In intuitive design steps, a fuzzy logic-based robust controller is designed to address the first 1990-1992 American Control Conference benchmark problem. Using a conceptual transformation of the original flexible body into a perpetual rigid body mode, a final design which succeeds 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.
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
control benchmark problems. This has led to several
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
robust control. One of these problems, referred to by
Probably due to the linearity of this problem, most
published solutions have appropriated linear controllers of
some sort, from H-infinity to game theory.
Some recent solutions, however, make use of qualitative
approaches capitalizing on fuzzy reasoning, which have
been shown to perform just as good as or even better than
the existing quantitative methods
In this paper, we build on a solution using fuzzy logic.
We start by generating a detailed model of the system and
highlight the required design objectives for the controller.
Next we obtain a reduced or simplified model of the system
in the rigid-body mode, where spring oscillations have
been effectively damped out using fuzzy logic heuristics
The benchmark plant shown in Figure 1 consists of two masses connected via a spring, with the following characteristics.
1) The system has a non-collocated sensor and
actuator; the sensor senses the position of m2
while the actuator accelerates m1. This introduces
extra phase lag into the system, making control of
the plant difficult
For the above system, consider a simplification, where m1 = m2 = 1 and k = 1 with the appropriate units. A control force acts on body 1 (m1), and x2, which is the position of body 2 (m2), is instead measured thus resulting in a noncollocated control problem. The state space representation of the system is given as
y = x2
where x1 and x2 are the positions of body 1 and body 2,
respectively; x3and x4 are the velocities of body 1 and body
2, respectively; u is the control input acting on body 1; y is
the sensor measurement, w is the disturbance acting on
body 2, and k is the spring constant. The transfer function
representation is
and the corresponding transfer function between a
disturbance to and plant output is
This paper considers only problems 1 and 2 as described
by
This problem addresses, primarily, a disturbance rejection
control problem in the presence of parametric uncertainty.
The plant has eigenvalues at (± j √(k(m1+ m2)/(m1m2)), 0,0),
and a single-input/single-output (SISO) controller must
close its loop around Tuy, which has a pole-zero surplus of
four
Fuzzy logic controller design was first started by
Since the rules specify the implication relationships between the input variables and output variables characterized by their corresponding membership functions, the choice of the rules along with the 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,
fuzzy sliding-mode controllers and fuzzy gain scheduling
controllers
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.
Using fuzzy logic, a process model of the spring, needed 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 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.
The above model is possible only if the states of the masses can be observed or correctly estimated. Due to the springLength
\ deltaSpringL ength neg sneg zero spos pos neg sneg zero spos pos nscn cfn cfn cfn nscn nscn nscn cfn nscn nscn nscn nscn nscn nscn nscn nscn nscn sfn nscn nscn nscn sfn sfn sfn nscn 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 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. With the system in a rigid-body mode, due to the damping effects on the interconnecting spring, it is evident that the position and velocity of body 2, x2 and , are fixed relative to body 1. Hence, measuring gives us , while the displacement of x1 from its initial position at rest is equivalent to the displacement of x2 from its own initial position. Essentially, the problem has been reduced to one that can be solved with a collocated controller on body 1. In robust control, collocation guarantees the asymptotic stability of a wide range of SISO control systems, even if the system parameters are subject to large perturbations, while also enabling the achievement of desired performance objectives.
We also use an additional Mamdani fuzzy controller 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 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. reasonable maximum value of u was obtained to be 1.262 units as shown in Figure 8.
The observed decision surface of Figure 6 shows that the 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.
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 units of the final value in 4.8s. System stability was obtained as required in the design specifications, and a
It is evident that the fuzzy logic-based controller solves the first two of the benchmark problems. It, however, achieves better settling time performance over other fuzzy logic solutions, while staying within the requirements of reasonable controller output. This is due to unique fuzzy membership function placements and tunings, especially for stability and robust tracking.
Also, as the value of the spring constant k is increased the peak time and peak value decreases simultaneously. This is due to the fact that an increase in the spring constant allows the system to exhibit more inherent natural dampness that ensures less oscillations or more rigidity. This, however, increases the settling time significantly as the controller has less “control” over the system. The designed controller has been optimized for the case where k=1. This can be repeated for other values of spring constants in order to achieve better performances.
This paper uses a superimposition of qualitative stability and tracking behaviors instantiated with fuzzy rules which have clear linguistic interpretations. The impressive performance of the fuzzy logic controller on the ACC robust control benchmark shows its suitability for designing and developing controllers for stability and performance robustness in view of plant uncertainties, and sensitivity to actuator/sensor noncollocation. Of significant interest is the fact that the developed control strategy leads to robust near time-optimal control while requiring a relatively small amount of control effort.
Further studies can be pursued to test and improve the controller presented herein for the vibration suppression of structures, such as beams, plates, shells, and those possessing very high modal densities at lower frequencies. Also, the effects of high frequency sensor noise can be modeled in to the system, and a stochastic robustness analysis, using Monte Carlo simulations, can be used to obtain performance metrics, as estimated probabilities of stability/performance.