A new approach to controller design in presence of constraints

Abstract

In this paper, we present a new approach to control design in presence of constraints. This approach relies on the reformulation of the controller design problem as a semi-infinite convex optimization program, and on the solution of this program by the scenario optimization technology. The approach is illustrated through a simple example of disturbance rejection subject to input saturation constraints.

Full text

A new approach to controller design in presence of constraints
Maria Prandini and Marco C. Campi
Abstract— In this paper, we present a new approach to
control design in presence of constraints. This approach relies
on the reformulation of the controller design problem as a
semi-infinite convex optimization program, and on the solution
of this program by the scenario optimization technology.
The approach is illustrated through a simple example of
disturbance rejection subject to input saturation constraints.
I. INTRODUCTION
In this paper, we propose a new approach to address robust
control design in presence of constraints in a systematic and
optimal way.
For ease of explanation, we illustrate this new approach
through a simple example where, given a linear system
affected by a disturbance belonging to some class, the goal
is to design a feedback controller that attenuates the effect
of the disturbance on the system output, while avoiding
saturation of the control action due to actuator limitations.
The proposed control design method relies on the re-
formulation of the problem as a robust convex optimiza-
tion program by adopting an appropriate parametrization
of the controller. A robust convex optimization problem is
expressed in mathematical terms as
min
θ∈ℜ
n
g(θ) subject to: (1)
f
δ
(θ) ≤ 0, ∀δ ∈ ∆,
where δ is the uncertain parameter, and g(θ) and f
δ
(θ) are
convex functions in the n-dimensional optimization variable
θ for every δ within the uncertainty set ∆. Convexity is
appealing since ‘convex’ - as opposed to ‘non-convex’ -
means ‘solvable’ in many cases, [1], [2]. In our context,
the uncertain parameter δ represents a realization of the
disturbance affecting the system, hence ∆ contains an infinite
number of instances. It is well known that semi-infinite opti-
mization problems, that is problems with a finite number n of
optimization variables and an infinite number of constraints,
are difficult to solve and they have even proven NP-hard in
some cases, [3], [4], [5], [6].
In [7], [8], an innovative technology called ‘scenario
approach’ has been introduced to deal with semi-infinite
convex programming at a very general level. The main thrust
This work was supported by MIUR (Ministero dell’Istruzione,
dell’Universit
`
a e della Ricerca) under the project Identification and adaptive
control of industrial systems.
M. Prandini is with Dipartimento di Elettronica e Informazione, Po-
litecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
prandini@elet.polimi.it
M.C. Campi is with Dipartimento di Elettronica per l’Automazione,
