Ph.D. Thesis - Introduction

Everything should be made as simple as possible, but not simpler.

Albert Einstein (1879-1955)

Disturbances caused by unmeasured inputs, plant perturbations or faulty actuators degrade the robustness and performance of both control and diagnostic systems. These disturbances can be known or unknown. When the disturbances are known, extensive techniques exist for accommodating them, while only weaker techniques exists for estimating and accommodating faults of unknown origin and unknown dynamics. This thesis contributes two original techniques for the accommodation of unknown disturbances. The first technique uses a variation of the linear state observer, the Proportional Integral (PI) observer, to estimate disturbances. The estimate is then used to reject the effect of the disturbances. The second new technique does not rely on the quantitative linear plant model needed by the linear observer, but rather relies on qualitative models of the plant and control behaviors. This innovative technique, Qualitative Robust Control (QRC), uses qualitative models, based on linguistic terms, which capture the structure of the plant and subsume perturbations and faults. The controller is then developed using a unique hierarchy of qualitative control behaviors, with separate behaviors achieving stability and tracking objectives. Validation of both methodologies is achieved through extensive simulation studies. The PI observer is shown to be the only known methodology to both estimate and accommodate unknown disturbances caused by either unmeasured inputs, plant perturbations or faulty actuators. The QRC methodology is validated with the popular 1992 American Control Conference (ACC) Robust Control Benchmark. The QRC controller is the first published controller for the Benchmark based on intelligent control techniques. It performs as well as, if not better than, any previous solution.

When the state of a system is otherwise inaccessible disturbances limit the ability to inferred the state of a system from observations of the system’s input. These inferences are made by state observers, which use measurements of the system’s input and output, in conjunction with a dynamic model of the system, to estimate the system’s hidden states and predict future system output. The complications arise when either system inputs are corrupted by disturbances, when inputs to the system are not accessible for measurement, or when precise models of the system dynamics are unavailable. Ideally the observer is robust, rejecting disturbances caused by unknown inputs, faulty actuators and plant perturbations.

Observers designed by engineers to implement controllers and fault detectors have historically been quantitative, utilizing precise numerical equations to model the system and characterize inputs and disturbances . With these quantitative methods an observer can be designed to reject any single disturbance that can be described quantitatively . However, if the disturbance disappears or changes, the observer can not dynamically compensate for the change.

The PI observer was developed by Shafai to extend the robustness of observers by including an integral action in the observer equation. This thesis extends this work by characterizing integral action for the purpose of disturbance rejection. First, integral action was extended to the adaptive case, where the parameters of the plant are unknown, with the new PI Adaptive Observer (PIAO). Next, the integral action of a PI observer was shown to effectively estimate and compensate for an arbitrary set of disturbances with n distinct injection points, if n independent state measurements are available. When used with a single output system, integral action allows the PI observer to reject any single unknown input, any single faulty actuator, or any single rank one perturbation, as long as the distribution matrix of the disturbance is known and the effect of the disturbance is slower than the time constant of the integral action. Increasing the integral gain allows for the rejection of faster perturbations, but with the negative side effect of decreasing the filter's stability margin.

The robustness of the PI observer can, however, be adversely effected by transitory disturbances with unmodelled distribution matrices. This can in fact severely limit the applicability of the PI observer; simple integral action cannot alone provide a robust solution to the 1992 ACC Benchmark. This thesis presents a new contribution to the field, generalization of the PI observer, the Proportional Fading-Integral (PFI) observer which discounts the integral term over time. Fading enables the rejection of these transitory events with unmodelled distribution matrices. Additionally, the fading term can improve the stability margin of the observer, allowing an unstable PI observer to become a stable PFI observer, yet with still sufficient integral action to reject disturbances.

The second approach to disturbance rejection, Qualitative Robust Control, is based on qualitative modeling. Qualitative models of the plant, the system to be controlled, and the control strategies are used to subsume any perturbations in the plant or actuators, and compensate for unmeasured inputs.

