=Paper=
{{Paper
|id=None
|storemode=property
|title=Fuzzy Logic-based Robust Control of a Flexible two-mass System (1990 ACC Benchmark Problem)
|pdfUrl=https://ceur-ws.org/Vol-841/submission_7.pdf
|volume=Vol-841
|dblpUrl=https://dblp.org/rec/conf/maics/AdedijiE12
}}
==Fuzzy Logic-based Robust Control of a Flexible two-mass System (1990 ACC Benchmark Problem)==
https://ceur-ws.org/Vol-841/submission_7.pdf
Fuzzy Logic-based Robust Control of a Flexible two-mass
System (1990 ACC Benchmark Problem)
Adekunle C. Adediji and John Essegbey
School of Electronics and Computing Systems
University of Cincinnati
497 Rhodes Hall
Cincinnati, Ohio 45221-0030
Abstract H2 design with a collection of existing controllers such as
Pole Placement, and Minmax Linear Quadratic Gaussian
In intuitive design steps, a fuzzy logic-based robust (LQG). (Hughes and Wu 1996) also presented an observer-
controller is designed to address the first 1990-1992 based extension of a passive controller design, due to the
American Control Conference benchmark problem. Using a
fact that strictly passive feedback could no longer
conceptual transformation of the original flexible body into
a perpetual rigid body mode, a final design which succeeds
guarantee stability for the given problem.
in stabilizing the system after a unit impulse disturbance is
developed. The simulation results are shown to achieve and
exceed the required design specifications of the benchmark
problem, as well as those of other fuzzy logic-based
solutions.
Introduction
As the complexity of engineered systems increased, it
became imperative that the American Controls Conference
(ACC) adopt a set of control design problems as robust Figure 1: The ACC benchmark problem consisting of a dual mass,
control benchmark problems. This has led to several single spring system.
attempts by authorities in the field to come up with the best
possible solutions, serving as a good basis for comparing
the various heuristics and methodologies in designing for Some recent solutions, however, make use of qualitative
robust control. One of these problems, referred to by (Wie approaches capitalizing on fuzzy reasoning, which have
and Bernstein 1992) as ACC benchmark problem 1, was been shown to perform just as good as or even better than
concerned with vibration control of a two-mass system the existing quantitative methods (Cohen and Ben Asher
with an uncertain spring constant (Figure 1). The flexible 2001). It is worth noting that the presence of design
tow-mass system addresses, primarily, a disturbance constraints, and plant, as well as parameter uncertainties,
rejection control problem in the presence of parametric drastically increases the complexity of modeling plant
uncertainty. This problem has been addressed in over 30 behavior, and makes the application of non-linear solutions
papers, including papers in special issues of the Journal of worthwhile.
Guidance, Control and Dynamics and the International In this paper, we build on a solution using fuzzy logic.
Journal of Robust and Nonlinear Control (Linder and We start by generating a detailed model of the system and
Shafai 1999). highlight the required design objectives for the controller.
Probably due to the linearity of this problem, most Next we obtain a reduced or simplified model of the system
published solutions have appropriated linear controllers of in the rigid-body mode, where spring oscillations have
some sort, from H-infinity to game theory. (Niemann et al been effectively damped out using fuzzy logic heuristics
1997) applied the μ-synthesis method for mixed (Linder and Shafai 1999). Finally, an additional fuzzy
perturbation sets using a modified D-K iteration approach, controller produces a superimposition of stability and
while (Wie and Liu 1992) proposed a solution using the tracking behaviors to ensure the achievement of stated
H∞ controller design methodology. In addition, (Farag and design objectives.
Werner 2002) compared the performance of his robust
Copyright © 2012, University of Cincinnati. All rights reserved.
Problem Description and Modeling 1. The closed-loop system is stable for m1 = m2 = 1
and 0.5 < k < 2.0.
The benchmark plant shown in Figure 1 consists of two 2. The disturbance w(t)=unit impulse at t=0 and y
masses connected via a spring, with the following has a settling time of 15sec for the nominal plant
characteristics. parameters m1 = m2 = 1 and k = 1.
3. Reasonable performance/stability robustness and
1) The system has a non-collocated sensor and reasonable gain/phase margins are achieved with
actuator; the sensor senses the position of m2 reasonable bandwidth.
while the actuator accelerates m1. This introduces 4. Reasonable control effort is used.
extra phase lag into the system, making control of 5. Reasonable controller complexity is needed.
the plant difficult (Cohen and Ben Asher 2001). 6. Settling is achieved when y is bounded by ± 0.1
2) The system is characterized by uncertainties in units.
the temporal plant (spring constant that varies
within a very wide range) This problem addresses, primarily, a disturbance rejection
3) The system exists in both the flexible body mode control problem in the presence of parametric uncertainty.