Universit
`
a degli Studi di Brescia, via Branze 38, 25123 Brescia, Italy
marco.campi@ing.unibs.it
of this technology is that solvability can be obtained through
random sampling of constraints provided that a probabilis-
tic relaxation of the worst-case robust paradigm of (1) is
accepted. Here, we propose to use the scenario technology
for determining a solution to control design problems that
would otherwise be hard to solve because of the presence
of constraints and of uncertain signals/disturbances affecting
the system. No extensions of the scenario approach itself are
developed. Randomized algorithms for system analysis and
control design have recently become a topic of great interest
for the control community (see [9] for a comprehensive
survey on the subject). Our contribution consists in the
introduction of a novel randomized algorithm for robust
control design in presence of constraints, which is based on
the scenario approach.
In our control set-up where the uncertain parameter δ
represents the disturbance realization, the implementation
of the scenario optimization requires to randomly extract a
certain number of disturbance realizations and to simulate the
system behavior with the extracted realizations as input. This
justifies the terminology we adopt to describe the proposed
approach to control design as a ‘simulation-based method’.
The problem of disturbance rejection has been addressed
in the literature based on the dynamic programming approach
[10], [11], [12], the l
1
-optimal control theory [13], [14],
the use of an upper bound on the l
1
-norm (the star-norm)
[15], and, more recently, through the invariant ellipsoids
technique [16]. In all these approaches, the disturbance is
only assumed to be bounded. Further possible knowledge on
the disturbance signal (such as, for example, its correlation
in time and main frequency components) is not exploited
in the design process, which may lead to sub-optimal and
conservative solutions for the problem at hand. Also, in the
approaches based on dynamic programming and l
1
-optimal
control, the order of the controller cannot be fixed a-priori
and the complexity of the resulting ‘optimal’ compensator
may be high.
Other methodologies for solving quite general control
design problems for linear systems affected by uncertain
signals/disturbances and subject to constraints are present
in the literature of receding horizon and model predictive
control, [17], [18], [19], [20]. Differently from what we
propose here, no structure is imposed to the feedback con-
troller in these papers and design is carried out by directly
optimizing over the control input samples in a time horizon
of interest. The resulting feedback controller suffers from
the problem to be difficult to implement, but it secures high
performance under certain hypotheses. Moreover, applicabil-
ity of standard methods in receding horizon model predictive
Proceedings of the
46th IEEE Conference on Decision and Control
New Orleans, LA, USA, Dec. 12-14, 2007
WeA17.1
1-4244-1498-9/07/$25.00 ©2007 IEEE. 530

