensators is reduced to a delay (neutral) differential equation, using established
relations between such equations and their abstract models, and specific details of
the computed solution of the operator Riccati equation.
In order to provide a flavor of this set of result, the following is an abbreviated
version of the main result in [72].
Theorem 1 Consider a system G in minimal LTI realization [A, B, C], cascaded
with the input delay e−τ s . Let X, Y > 0 be the stabilizing solutions of the algebraic
Riccati equations
A0 X + XA − XBB 0 X + C 0 C = 0,
AY + Y A0 − Y C 0 CY + BB 0 = 0. (1)
I
Let γ0 be the infimal achievable induced norm of FK = K
(I + KG)−1 [I K] over
all stabilizing linear feedback compensators K. Then γ0 > 1, and γ > γ0 if and only
if there exists a bounded, positive semidefinite solution of the following differential
Riccati equation over [0, τ ]
Ż = ZA + A0 Z +
1
0
1
γ 2 −1 ZC CZ
+ BB 0 ,
Z(0) = Y
(2)
1
such that (γ 2 − 1)I > X 2 Z(τ )X 2 . Given such such γ and the associated Z, a
parameterization of stabilizing compensators that guarantee that performance level
is in terms of
ẋc (t) = Axc (t) − Bv(t − τ ) + Y C 0 (e(t) − Cxc (t))
R0
v(t) = U 0 xc (t) + −τ U 1 (s)v(t + s)ds + b(t)
R0
c(t) = V 0 xc (t) + −τ V 1 (s)v(t + s)ds − e(t);
b = K 4c
where the compensator’s input is the signal “e”, where U 0 , V 0 , U 1 (·) and V 1 (·)
are defined in terms of the system matrices and the solutions to the said Riccati
equations,as specified in the article, and where K 4 is a stable system with an
induced norm smaller than γ 2 − 1.
4. Frequency-domain methods
We review two transform-based methods, where state space machinery is restricted
to computational purposes. Again, the emphasis is on exploitation of the special
structure of the delay block to simplify solutions.
4.1
J-spectral factorization and the Smith predictor
A natural way to address delay is via the Smith predictor [62]. The predictor
compensates for the delay with a specially constructed internal feedback loop in the
H∞ control of system with i/o delay: review
7
controller (see §5.1). Consequently, it reduces some dead-time problems to delayfree forms [54]. This idea, however, had little impact on H∞ methods. While a
controller structure, reminiscent of the Smith predictor, can be recovered from some
H∞ frequency-domain solutions (e.g., [19]), and the possibility to recast controllers
in a generalized Smith predictor form was pointed out by Naimark and Cohen
[48], the idea to use a special controller transformation to reduce a problem to an
equivalent finite-dimensional one had not been used in the H∞ literature till the
work of Meinsma and Zwart [38, 40, 39], where it was applied first to a mixed
sensitivity (two-block) problem, and then generalized to the standard (four-block)
H∞ problem with a single delay.
Here are some details of the developments in [40, 39]. The core step is a Jspectral factorization: given a self-adjoint transfer matrix M , find a (bi-proper,
outer, etc) transfer matrix L satisfying
M1 M3∼
I
0
M=
= L∼
L
(3)
M3 M2
0 −I
Solutions are well understood in the rational case [27]. In dead-time problems the
diagonal blocks remain rational, and M3 = e−τ s Mr,3 , with rational Mr,3 . This
structure can be exploited in solving (3), as follows. The transfer matrix M3 is (nonuniquely) expanded as e−τ s M2−1 Mr,3 = R − ∆s where R is rational and ∆s ∈ H∞ .
In these terms one has M = Ψ ∼ Mr Ψ where Mr (determined
the
by
rational blocks
I 0
of M and by R) is a rational and self adjoint, and Ψ = −∆
is invertible in
s I
H∞ . Thus, a J-spectral factorization is provided by L = Lr Ψ , where Lr is the
rational J-spectral factorization of Mr . Necessary to complete the reduction is
an additional (stability) test, involving L, which cannot be expressed in terms of
rational matrices. That test, however, involves only solvability conditions and does
not affect the resulting controller.
To see the connection with the Smith predictor, consider the J-spectral
factorization controller. As is well known, proper compensators C, providing
γ-suboptimal
of
solutions
the
corresponding dead-time H∞ problem are those
satisfying I −C L−1 Q
= 0, for some Q ∈ H∞ , kQk∞ < γ. Substituting
I
−1
the explicit form
of
L
into
this
−1 Q expression, C = Cr (I + ∆s Cr ) , where the rational
Cr satisfies I −Cr Lr I = 0, i.e., it is the controller corresponding to the
“rationalized” version of (3). Thus, the resulting controller has the Smith predictor
structure where ∆s serves as a dead-time compensator block (in fact, C above can
be presented in the form shown in Fig. 2 on p. 9, where Gr depends on Lr only).
