Checking if controllers are stabilizing using closed-loop data
Alexander Lanzon, Andrea Lecchini, Arvin Dehghani and Brian D. O. Anderson
Abstract—Suppose an unknown plant is stabilized by a
known controller. Suppose also that some knowledge of the
closed-loop system is available and on the basis of that knowl-
edge, the use of a new controller appears attractive, as may arise
in iterative control and identification algorithms, and multiple-
model adaptive control. The paper presents tests using a limited
amount of experimental data obtained with the existing known
controller for verifying that introduction of the new controller
will stabilize the plant.
Let [P, C0] be a feedback control interconnection. The
symbols P and C0 denote respectively the plant and the
controller. The Multiple Input Multiple Output (MIMO) case
is considered here. The transfer function P(s) is not known
while the transfer function C0(s) is known. The closed-loop
interconnection [P, C0] is known to be internally stable and
is available for experiments. Let C1denote a new controller
which has been designed to replace C0in the loop. In this
paper, we develop tests to check whether C1 (instead of
C0) stabilizes the feedback loop. These tests are based on
the knowledge of C0(s) and C1(s) and on data obtained
from experiments on the closed-loop system [P, C0], but
not directly on P. The tests are based on gross properties
of the behaviour of the closed-loop, and so should exhibit
significant tolerance of noise.
It should be noticed that many iterative control design
methods have been developed to use closed-loop data ob-
tained from an existing closed loop system in order to update
the current controller with a controller with better perfor-
mance , , . Iterative data based control methods are
mainly focused on the objective of performance improvement
which is typically an objective competing with the robust
stability of the designed closed loop , . Therefore,
alongside data based iterative control design methods a num-
ber of stability tests have been developed to ascertain stability
of the new controller before implementing the controller in
the loop. Existing tests are based either on the identification
of a parametric ‘full order’ model of the current closed-loop
transfer function or on the estimation of frequency bounds
Corresponding author. Andrea Lecchini is with the Department of Engi-
neering, University of Leicester, LE17RH, UK, firstname.lastname@example.org
Alexander Lanzon, Arvin Dehghani and Brian D. O. Anderson are
with the Research School of Information Sciences and Engineering,
The Australian National University, Canberra ACT 0200, Australia and
the National ICT Australia Ltd., Locked Bag 8001, Canberra, ACT
This work was supported by an ARC Discovery-Projects Grant
(DP0342683) and National ICT Australia. National ICT Australia is funded
through the Australian Government’s Backing Australia’s Ability initiative,
in part through the Australian Research Council.
on the magnitude of the current closed-loop transfer function
, , , .
One may argue that a mismatch exists between the nature
of these tests and their usual application. Iterative methods
as , , ,  are based on limited closed loop
experiments which are intended to obtain information for the
design of small controller changes, see also , , , .
The existing validation tests are based on the identification of
the full dynamics of the current closed-loop system. Hence
the amount of experimental effort required for validation
purposes, can apparently be much larger that the amount of
experimental effort required for the design of the controller
update. In contrast to this fact we will show in Section IV
that our validation test requires gathering of information only
on a limited known frequency region whose size depends on
the size of the controller change. Hence the experimental
effort is linked to the size of the controller update.
In this paper we put forward the use of phase information.
Our validation tests rely on estimating the phase of the
current closed-loop transfer functions. The use of the phase
information to ascertain closed-loop stability derives from
the Nyquist stability criterion and leads to validation tests
which assess necessary and sufficient stability conditions.
This is in contrast with methods based on magnitude bounds
from which only sufficient conditions can be derived. We
will show that our validation experiments have can reflect
the limitation on the size of the controller update imposed
by the closed loop experimental setting. In particular it will
be shown that if the controller change has limited size then
it is sufficient to obtain an estimate of the phase of the
current closed loop system only up to a certain known finite
frequency. This fact makes the validation tests practical from
the experimental point of view.
The paper is organized as follows. In Section II we
recall coprime factors representations and stability results
in this framework. In this work we adopt coprime factors
representations because they allow us to obtain very neat
statements and simple derivations. In Section III we present
the result which defines the experimental setting for a
stability test based on phase information. Some stability
falsification and validation tests are derived in Section IV.
Numerical illustrations and conclusions complete the paper.
II. COPRIME FACTOR REPRESENTATIONS AND STABILITY
We shall denote by H∞the space of functions bounded
and analytic in the open right-half complex plane, and the
same function spaces with prefix R their real-rational proper
subspaces. The plant is assumed to be a MIMO linear time-
invariant system with m inputs and p outputs. The transfer
Proceedings of the 45th IEEE Conference on Decision & Control
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1-4244-0171-2/06/$20.00 ©2006 IEEE. 3660
(a) Magnitude Response of T?= T − 1
Phase (deg): −51.4
(b) Phase Response of T
Fig. 7.Example 2: Magnitude and Phase responses
satisfying Assumption 7,˜V1(j∞) = 1, and
˜U1=0.33(s + 0.586)(s + 2.99)(s + 3.416)
(s + 1.87)(s2+ 2.81s + 3.712)
Setting up the experimental configuration of Fig. 4 for
simulation and utilizing Theorem 6 to check if C1 is
stabilizing, we perform experiments with reference signal
r(t) = step(t) and the step response is measured at the
output z. The steady state of T : r → z is ¯ z = 4.74 > 0
which does not falsify the stability of [P, C1]. Thus, we
shall use the results of Theorem 11 to check if C1 is
stabilizing. As shown in Fig. 7a, the simulation reveals that
|T − 1| ≤ 1 ∀ω ≥ 1.27 rad/s. Given that argT(j0) = 0
and argT(jω0) = −0.285π as shown in Fig. 7b, then the
condition in Theorem 11 holds and hence C1is stabilizing.
Indeed, computing H(P, C1) shows that C1is stabilizing.
We have proposed tests for MIMO and SISO systems
to validate for stability the closed-loop system formed by
a controller and a plant, whose exact transfer function is
not known, a priori of the actual physical connection of the
controller to the plant. The tests assume that the plant is
connected to a stabilizing controller and that the resulting
closed loop system is available for experiments. The general
framework for our validation tests has been established in
Theorem 6. The result of Theorem 11 shows, for the SISO
case, that our validation tests require to gather information
on the frequency response of the current control system over
a limited bandwidth. Current research effort is set on the
extension of this result to the MIMO case. We will also
investigate which system identification methods are more
appropriate to gather the information required.
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