control requires that uncertainty is quite structured (typically,
the uncertain signals/disturbances are characterized through
some polytopic or ellipsoidal bound on their instantaneous
value), a limitation which is largely overcome by the ap-
proach proposed here.
The rest of the paper is organized as follows. In Section
II, we precisely describe the control problem addressed and
its reformulation as a semi-infinite convex optimization pro-
gram. The application of the scenario technology to solve this
optimization program is then explained in Section III, and a
numerical example is provided in Section IV to illustrate the
effectiveness of the resulting randomized method for control
design. Some concluding remarks are drawn in Section V.
II. CONTROL PROBLEM FORMULATION
We consider a discrete time linear system with scalar input
and scalar output, u(t) and y(t), governed by the following
equation:
y(t) = G(z)u(t) + d(t), (2)
where G(z) is a stable transfer function and d(t) is an
additive disturbance.
Our objective is to determine a feedback control law
u(t) = C(z)y(t) (3)
(see Figure 1) such that the disturbance d(t) is optimally
attenuated for every realization of d(t) in some set of
possible realizations D, and such that the control input keeps
within certain saturation limits. For example, D can be the set
of step functions with specified maximum amplitude or the
set of sinusoids with frequency in a certain range. A precise
formalization of the optimization problem is next given.
Fig. 1. The feedback disturbance compensation scheme.
Consider the finite-horizon 2-norm
P
M
t=1
y(t)
2
of the
closed-loop system output. This norm quantifies the effect of
the disturbance d(t). For simplicity, we here consider (2) and
(3) initially at rest, namely G(z)u(t) represents an infinite
backwards expansion
P
∞
j=1
g
j
u(t − j) where u(t − j) = 0
for t − j ≤ 0, and similarly for C(z)y(t).
The goal is to minimize the worst-case disturbance effect
max
d(t)∈D
M
X
t=1
y(t)
2
, (4)
while maintaining the control input u(t) within a saturation
limit u
bound
:
max
1≤t≤M
|u(t)| ≤ u
bound
, ∀d(t) ∈ D. (5)
Controller C(z) is expressed in terms of an Internal Model
Control (IMC) parametrization, [21]:
C(z) =
Q(z)
1 + Q(z)G(z)
, (6)
where G(z) is the system transfer function and Q(z) is a
free-to-choose transfer function (see Figure 2).
Fig. 2. The IMC parameterization of the controller.
Expression of C(z) in (6) is totally generic, in that, given
a C(z), a Q(z) can be always found generating that C(z)
through expression (6). The advantage of (6) is that the set
of all controllers that closed-loop stabilize G(z) is simply
obtained from (6) by letting Q(z) vary over the set of all
stable transfer functions (see [21] for more details).
With (6) in place, the control input u(t) and the controlled
output y(t) are given by:
u(t) =
C(z)
1 − C(z)G(z)
d(t) = Q(z)d(t) (7)
y(t) = G(z)u(t) + d(t) = [G(z)Q(z) + 1]d(t). (8)
The distinctive feature of these expressions is that u(t)
and y(t) are affine in Q(z). Consequently, if Q(z) is selected
from a family of stable transfer functions linearly parame-
terized in γ := [γ
0
γ
1
. . . γ
k
]
T
∈ ℜ
k+1
, i.e.
Q(z) = γ
0
β
0
(z) + γ
1
β
1
(z) + γ
2
β
2
(z) + · · · + γ
k
β
k
(z), (9)
where β
i
(z)’s are pre-specified stable transfer functions, then
the cost (4) and the constraints (5) are convex in γ.
A common choice for the β
i
(z)’s functions is to set them
equal to pure ‘delays’: β
i
(z) = z
−i
, leading to
Q(z) = γ
0
+ γ
1
z
−1
+ γ
2
z
−2
+ · · · + γ
k
z
−k
.
Another possibility is to let β
i
(z)’s be Laguerre polynomials,
[22], [23].
The control design problem can now be precisely formu-
lated as follows:
min
γ,h∈ℜ
k+2
h subject to: (10)
M
X
t=1
y(t)
2
≤ h, ∀d(t) ∈ D, (11)
max
1≤t≤M
|u(t)| ≤ u
bound
, ∀d(t) ∈ D. (12)
Due to (11), h represents an upper bound to the output 2-
norm
P
M
t=1
y(t)
2
for any realization of d(t). Such an upper
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bound is minimized in (10) under the additional constraint
(12) that u(t) does not exceed the saturation limits.
We now rewrite problem (10)–(12) in a more explicit form.
By (7) and (8) and the parametrization of Q(z) in (9),
the input and the output of the controlled system can be
expressed as
u(t) =
¡
γ
o
β
0
(z) + . . . + γ
k
β
k
(z)
¢
d(t) (13)
y(t) = G(z)
¡
γ
o
β
0
(z) + . . . + γ
k
β
k
(z)
¢
d(t) + d(t). (14)
Let us define the following vectors containing filtered
versions of the disturbance d(t):
φ(t) :=