Note, that thechoice of
∆s and. R is non unique. In [40] ∆s was chosen in the
form ∆s = π τ e−τ s Pa for Pa = M2−1 Mr,3 . When Pa is stable, R can be chosen
either as R = 0 (in the spirit of the Morari’s internal model control) or R = Pa (in
the spirit of the Smith predictor). In the latter case Mr is equal to the delay-free
(τ = 0) version of M , which means that if Pa is stable, then the dead-time problem
is solvable iff so is its delay-free counterpart. Typically, however, Pa is unstable.
Comparing with the time domain methods in Section 3, the clear advantage is
in a simpler and more transparent controller structure. The disadvantage lies in a
somewhat less transparent structure of the solvability conditions, which are based
8
L. Mirkin & G. Tadmor
upon two H∞ AREs, one of which depends on the delay τ (in [47] only the coupling
condition depends on τ ).
4.2
Extraction of dead-time controller from delay-free parameterizations
Conventionally, the delay block e−τ s is treated as a part of a plant. This results
in an infinite-dimensional plant, and as seen above, typically, a rather involved
controller structure. In [41, 42] it is proposed to treat the delay as a part of the
controller. This leads to the design of a constrained-structure controller for a finitedimensional plant. The challenge here is to extract admissible (delay) controllers
from the well known parameterization of all suboptimal controllers for the rational
plant.
The idea is borrowed from [45], where the structure of sampled-data controllers
in the lifted domain (see §3.1) was studied. It was shown that a controller can
be implemented as the cascade of a generalized sampler, a discrete-time controller
and a generalized hold iff its lifted “transfer function” is strictly proper. Using this
fact, the sampled-data H∞ standard problem was then solved by extracting strictly
proper transfer functions from the lifted version of the DGKF [15] parameterization
of all continuous-time H∞ controllers. This procedure yields simple formulae
both for the digital part of the controller and for the generalized sampler and
hold. Moreover, an attractive byproduct of this procedure is that the solvability
conditions are naturally separated into the existence conditions for a continuoustime controller and an additional condition imposed by the sampled-data structure.
The constraints imposed upon dead-time controllers in the lifted domain are
more complicated than in the sampled-data case. For that reason, a direct extension
of the technique of [45] to delay systems is much more involved. The extraction
approach, however, can be adopted to dead-time systems by translating the deadtime constraints into the frequency domain. This approach was followed in [42]
resulting in simple and intuitively appealing solvability conditions and the controller
structure for the dead-time H∞ standard problem. Below, the solution procedure
in [42] is outlined.
The standard (four-block) H∞ problem with a single delay e−τ s in the loop
is considered. The solution starts with the parameterization of all γ-suboptimal
controllers K for the rational part of the plant. This parameterization is well-known
[15] and has the form of an LFT interconnection of a 2 × 2 block transfer matrix G
with an arbitrary Qa ∈ H∞ such that kQa k∞ < γ. The next step is then to extract
all Qa which bring the LFT in the form e−τ s Kτ for a proper Kτ . This can easily be
done by the loop-shifting arguments of [44] yielding the following characterization
of all admissible Qa : Qa = ∆22 + e−τ s Qb , where Qb ∈ H∞ is arbitrary and ∆22
is the FIR truncation of the (2, 2) subblock of G−1 . Now, taking into account the
norm constraint on Qa one can see that the original (four-block) problem is reduced
to the (one-block) problem of the characterization of all Qb ∈ H∞ such that:
k∆22 + e−τ s Qb k∞ < γ.
(4)
According to the Nehari Theorem (see [19, Theorem 15]), (4) is solvable iff γ is
larger than the norm of the Hankel operator with symbol eτ s ∆22 that, in turn,
H∞ control of system with i/o delay: review
u
- ∆s
Gr
9
y ?y
e i
- Q
Fig. 2. DTC structure of delay controllers Kτ .
is equivalent to the L2 [0, τ ] induced norm of ∆22 . Thus, γ > k∆22 kL2 [0,τ ] is the
additional solvability condition, as compared with the delay-free case, capturing the
effect of delay on achievable performance. Note, that this condition can be expressed
either in terms of the matrix exponential of a Hamiltonian matrix or in terms of
the solvability of a differential Riccati equation defined over [0, τ ], see [8, 28], where
several computational schemes are discussed (as a matter of fact, when expressed
in terms of a Riccati equation, these conditions coincide with those obtained by
Nagpal and Ravi [47]). Then, for any admissible γ the set of all solutions to (4) can
be parameterized in terms of an arbitrary Q ∈ H∞ such that kQk∞ < γ. Finally,
the latter is substituted into the original delay-free parameterization yielding, after
some elaborate simplification steps, the characterization of all dead-time controllers
solving the original problem.