In contrast to the quantitative techniques used with the PI Observer, people in daily life often utilize symbolic reasoning and a qualitative abstraction of a system to ascertain a system’s hidden states (e.g. shaking a box to ascertain its contents). These qualitative abstractions are frequently created using fuzzy logic and are represented as symbolic linguistic models. These models provide a convenient mechanism for tuning quantitative observers and improving their disturbance rejection properties. In addition to providing a mechanism for tuning quantitative observers fuzzy logic can be used to build fully fuzzy observers . However, these existing fuzzy observers rely on machine learning to model a system, rather than using symbolic linguistic models of the system. An observer based on quantitative abstractions of the system and its disturbances will function over the entire range of system and disturbance configurations described by the qualitative model instead of the narrow range of systems described by the more precise quantitative models used by classical quantitative observers . The symbolic linguistic model effectively improves the robustness of the derived observer.

The use of symbolic linguistic models allows symbolic reasoning to complement quantitative reasoning. Symbolic reasoning is preferred for many tasks because, in general, people relate better to reasoning based on alphanumeric symbols . Since planning and goals can easily expressed in a expressed in high-level symbolic language, symbolic reasoning lends itself to the design of controllers with conflicting robustness requirements. With symbolic reasoning the various robustness requirements can be implemented in a hierarchical structure and tradeoffs between requirements can be made at a symbolic level.

The QRC methodology for designing fuzzy controllers transcends the quantitative methods that are utilized by the majority of fuzzy controllers described in the literature and contributes a new qualitative methodology. Instead of using the fuzzified versions of quantitative controllers, our methodology relies on qualitative models of the plant behavior. As in conventional control where the designer must make tradeoffs between different performance metrics with the fuzzy controller, the designer must weigh the effects of different localized qualitative behaviors on the overall performance of the controller.

Our contribution to the field of fuzzy set-point control shows that robust set-point control is achieved if a qualitative plant model that subsumes all specified plant perturbations is utilized in designing the controller. A rule-based fuzzy controller is then incrementally designed based on the "qualitatively robust" plant model. First, stability behaviors are developed and characterized. Then tracking behaviors are developed to augment the controller. Because these behaviors are based on a qualitatively robust model of the plant, the stability and tracking behaviors are robust over the extent of plant configurations that are subsumed by the qualitative plant model. The resulting fuzzy controller supports both stability and performance robustness and allows for simple compensator tuning by changing linguistically interpretable rule parameters.

The following sections provide a brief overview of the background material for this research. The reader is then shown the motivation for this research and a concise goal statement is given. The introduction continues with a solution statement that briefly describes the new techniques for improving disturbance rejection and a validation procedure for these new techniques.

This section gives a brief overview and provides a context for our results. Detailed background material, complete with equations, is given in Chapter 2 of this thesis. We begin with quantitative techniques and conclude with qualitative approaches to rejecting disturbances.

Quantitative approaches to rejecting disturbances for state observes have been under development for the last half century. This section begins with a brief overview of the three major types of observers: Luenberger observer, adaptive observer and Kalman filter. Then the existing disturbance rejection techniques, which involve the modeling of the disturbance or the use of multiple models are described. The section concludes with a description of two applications of state observers: state-based feedback control and fault detection.

A device that estimates or observes state variables of a system is called a state observer. A state observer utilizes measurements of the system inputs and outputs and a model of the system based on differential or difference equations. Three main quantitative state observers are: Luenberger observer, adaptive observer and Kalman filter.

In the deterministic case, when no random noise is present, the Luenberger observer and its extensions are used for time-invariant systems with known parameters. The equation for the Luenberger observer contains a term that corrects the current state estimates by an amount proportional to the prediction error: the estimation of the current output minus the actual measurement. Inclusion of this correction ensures stability and convergence of the observer even when the system being observed is unstable.

When the parameters of the system are unknown or time varying, an adaptive observer must be used. The adaptive observer, in addition to estimating the system states, must now also estimate the system parameters. Achieving this added requirement, while maintaining stability, has resulted in the development of significantly complex observer structure. Because prediction error can no longer be unambiguously associated with errors in estimating state, a persistently exciting signal must be generate to insure the stability of the adaptive observer. Even with this persistently exciting plant signal, the adaptive observer has significant difficulty distinguishing between the effects of inaccurate parameter estimates and measurement disturbances.