(due to the spring) and rigid-body mode (when The plant has eigenvalues at (± j √(k(m1+ m2)/(m1m2)), 0,0),
relative movements due to the spring are damped and a single-input/single-output (SISO) controller must
out). close its loop around Tuy, which has a pole-zero surplus of
four (Stengel and Marrison 1992).
For the above system, consider a simplification, where m1
= m2 = 1 and k = 1 with the appropriate units. A control Robust Design Solution using Fuzzy Logic
force acts on body 1 (m1), and x2, which is the position of
body 2 (m2), is instead measured thus resulting in a non- Fuzzy logic controller design was first started by (King and
collocated control problem. The state space representation Mamdani 1977) on the basis of the fuzzy logic system
of the system is given as generalized from the fuzzy set theory of (Zadeh 1965). It
has gained wide practical acceptance providing a simple,
intuitive, and qualitative methodology for control
(Jamshidi, Vadiee, and Ross 1993), (Yen, Langari, and
Zadeh 1992), (Zadeh 1994). In a typical implementation, a
fuzzy controller consists of a set of if-then rules, where the
controller output is the combined output of all the rules
evaluated in parallel from the antecedents of the inputs.
y = x2 The inference engine, of a fuzzy logic controller, plays the
role of a kernel that explores the fuzzy rules pre-
where x1 and x2 are the positions of body 1 and body 2, constructed by experts to accomplish inferences.
respectively; x3and x4 are the velocities of body 1 and body Since the rules specify the implication relationships
2, respectively; u is the control input acting on body 1; y is between the input variables and output variables
the sensor measurement, w is the disturbance acting on characterized by their corresponding membership
body 2, and k is the spring constant. The transfer function functions, the choice of the rules along with the
representation is membership functions makes significant impacts on the
final performance of the controller and therefore becomes
the major control strategy in Fuzzy Logic Controller
design.
Common classifications of fuzzy controllers include
fuzzy Proportional Integral Differential (PID) controllers,
and the corresponding transfer function between a fuzzy sliding-mode controllers and fuzzy gain scheduling
disturbance to and plant output is controllers (Driankov, Hellendoom, and Reinfrank 1996),
(Jang and Sun 1995). Even though all three categories
realize closed-loop control action and are based on
quantitative control techniques, the first and second are
implementations of the linear quantitative PID controller
This paper considers only problems 1 and 2 as described and a nonlinear, quantitative sliding-mode controller. The
by (Wie and Bernstein 1992) and ignores the effect of last category, however, utilizes Sugeno fuzzy rules to
sensor noise (full state feedback) and disturbance acting interpolate between several control strategies, and are
on body 1. The constant-gain linear feedback controller suitable for plants with time varying or piecewise linear
parameters (Jang and Sun).
design requirements are stated as
A. Fuzzy Logic for Benchmark
For the robust control problem described above, plant
stabilization is required first before performance
objectives. Ensuring stability, however, entails the
dampening of vibrations after an external disturbance is
applied. (Linder and Shafai 1999) described an approach
using Qualitative Robust Control (QRC) methodology,
where stability and tracking behaviors are separately
developed, and the superimposition of these behaviors
achieves the final control objective. These behaviors
exploit the rigid body mode of the plant, where the plant
behaves as if the masses are rigidly connected. The
stability behavior is derived from the heuristic that a
control action is more effective in suppressing plant
vibration if it is applied when the spring is neutral, and the
control action opposes the motion of the spring.
Using fuzzy logic, a process model of the spring, needed (a)
to provide the qualitative state information that dampens
plant vibrations and achieve stability, is achieved by
abstracting the system to a state that indicates whether the
spring is at its neutral length and whether the spring is in
the process of compressing or elongating. In modeling the
spring, the length of the spring and its rate of stretching or
contraction are used as input parameters and the output, its
state. The process utilizes a qualitative spring state that is
specified by a qualitative partition of the spring length
L = x2 – x1 and the spring length velocity .
These parameters are partitioned using five membership
functions as shown in Figure 2. A Mamdani Fuzzy
Inference System (FIS) applies 25 rules, shown in the
Fuzzy Association Memory (FAM) of Figure 3, to infer
the qualitative spring state from inputs L and The fuzzy
controller is developed using the minimum operator to
represent the “and” in the premise, and the Center of (b)
Gravity (COG) defuzzification as the implication.