β
0
(z)d(t)
β
1
(z)d(t)
.
.
.
β
k
(z)d(t)





and ψ(t) =





G(z)β
0
(z)d(t)
G(z)β
1
(z)d(t)
.
.
.
G(z)β
k
(z)d(t)





. (15)
Then, (13) and (14) can be re-written as
u(t) = φ(t)
T
γ
y(t) = ψ(t)
T
γ + d(t),
and
P
M
t=1
y(t)
2
= γ
T
Aγ + Bγ + C, where
A =
M
X
t=1
ψ(t)ψ(t)
T
, B = 2
M
X
t=1
d(t)ψ(t)
T
, C =
M
X
t=1
d(t)
2
(16)
are matrices that depend on d(t) only.
With all these positions, (10)–(12) rewrites as
min
γ,h∈ℜ
k+2
h subject to: (17)
γ
T
Aγ + Bγ + C ≤ h, ∀d(t) ∈ D
− u
bound
≤ φ(t)
T
γ ≤ u
bound
, ∀t ∈ {1, 2, . . . , M },
∀d(t) ∈ D.
Compared with the general form (1), the optimization
variable θ is here (γ, h) and has size n = k + 2, and
the uncertain parameter δ is the disturbance realization d(t)
taking value in the set ∆ = D. Note that, given d(t),
quantities A, B, C, and φ(t) are fixed so that the first
constraint in (17) is quadratic, while the others are linear.
Typically, the set D of disturbance realizations has infinite
cardinality. Hence, problem (17) is a semi-infinite convex
optimization problem.
III. RANDOMIZED SOLUTION THROUGH THE SCENARIO
TECHNOLOGY
As already pointed out in the introduction, semi-infinite
convex optimization problems like (17) are difficult to solve.
The idea of the scenario approach is that solvability can
be recovered if some relaxation in the concept of solution
is accepted. In the context of our control design problem,
this means requiring that the constraints in (17) are satisfied
for all disturbance realizations but a small fraction of them
(chance-constrained approach).
The scenario approach goes as follows. Since we are
unable to deal with the wealth of constraints in (17), we
concentrate attention on just a few of them and extract at
random N disturbance realizations d(t) according to some
probability distribution P introduced over D. This proba-
bility distribution should reflect the likelihood with which
the disturbance realizations occur or the relative importance
that is attributed to different disturbance realizations. If no
hint is available on which realization is more likely to occur
and none of them is more critical than the others, then the
uniform distribution can be adopted. A discussion on the
use of the uniform distribution in randomized methods can
be found in [24].
Only the extracted instances (‘scenarios’) are considered
in the scenario optimization:
SCENARIO OPTIMIZATION
extract N independent identically distributed realizations
d(t)
1
, d(t)
2
, . . . , d(t)
N
from D according to P . Then,
solve the scenario convex program (SCP
N
):
min
γ,h∈ℜ
k+2
h subject to: (18)
γ
T
A
i
γ + B
i
γ + C
i
≤ h, i = 1, . . . , N,
− u
bound
≤ φ(t)
T
i
γ ≤ u
bound
,
∀t ∈ {1, 2, . . . , M }, i = 1, . . . , N,
where A
i
, B
i
, C
i
, and φ(t)
i
are as in (16) and (15) for
d(t) = d(t)
i
.
Letting (γ
∗
N
, h
∗
N
) be the solution to SCP
N
, γ
∗
N
returns
the designed controller parameter, whereas h
∗
N
quantifies the
performance of the design compensator over the extracted
disturbance realizations d(t)
1
, d(t)
2
, . . . , d(t)
N
.
The implementation of the scenario optimization requires
that one picks N realizations of the disturbance and com-
putes A
i
, B
i
, C
i
, and φ(t)
i
in correspondence of the
extracted realizations. Since these quantities are artificially
generated (that is they are not actual measurements coming
from the system, but, instead, they are computer-generated),
the proposed control design methodology can as well be seen
as a simulation-based approach.
SCP
N
is a standard convex optimization problem with a
finite number of constraints, and therefore easily solvable.
On the other hand, it is spontaneous to ask: what kind of
solution is one provided by SCP
N
? Specifically, what can we
claim regarding the behavior of the designed control system
for all other disturbance realizations, those we have not taken
into consideration while solving the control design problem?
Answering this question is necessary to provide performance
guarantees.
The above question is of the ‘generalization’ type in a
learning-theoretic sense: we want to know how the solution
(γ
∗
N
, h
∗
N
) generalizes in constraints satisfaction, from seen
disturbance realizations to unseen ones. Certainly, any gener-
alization result calls for some structure as no generalization
is possible if no structure linking what has been seen to
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 WeA17.1
532