The corresponding observer for a stochastic system containing additive noise processes, with known parameters, is a stochastic observer with a structure attributed to Kalman . This Kalman filter is a recursive solution to Gauss’s original least squares estimation problem and builds on the work of Norbert Wiener in estimating the underlying signal from a noisy time series. The Kalman filter is the optimum estimator when the corrupting noise has a Gaussian probability distribution. Like the Luenberger observer, the Kalman filter also includes a correction factor to insure stability and convergence, but for the Kalman filter it is based on the variances of the noise processes. If accurate estimates of the variance are not available, optimal observer performance is not obtained .

All three of these observers have degraded performance in the presence of input disturbances. The following section discuss the two major techniques for rejecting known input disturbances.

Disturbances can be either stochastic or deterministic. While stationary stochastic input processes with a zero-mean Gaussian distribution can be effectively rejected by a Kalman filter when accurate noise statistics are available, fixed non-stochastic disturbances can only be rejected when the observer is augmented with a dynamic model of the disturbance . Time varying disturbances of either type that can not be modeled as a linear system are difficult if not impossible to reject. Any ability to compensate for either stochastic or deterministic disturbances is called disturbance rejection. However, this dissertation will use the term disturbance rejection only for the rejection of deterministic disturbances, while the term noise rejection will used for the rejection of stochastic disturbances.

Disturbances severely degrade observer performance. In the tracking domain, where disturbances are often referred to as maneuvers, a typical radar tracker tracks an aircraft using a parsimonious constant-velocity model and switches to a constant-acceleration model only after detecting an acceleration disturbance . The tracker cannot measure acceleration directly, but must instead infer that an acceleration disturbance has occurred when the predicted position, calculated by the observer, begins to diverge from the measured position. This difference, the measurement residual, increases rapidly as the maneuver progresses and indicates poor state estimation performance. Only if the disturbance is detected and rejected promptly can the divergence of state estimation be prevented.

Target trackers often use multiple observers to ensure prompt detection and rejection of disturbances and faults . Individual observers are tuned to include the dynamics of a known disturbance. When the set of disturbances is finite, complete coverage of the disturbance set can be achieved by the tracker with a finite set of observers. Rule-based or statistical reasoning is then used to analyze the measurement residuals generated by the observer set with the results of the analysis used to accommodate the maneuver or fault . This technique fails if any unmodelled disturbance is present.

Rejection of a single know disturbance can be achieved in a straight forward manner with a single observer when the system has known parameters. However, no known disturbance rejection methodologies exist for adaptive observers. Even with a persistently exciting plant input the adaptive observer has difficult distinguishing between the effect of inaccurate parameter estimates and disturbances. Additional difficulties exist when the adaptive observer is utilized in a closed loop regulator because the controller suppresses the persistently exciting input signal needed for the convergence of parameter estimates.

The two main applications of state observers are observer-based state feedback control and fault detection. Both applications rely on accurate state estimation and suffer from performance degradation when input disturbances corrupt the observer’s state estimates.

State feedback controllers designed as Linear Quadratic Regulators (LQR) have impressive robustness properties , including the rejection of disturbances from unknown inputs, actuator faults and plant perturbations. A LQR however requires access to system state variables. When state variables are unavailable a state observer must be included in the feedback loop, but this drastically reduces the robustness of state feedback control. The Linear Quadratic Gaussian/ Loop Transfer Recovery (LQG/LTR) techniques were introduced to recover some of the robustness properties of LQR, but these techniques often do not achieve full loop transfer recovery, and do not have disturbance rejection properties of the original LQR.

The PI observer, however, can be used to enhance the LTR procedure; the integral action of the PI observer allows additional freedom in adjusting the controller. A PI observer-based controller can be designed with frequency and time recovery properties approaching that of full state feedback control , and therefore recover the robustness property of a full state feedback controller.

An equivalent loop transfer recovery procedure, however, does not exist for the adaptive observer-based controller. Therefore, LTR techniques can not be used to recover the disturbance rejection properties of the original LQR. Instead, the adaptive control community has developed Model Reference Adaptive Control (MRAC) for linear systems with unknown parameters . MRAC does not use state information in the controller implementation, but rather relies only on measurement of the input and output of the system and a separate reference model to adapt the control gain. While MRAC can adapt to known disturbances it does not provide estimate of the disturbance or states.