5. Discussion
Perhaps the most serious disadvantage of the early H∞ methods for delay systems is
the lack of transparency. Design procedures typically involved several intermediate
steps, so the controller structures and the achievable cost are not easily recoverable.
On the other hand, the methods described in Sections 3 and 4 enable one to compute
both the (sub)optimal controllers and the achievable cost explicitly for general
problems in terms of the original data. Among other advantages, this enables one to
gain deeper insight into the controller structures and the limitations on performance
imposed by delay elements. These issues will be discussed below.
5.1
Controller structure
One of the most widely used controller configurations to control delay systems is
the so-called dead-time compensator (DTC) depicted in Fig. 2. DTC consists of a
rational part Gr and a stable internal feedback ∆s containing the delay (the free
parameter Q is typically zero). This scheme was proposed by Smith [62], whose idea
was to use the internal feedback with ∆s = P22 − e−τ s P22 to convert dead-time
problems to their delay-free versions. The rationale behind this approach becomes
.
apparent when the signal ye = y + ∆s u entering the rational part of the controller
10
L. Mirkin & G. Tadmor
in Fig. 2 is considered. Indeed, it is readily seen that
ye = e−τ s P21 w + P22 u + P22 − e−τ s P22 u = P21 e−τ s w + P22 u
and thus the resulting feedback loop (u ye ) does not contain any delay and is
exactly the feedback loop for τ = 0. In other words, ∆s “compensates” the delay
in the loop. The scheme of Smith can be applied to stable systems only. Yet it can
easily be modified to cope with the general case by choosing ∆s = P̃22 − e−τ s P22
with any P̃22 making ∆s stable. In particular, it is always possible to make ∆s FIR
[79]. Some other possible choices are discussed in [40]. Moreover, as shown in [44]
the set of all stabilizing dead-time controllers can be parameterized in the DTC
form shown in Fig. 2.
In the context of H∞ control, the DTC controller structure was derived by
Meinsma and Zwart [40] and Mirkin [41]. The version of the H∞ DTC proposed in
[41], in which the emphasis is on transparency, is discussed below (for more details
see [43]).
The set of all γ suboptimal dead-time controllers is parameterized in [41, 42]
in the form depicted in Fig. 2, subject to a rational Gr , similar to that for the
delay free
Q ∈ H∞ such
that kQk∞ < γ, and the FIR
problem,
an arbitrary
.
∼
∆s = π τ e−τ s Pa , where Pa = Fu P, γ −2 P11
. Unlike the Smith predictor and its
modifications, the delay in the loop is now not canceled (since e−τ s (P22 − Pa ) 6= 0).
Yet it is easily seen that the relationship y = Pa u can be equivalently written as
y = P21 w∗ + P22 u,
where the “disturbance” w∗ satisfies:
w∗ =
∼
1
γ 2 P11
P11 w∗ + P12 u .
Thus, if the
w in Fig. 1 were equal to the w∗ above, then the block
disturbance
∆s = π τ e−τ s Pa would just compensate the dead time in both feedback and
feedforward loops (“predict” y subject to a given w). This agrees well with the result
of Palmor and Powers [55], who proposed to transmit measured disturbances into
the Smith predictor to improve its disturbance attenuation properties (feedforward
Smith predictor). Although the H∞ DTC does not measure the disturbance, it
generates it artificially.
Taking into account the worst-case nature of the H∞ methodology, one would
expect that the disturbance w∗ is generated on the basis of a worst-case scenario. It
turns out that this indeed happens. To see this, consider the relationship y = Pa u
in the state-space setting. We have:
−λ̇ = A0 λ + C10 z
ẋ = Ax + B1 w∗ + B2 u
and
,
y = C2 x + D21 w∗
w∗ = γ12 B10 λ
where z = C1 x+D12 u. Using the standard arguments from the calculus of variations
[26], the w∗R above can roughly be thought of as the maximizing disturbance for the
∞
index J = 0 (z 0 z − γ 2 w0 w)dt subject to any fixed u.
H∞ control of system with i/o delay: review
11
Thus, the H∞ DTC attempts to compensate the dead time h assuming that the
disturbance w is the worst-case one for the open-loop problem. The fact that w∗ is
worst-case for an open-loop problem agrees well with the open-loop nature of the
dead-time compensation.