The qualitative behavior of the spring is based on a
sense of direction and rate. Thus the parameters are
defined on a bivalent range or universe of [-1, 1], and the
outputs are described as follows;
NSCN: Not Stretching or Compressing with Neutral spring
CFN: Compressing Fast with Neutral spring
SFN: Stretching Fast with Neutral spring
The decision surface of Figure 4 is such that a vibration is
observed when L is Small_positive or Small_negative, and
is Negative_large or Positive_large. A similar situation
occurs when L is Zero and is Small_negative or
Small_positive.
(c)
B. State Observers
Figure 2: Fig. 2(a) Membership functions of Spring Length L =
The above model is possible only if the states of the x2 – x1 Fig. 2(b) Membership functions of the velocity of spring
masses can be observed or correctly estimated. Due to the contraction or stretching Fig. 2(c) Membership
functions of the output, spring state.
C. Robust Tracking and Stability
With the system in a rigid-body mode, due to the damping
springLength effects on the interconnecting spring, it is evident that the
neg sneg zero spos pos position and velocity of body 2, x2 and , are fixed relative
\ to body 1. Hence, measuring gives us , while the
deltaSpringL displacement of x1 from its initial position at rest is
equivalent to the displacement of x2 from its own initial
ength position. Essentially, the problem has been reduced to one
that can be solved with a collocated controller on body 1.
neg nscn nscn nscn nscn nscn
In robust control, collocation guarantees the asymptotic
sneg cfn nscn nscn nscn sfn stability of a wide range of SISO control systems, even if
the system parameters are subject to large perturbations,
zero cfn cfn nscn sfn sfn while also enabling the achievement of desired
spos cfn nscn nscn nscn sfn performance objectives.
We also use an additional Mamdani fuzzy controller
pos nscn nscn nscn nscn nscn which receives the position and velocity of body 1, x1 and
, as inputs and outputs an appropriate control action. This
output is superimposed directly on the output of the spring
Figure 3: Fuzzy association memory of the spring model. controller to obtain the final control action on the system.
The controller utilizes a qualitative partitioning of x1 and
using five membership functions as shown in Figure 5. The
input partitions of negbig (Negative), negsm
(Negative_small), Zero, possm (Positive_small) and posbig
(Positive) produce output partitions of nb (Negative), ns
(Negative_small), Zero, ps (Positive_small) and pb
(Positive), which represent the control force on body 1.
Figure 4: Output surface of the spring fuzzy process model.
non-collocated nature of this problem, designing for robust
disturbance rejection requires the use of state observers to
model disturbances and other uncertainties, such as
(a)
position of the masses. In the deterministic case, when no
random noise is present, the Luenberger observer and its
extension may be used for time-invariant systems with
known parameters. When parameters of the system are
unknown or time varying, an adaptive observer is
preferred. The corresponding optimum observer for a
stochastic system with additive white noise processes, with
known parameters, is the Kalman filter. As indicated
earlier, this project assumes full state feedback of masses
1& 2.
reasonable maximum value of u was obtained to be 1.262
units as shown in Figure 8.
(b)
Figure 6: Output surface of the controller.
(c)
Figure 5: Fig. 5 (a) Membership functions of position of body 1
x1 Fig. 5 (b) Membership functions of the velocity of body 1 Figure 7: Time series of position of body 2, x2, after a unit
Fig. 5 (c) Membership functions of output.
impulse disturbance on m2 for nominal plant parameters m1 = m2 =
1 and k = 1. Settling time (Ts) = 4.8 seconds, Peak time (Tp) =
The observed decision surface of Figure 6 shows that the 2.2 seconds and Peak Value (Pv) = 1.068 units.
corresponding output produced, for a given set of inputs,
has a somewhat inverse linear relationship to those inputs.
Two special membership functions, movingN and
movingP, with output membership functions of guardP and
guardN respectively, were also added to (velocity of
body 1) to ensure full stability.
Simulation Results
The performance of our fuzzy controller was investigated
using computer simulations in Simulink® and
MATLAB®. Figure 7 shows the response to a unit
impulse disturbance to m2, w(t) at t=0, for the nominal
plant parameters m1 = m2 = 1 and k = 1. The controller
shows excellent vibration suppression properties as the
position initially increases from 0 to 1.068 units before
returning and staying bounded within the required ± 0.1 Figure 8: Time series of cumulative controller output u after a
units of the final value in 4.8s. System stability was unit impulse disturbance on m2 for nominal plant parameters m1 =
obtained as required in the design specifications, and a m2 = 1 and k = 1. Maximum value of u = 1.262 units.
Figure 9 shows the stability of the system to varying Table 1. Controller performances for nominal plant parameters
spring constants in the range 0.5