Fault detection is the second major application of state observers. Faults are often detected by monitoring the measurement residuals of state observers. Excessive measurement residuals are interpreted as being indicative of a fault. A tradeoff, however, must be made between detecting all faults and creating an excessive number of false alarms since measurement residuals can also be generated by unmodelled plant dynamics, parameter mismatch or plant input disturbances. Disturbance decoupling is required in order to distinguish between true faults and the effects of disturbances .

Accurate estimation and robust accommodation of actuator faults can greatly increase the reliability and flexibility of control systems. Estimation allows for the characterization and classification of the fault, while actuator fault accommodation increases the robustness of the control system and allows time for diagnostic evaluation of the fault mechanism. With sufficiently accurate fault estimates and sufficiently robust accommodation, the dynamics of a fault can be closely monitored and used for the preemptive scheduling of repairs, without interrupting normal plant operation. Robust accommodation also allows for the utilization of less expensive actuators . High accuracy and performance can thus be achieved with components that previously were not precise enough and did not have sufficiently stable performance characteristics.

Currently however, no simple scheme exists for the design of a controller that both estimates and accommodates for unknown actuator faults. If the type of fault is known, and has a priori been characterized by a piecewise linear model, adaptive techniques exist for estimating the parameters of the fault model . Other techniques can accommodate, but not estimate, a class of faults with a known H¥ bound and much work has been done in accommodating actuator faults in systems with redundant actuators . More complex methodologies have also been developed that use computer-automated reconfiguration of control laws to accommodate for a set of known actuator faults . Results have also been obtained for a single known actuator fault; as an example, accommodation of saturation failures caused by faulty aircraft control surfaces are covered in .

Qualitative approaches to observer design, fault detection and control requires the creation of qualitative models of the underlying quantitative system. These qualitative models must be designed so they support reasoning that is consistent with the quantitative system. This coupling of consistent qualitative models with a continuous, quantitative plant is called a hybrid system . In contrast, many intelligent systems do not use qualitative models of the underlying quantitative system. As an example, intelligent systems based on neural networks often learn to recognize patterns, and reason about the recognized patterns, but never develop a qualitative model of the patterns.

This section explains how qualitative models are abstracted from quantitative systems and then introduces the implementation of fuzzy systems from qualitative models. It concludes with an overview of fuzzy tuning systems for linear observers, linear observer-based fault detectors and Proportional, Integral and Differential (PID) controllers, and a discussion of full fuzzy controllers.

When a qualitative model is created of a continuous plant, perceptual chunks of information are abstracted . Multiple points in the continuous domain are condensed into perceptual chunks that represent single symbolic concepts. However, an arbitrary mapping from the continuous quantitative plant to a symbolic qualitative model may create a model that looses many important details about the plant. Indeed, so many details may be lost that the model can not support consistent reasoning about the plant. The qualitative reasoning community therefore has focused on developing chunks that promote consistent reasoning. Consistency must be preserved in order to have confidence in any performed symbolic reasoning .

A typical approach to translating quantitative variables into qualitative symbols is to partition the quantitative variables with some critical values of the quantitative variables. When the value of the quantitative variable crosses a critical value a qualitative event is generated . These events can then be used as input to a finite state automaton; an automaton that qualitatively models the state transitions of the continuous plant.

The continuous plant of a hybrid system has traditionally been partitioned with orthogonal, hyperplanes to form hyperboxes . However, hyperboxes, with their orthogonal boundaries, do not take into account the interactions between plant variables produced by the dynamics of the plant and therefore do not produce consistent finite qualitative abstraction of the underlying continuous system . Hyperbox based hybrid systems can be forced to behave consistently by using smaller hyperboxes in areas of the system space that contain complex behavior , an approach similar to that utilized in finite-element analysis. This approach has the drawback of creating many additional qualitative states.

Kokar, in his earlier papers on dimensional analysis , suggested a different approach for developing consistent qualitative abstractions that limits the number of partitions. Instead of utilizing a grid of hyperboxes, it would be better to utilize hyper-surfaces suggested by the physical model of the plant. This work resulted in the Q² methodology for constructing consistent symbolic models of continuous noise-free dynamic plants . Q² focuses on how to partition a system’s space into a finite number of qualitative chunks that preserve consistency. Q²'s partitions allow the construction of a symbolic representation of the underlying plant that is provably consistent for noise-free general dynamic systems and provides superior qualitative models for noisy plants . However, the Q² methodology is difficult to extend to complex systems because it requires a complete quantitative model for generating partitions.