what has not been seen is present. The formidable fact in
the context of convex optimization is that the solution of
SCP
N
always generalizes well, with no extra assumptions.
We have the following theorem (see Corollary 1 in [8]).
Theorem 1: Select a ‘violation parameter’ ǫ ∈ (0, 1) and
a ‘confidence parameter’ β ∈ (0, 1). Let n = k + 2.
If
N =
»
2
ǫ
ln
1
β
+ 2n +
2n
ǫ
ln
2
ǫ
¼
(19)
(⌈·⌉ denotes the smaller integer greater than or equal to the
argument), then, with probability no smaller than 1 − β, the
solution (γ
∗
N
, h
∗
N
) to (18) satisfies all constraints of problem
(17) with the exception of those corresponding to a set of
disturbance realizations whose probability is at most ǫ. ¤
Let us read through the statement of this theorem in some
detail. If we neglect the part associated with β, then, the
result simply says that, by sampling a number of disturbance
realizations as given by (19), the solution (γ
∗
N
, h
∗
N
) to (18)
violates the constraints corresponding to other realizations
with a probability that does not exceed a user-chosen level
ǫ. This corresponds to say that – for other, unseen, d(t)’s
– constraints (11) and (12) are violated with a probability
at most ǫ. From (11) we therefore see that the found h
∗
N
provides an upper bound for the output 2-norm
P
M
t=1
y(t)
2
valid for any realizations of the disturbance with exclusion of
at most an ǫ-probability set, while (12) guarantees that, with
the same probability, the saturation limits are not exceeded.
As for the probability 1−β, one should note that (γ
∗
N
, h
∗
N
)
is a random quantity because it depends on the randomly
extracted disturbance realizations. It may happen that the
extracted realizations are not representative enough (one can
even stumble on an extraction as bad as selecting N times the
same realization!). In this case no generalization is certainly
expected, and the portion of unseen realizations violated by
(γ
∗
N
, h
∗
N
) is larger than ǫ. Parameter β controls the probabil-
ity of extracting ‘bad’ realizations, and the final result that
(γ
∗
N
, h
∗
N
) violates at most an ǫ-fraction of realizations holds
with probability 1 − β.
In theory, β plays an important role and selecting β = 0
yields N = ∞. For any practical purpose, however, β has
very marginal importance since it appears in (19) under the
sign of logarithm: we can select β to be such a small number
as 10
−10
or even 10
−20
, in practice zero, and still N does
not grow significantly.
It is worth mentioning that improved bounds on the sample
complexity N have been developed very recently in [25] and
[26]. In particular, the bound derived in [26] is exact for the
class of the so-called fully-supported problems.
IV. NUMERICAL EXAMPLE
A simple example illustrates the controller design proce-
dure.
With reference to (2), let
G(z) =
0.2
z − 0.8
,
and let the additive output disturbance be a piecewise con-
stant signal that varies from time to time, at a low rate, of
an amount bounded by some given constant. Specifically,
let the set of admissible realizations D consists of piecewise
constant signals changing at most once over any time interval
of length 50, and taking value in [−1, 1].
As for the IMC parametrization Q(z) in (9), we choose
k = 1 and Q(z) = γ
0
+ γ
1
z
−1
.
A control design problem (10)–(12) is considered with
M = 300, and for two different values of the saturation
limit u
bound
: 10 and 1. Probability P is implicitly assigned
by the recursive equation
d(t + 1) =
¡
1 − µ(t)
¢
d(t) + µ(t)v(t + 1),
initialized with d(1) = v(1), where µ(t) is a {0, 1}-valued
process (µ(t) = 1 at times where a jump occurs), and v(t) is
a sequence of i.i.d. random variables uniformly distributed
in [−1, 1] (v(t) is the new d(t) value). µ(t) is generated
according to
µ(t) = α(t)
50
Y
k=1
¡
1 − µ(t − k)
¢
,
initialized with µ(0) = µ(−1) = · · · = µ(−49) = 0,
where α(t) is a sequence of i.i.d. {0, 1}-valued random
variables taking value 1 with probability 0.01. An admissible
realization of d(t) in D is reported in Figure 3.
1 50 100 150 200 250 300
−1.5
−1
−0.5
0
0.5
1
1.5
Fig. 3. A disturbance realization.
In the scenario approach we let ǫ = 5 · 10
−2
and β =
10
−10
. Correspondingly, N given by (19) is N = 1370.
From Theorem 1, with probability no smaller than 1 −
10
−10
, the obtained controller achieves the minimum of
P
M
t=1
y(t)
2
over all disturbance realizations, except a frac-
tion of them of size smaller than or equal to 5%. At the
same time, the control input u(t) is guaranteed not to exceed
the saturation limit u
bound
except for the same fraction of
disturbance realizations.
A. Simulation results
For u
bound
= 10, we obtained Q(z) = −4.993 + 4.024z
−1
and, correspondingly, the transfer function F (z) = 1 +
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 WeA17.1
533