5.2
The “cost of delay”
Another important question raising in H∞ control of delay systems concerns the
quantification of the effect of delay on achievable performance. Obviously, any delay
problem is solvable only if so is its delay-free counterpart. One therefore would
expect that the achievable performance level for any dead-time problem can be
presented as a sum of the delay-free performance and a term that reflects the cost
of delay. The knowledge of the latter may be of value in numerous applications
where delay tolerance is important. Unfortunately, in most early treatments, this
“cost of delay” cannot be easily recovered from the solvability conditions. This
complicates any further analysis considerably. In this respect, [47, 74, 41] offer a
clear advantage since the solvability conditions there are explicitly presented in
such a two-term form with a transparent “cost of delay” term. In particular, the
following result can be summarized:
Lemma 2 Consider the system in Fig. 1 with P11 = C1 (sI−A)−1 B1 . The following
three conditions are equivalent (below, X, Y > 0 stand for the stabilizing solutions
to the matrix Riccati equations associated with the delay-free problem):
(a) There exists a stabilizing Kτ so that kF` P, e−τ s Kτ k∞ < γ
(b) The problem is solvable for τ = 0 and, in addition, Π(t) defined on [0, τ ] as
Π̇ + ΠA + A0 Π + C10 C1 +
0
1
γ 2 ΠB1 B1 Π
= 0,
Π(τ ) = X,
(5)
exists and satisfies ρ(YΠ(0)) < γ 2 ;
(c) The problem is solvable for τ = 0 and, in addition, kP11 kL2 [0,τ ] < γ and
−1
ρ XΣ12 Σ22
< 1 and ρ (Σ22 − XΣ12 )−1 (Σ21 − XΣ11 )Y < γ 2 , where
Σ11
Σ21
Σ12
Σ22
.
= exp
A
−C10 C1
1
0
γ 2 B1 B1
0
−A
τ .
Condition (b) above is from [47] while condition (c) is from [41]. Although the
solvability conditions in [74] appear slightly different, they can be brought to a form
equivalent to that in [47] using standard manipulations of Hamiltonian matrices.
Some observations are in order. First, it appears that in some situations (small
enough) delay does not contribute to the final cost, i.e., any performance achievable
with τ = 0 can be achieved with “small” τ as well. Indeed, it is known [23] that the
delay-free test fails if either (i) ρ(YX) = γ 2 or (ii) X or Y is no longer stabilizing.
Assume for simplicity that the triple (C1 , A, B1 ) is observable and controllable.
Then the solution Π(t) of (5) is monotonically increasing as t passes from τ to
0. Therefore, if (i) happens prior to (ii) in the delay-free case, then the condition
12
L. Mirkin & G. Tadmor
ρ(YΠ(0)) < γ 2 fails for all τ > 0. Yet if the opposite happens for some γ = γ ∗ , then
there exists a τ0 > 0 such that for every γ > γ0 condition (b) of Lemma 2 holds
true ∀τ < τ0 . Moreover, if the observability or/and controllability of (C1 , A, B1 ) are
relaxed, then small enough delay might not affect the achievable performance even
when the spectral radius condition fails first. This, for instance, occurs in the case
when A is Hurwitz and P11 = 0, see [44].
Another interesting observation is that exactly the same condition appears in
the context of sampled-data control when both sampler and hold are the design
parameters [69, 45]. In other words, it turns out that the cost of delay is equal to
the cost of sampling. This fact is surprising since the constraints imposed by the
delay are more restrictive, than those imposed by the sampling. Indeed, as was
mentioned in §4.2, a controller K contains the delay e−τ s iff its kernel k(t, s) = 0
whenever t < s + τ . On the other hand, it follows from [45] that a controller is
the cascade of a generalized sampler, a discrete-time controller and a generalized
hold with the period τ iff its kernel k(t, s) = 0 whenever t < b τs cτ + τ , where
b·c stands for the “floor” (round towards −∞) operation. Thus, the sampled-data
controller “receives odds” on the intervals b τs c 6 τt − 1 < τs but is not able to
take any advantage of it. A possible explanation may be that on this interval the
sampled-data controller can act only in an open-loop fashion. Then, the disturbance
is “smart enough” to outplay the control.
6. Conclusions
In this paper several problem-oriented methods for the H∞ control of continuoustime systems with a single delay in the feedback loop have been reviewed. Reviewed
methods include operator interpolation methods (Section 2) and game-theoretic
state-space (Section 3) treatments in the time domain and a J-spectral factorization
(Subsection 4.1) and extraction (Subsection 4.2) approaches in the frequency
domain. Some properties of the resulting solutions and solvability conditions have
also been discussed in Section 6.
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