Fuzzy linguistic models hold the promise of providing a finite qualitative partition of a quantitative dynamic system while being applicable to any system that can be described in linguistic terms. Fuzzy models provide a succinct and robust representation of systems that lack a complete quantitative model or have uncertain system perturbations. Consistency in reasoning, however, has not yet been proven for a fuzzy linguistic representation of a quantitative system.

Fuzzy linguistic models use fuzzy sets to create a finite number of partitions, membership functions, of the inputs, outputs and states of a quantitative system. Currently most fuzzy models are implemented as a set of if-then rules, where the system input is used to evaluate the rules’ antecedents and the model’s output is the combined output of all the rules evaluated in parallel . This simple logical system, a Fuzzy Inference System (FIS) , does not implement inference chaining and can only evaluate a simplified qualitative model of a plant. Recent work has expanded the usefulness of this structure by providing machine learning methodologies to adapt and tune fuzzy linguistic models and to automatically generate new models through self-organization.

Learning or tuning allows the initial linguistic fuzzy model developed from heuristic domain knowledge to be optimized. Learning is achieved by using a neuro-fuzzy structure and exploiting the supervised learning strategies originally developed for neural networks. These strategies include gradient descent back-propagation , least-mean-squares, and a hybrid methodology that combines least-squares to optimize linear parameters and back-propagation to optimize the nonlinear parameters .

These same supervised learning methodologies can automatically learn any arbitrary nonlinear mapping between input and output without an initial linguistic fuzzy model . The resulting self-organized fuzzy models do not necessarily have a linguistic interpretation that would be recognized by a human expert. Often systems developed through self-organization are never interpreted linguistically, but are utilized effectively for pattern matching and curve fitting. Fuzzy networks are often preferred for curve fitting because the fuzzy rules used by the network have only a local effect, in effect providing an adaptive mechanism for implementing B-splines . As an example application, neuro-fuzzy curve fitting is used by Roberts, Mills, Charnley and Harris to extract estimates from a noisy time series. These estimates then initialize a Kalman filter, resulting in improved overall performance of the filter. Self-organized fuzzy models have also found application in fuzzy implementations of state estimators.

Moore, Harris and Rogers have used least-mean-squares learning to train a set of fuzzy networks, where each individual fuzzy network is trained with noisy data to track a target when perturbed by a single unique acceleration disturbance/maneuver. These fuzzy networks are then used in a hybrid scheme to detect and identify maneuvers and estimate target position.

A similar approach to solving the least-mean-squares estimation problem has been developed by Chao and Teng . A fuzzy network is trained off-line to estimate the state of a non-linear process from a sequence of noisy measurements. An estimation correction term similar to that utilized by a Kalman filter is included in the observer to ensure stability and convergence. As in the work by Moore, Harris and Rogers no linguistic qualitative model is employed in the design of the observer.

In contrast to these last two examples of fuzzy estimators where the fuzzy models do not have a qualitative interpretation, the next section will give examples of how qualitative linguistic models are used to automate the tuning of classical linear observers and controllers. In addition, the application of linguistic models to fuzzy control will be discussed.

Qualitative models and machine learning techniques are used extensively in industry to tune linear systems. Undergraduate control courses introduce tuning by teaching the Ziegler-Nichols tuning rules for PID controllers . These heuristics adapt the three PID controller parameters based on the step response of the compensated system. When first developed in 1942, these heuristics were manually employed by an engineer to tune a PID controller. Recently, most of the tuning systems developed for industrial controllers rely on qualitative reasoning or machine learning techniques to automate the pattern recognition needed for tuning .

Qualitative tuning is also used to adapt many other types of linear and non-linear dynamic systems. Most relevant to this dissertation is the work on tuning fault detectors and Kalman filters. The fault detection domain focuses on detecting frictional faults where even small amounts of unwanted friction severely compromises the positional accuracy of robots and actuators .