Q(z)G(z) between d(t) and y(t) (closed-loop sensitivity
function) was
F (z) = 1 + (−4.993 + 4.024z
−1
)
0.2
z − 0.8
≃ 1 − z
−1
.
The pole-zero plot of F (z) is in Figure 4.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Fig. 4. Pole-zero plot of F (z) when u
bound
= 10. The poles are plotted
as x’s and the zeros are plotted as o’s.
Since y(t) = F (z)d(t) ≃ d(t) − d(t − 1), then, when
d(t) has a step variation, y(t) changes of the same amount
and, when the disturbance gets constant, y(t) is immediately
brought back to zero and maintained equal to zero until
the next step variation in d(t) (see Figure 5). The obtained
solution that F (z) is approximately a FIR (Finite Impulse
Response) of order 1 with zero DC-gain is not surprising
considering that d(t) varies at a low rate.
1 50 100 150 200 250 300
−1.5
−1
−0.5
0
0.5
1
1.5
d(t)
y(t)
Fig. 5. Disturbance realization and corresponding output of the controlled
system for u
bound
= 10.
In the controller design just described, the limit u
bound
=
10 played no role in that constraints −u
bound
≤ φ(t)
T
i
γ ≤
u
bound
in problem (18) were not active at the found solution.
As u
bound
is decreased, the saturation limits become more
stringent and affect the solution.
For u
bound
= 1, the following scenario solution was found
Q(z) = −0.991 + 0.011z
−1
, which corresponds to the
sensitivity function:
F (z) = 1 + (−0.991 + 0.011z
−1
)
0.2
z − 0.8
≃
z − 0.996
z − 0.8
.
The pole-zero plot of F (z) is in Figure 6, while Figure 7
represents y(t) obtained through this new controller for the
same disturbance realization as in Figure 5. Note that the
time required to bring y(t) back to zero after a disturbance
jump is now longer than 1 time unit, owing to saturation
constraints on u(t).
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Fig. 6. Pole-zero plot of F (z) when u
bound
= 1. The poles are plotted as
x’s and the zeros are plotted as o’s.
1 50 100 150 200 250 300
−1.5
−1
−0.5
0
0.5
1
1.5
d(t)
y(t)
Fig. 7. Disturbance realization and corresponding output of the controlled
system for u
bound
= 1.
The optimal control cost value h
∗
N
is h
∗
N
= 9.4564 for
u
bound
= 10 and h
∗
N
= 27.4912 for u
bound
= 1. As expected,
the control cost increases as u
bound
becomes more stringent.
The numerical example of this section is just one instance
of application of the scenario approach to controller selec-
tion. The introduced methodology is of general applicability
to diverse situations with constraints of different type, pres-
ence of reference signals, etc.
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 WeA17.1
534

V. CONCLUSIONS
In this paper, we considered an optimal disturbance re-
jection problem with limitations on the control action and
showed how it can be effectively addressed by means of
the so-called scenario technology. This approach basically
consists of the following main steps:
- reformulation of the problem as a robust (usually with
infinite constraints) convex optimization problem;
- randomization over constraints and resolution (by
means of standard numerical methods) of the so ob-
tained finite optimization problem;
- evaluation of the constraint satisfaction level of the
obtained solution through Theorem 1.
Extensions to tracking of some class of reference signals, and
to control problems where the initial condition is uncertain
or the output of the system is subject to some constraint are
quite straightforward.
The applicability of the scenario methodology is not
limited to optimal control problems with constraints and,
indeed, this same methodology has been applied to a number
of different endeavors in systems and control, [27], [28], [29],
[30].
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