A fault detection system has been developed by Schneider and Frank for an industrial robot that increases fault detection accuracy by adapting the detection threshold. A fuzzy rule base adapts the detection threshold with heuristic qualitative rules that correlate nominal disturbance levels with the robot’s joint velocities and acceleration. In general, the higher the expected disturbance level for a particular robot configuration, the higher the fault detection threshold.

Rule-based adaptive compliance to frictional faults in positional actuators has been investigated by Isermann, Keller and Raab . A complete autonomous controller was implemented with an 8-bit microcontroller that with 15 heuristic rules adaptively determined the optimal control parameters from the step response of the actuator.

Improved estimation accuracy for automobile velocity estimation has been shown by Daiß and Kiencke using a fuzzy inference system based on a qualitative model of sensor performance. The qualitative model guides the fusion of measurements from several sensors, and results in overall improvement in velocity estimates utilized in antilock braking and traction control systems. Another approach to improving velocity estimates is shown by Kobayashi, Cheok and Watanabe . A fuzzy inference system is used to tune the covariance matrices of the Kalman filter. When a skid or wheel slip is detected, the noise variance for the velocity sensor is increased, biasing the filter to rely less on the velocity sensor and more heavily on the acceleration sensor.

Fuzzy tuning of quantitative systems is a large field, but by far the largest application of fuzzy systems is control. The next section describes fuzzy derivatives of quantitative controllers and controllers based on qualitative models.

Table 1. The number of positive responses to 9 survey questions for 311 polled companies that implemented fuzzy controllers.

A survey of 311 companies that recently implemented fuzzy controllers was made by von Altrock . Table 1 tallies the number of positive response to 9 survey questions. The survey shows that fuzzy control did not displace many conventional control designs. Rather fuzzy control was used in new products where the faster implementation time of fuzzy control and the ability to add new features to a product line convinced engineers and managers to try fuzzy control. 97% of the companies had a positive experience with fuzzy control and would use fuzzy control again.

The vast majority of the controllers covered by this survey were of medium complexity. These controllers were implemented with between 20 and 80 rules, with each rule having up to 7 terms and usually one output. The survey did not further elaborate on the types of implementations used. However the majority of fuzzy controllers described in the literature fall into three main categories :

All three compensators realize close-loop control action and are based on quantitative control techniques. The fuzzy PID controllers and fuzzy slide mode controllers are fuzzy implementations of the linear quantitative PID controller and a nonlinear quantitative sliding mode controller. Both controllers use the error term and its derivatives and integrals as input into a fuzzy rule base. Coleman and Godbode compare the robustness of a fuzzy PID controller with that of a conventional PID and sliding mode controller and conclude that the fuzzy controller has equivalent robustness characteristics .

The fuzzy gain scheduler uses Sugeno type fuzzy rules to interpolate between several control strategies . This methodology is useful for controlling nonlinear plants that are piece wise linear or for linear plants that have a time varying parameter. An example of this is a controller built for an inverted pendulum with a variable length pendulum. Measurements of the pendulum length are used as inputs into a fuzzy rule base that interpolates the output of a small number of controllers that are optimized for controlling short, medium length and long pendulums .

Additionally, fuzzy controllers have been developed using qualitative models of target behavior. These controllers are often developed using the following steps:

Ruspini, Saffiotti and Konolige utilize these steps and show that only 15 rules, representing 6 elemental behaviors, are needed to navigate a simple maze. These rules reasoned about three sonar sensor outputs and direction, and implement a "reactive navigation" behavior. Fuzzy logic achieves a consistent global behavior by interpolating and superimposing the elemental behaviors.

This dissertation is motivated by a desire to develop new disturbance rejection methodologies that are simpler and more robust than existing methodologies. Existing methodologies for rejecting unknown disturbances use complex paradigms. A quantitative model of each possible disturbance is required and a complex reasoning method most coordinate the accommodation of the disturbance. Even when using fuzzy control techniques researchers rely on fuzzy version of quantitative controllers to achieve robustness. Qualitative models and behaviors are rarely used.

Current techniques for disturbance rejection require either a precise quantitative model of a disturbance or extensive supervised learning. This requirement limits the disturbance rejection capabilities of a single observer to a single narrow class of quantifiable disturbance. If a set of several disturbances need to be rejected a corresponding set of observers is required, with each individual observer designed to reject only one of the know possible disturbances. Optimal rejection of the entire set of disturbance then relies on an autonomous supervisor which selects the appropriate observer outputs to use in construction the state estimation.

Both the quantitative and learning approach suffer from some serious limitations, including:

New techniques for disturbances reject must be developed that allow the rejection of unknown set of disturbances with only one observer.

Robust setpoint control is implemented almost exclusively using conventional techniques relying on precise quantitative models of plant and disturbances, while qualitative and intelligent techniques have focused on task oriented control of complex plants. The conventional robust control techniques first define a set of controllers that can stabilize the plant, including any disturbances that might be present, and then search for a controller that minimizes a robustness cost function. After a search process finds the optimal design, direct incremental tuning of the initial design is difficult because compensator parameters do not necessarily have a linguistic interpretation to guide the refinement, nor does the designer know the precision that must be preserved in the individual parameter in order to preserve the specified performance. These are severe shorting comings for the existing qualitative approaches to robust control.

Current approaches to fuzzy control are limited by their almost exclusive use of output feedback. It is imperative that fuzzy control be expanded beyond the prevalent PID paradigm. Iterative, incremental, design of qualitative models for use in the design of both the observer and the controller will help in developing robust fuzzy controllers. Rule-based fuzzy systems can be augmented directly by the addition of new rules and tuned by adjusting parameters with clear linguistic interpretations. As shown by Ruspini, Saffiotti and Konolige fuzzy logic can achieve a consistent robust global behavior by interpolating and superimposing elemental behaviors. We need to extend this technique to include the development of fuzzy controllers which are robust to disturbance.

In addition to rejecting these disturbances the observers and controllers should perform robustly with respect to

Two solutions are presented. The first uses the PI Observer and its new variants to estimate and accommodate unknown disturbances. The second uses fuzzy control based on qualitative models to accommodate for disturbance. The solution steps involving the integral action are:

The following section will give a brief overview of the results achieved for each of the six solution steps.

The PI observer and PI Kalman filter has been shown to estimate and reject disturbance for systems with known parameters and when the injection point of all disturbances are known. In the presence of faulty actuators, a observer-based regulator with integral action can out perform even a LQR using full state feedback.

The PI Adaptive Observer allows for the estimation and rejection of disturbances when plant parameters are unknown. Convergence of disturbance estimates are compromised only slightly by the concurrent estimation of plant parameters. This dissertation shows that only with integral action can the parameters estimates of an adaptive observe converge in the presence of disturbances.

The PFI observer, which discounts the integral term over time, enables the rejection of transitory events with unmodelled distribution matrices. Additionally, the fading term can improve the stability margin of an observer, allowing an unstable PI observer to become a stable PFI observer, yet with still sufficient integral action to reject disturbances. In fact, only with the addition of fading, does integral action provide a robust solution to the 1992 ACC Benchmark.

Robust rejection of disturbances has been achieved using qualitative behaviors with the Qualitative Robust Control (QRC) technique. Initially behaviors are developed and evaluated that stabilize the plant and then the fuzzy system is incrementally augmented with tracking behaviors that achieve the final performance objectives. Behaviors can be tuned with the addition of new rules and by the adjustment of parameters that have clear linguistic interpretations. This methodology produce an extremely robust controller for the ACC Benchmark, but with the use of full sate feedback.

Instead of directly designing a QRC controller using only output feedback, the QRC is first developed using full state feedback, and then the design is migrated to the output feedback configuration. A robust PFI observer is used to estimate the state of the plant. This hybrid solution performed as well if not better than existing ACC Benchmark solutions.

The theory and methodology developed for this dissertation is benchmarked using the very popular 1992 American Control Conference (ACC) Robust Control Benchmark . The ACC Benchmark has been utilized in over 40 papers and has been the theme of several journal issues. Stengel and Marrison provide an excellent comparison of solutions from numerous authors in their 1992 review . Additional validation was accomplished using our proposed benchmark based on the plant from Kudva and Narendra’s seminal 1973 paper on adaptive observers . Comparisons are made between the PI Observer, the P Observer and the Linear Quadratic Regulator (LQR). The LQR was selected because it shows the optimal robustness to plant perturbations of any linear .

The theoretical results obtained for PI observer and its variants were validated with both the Kudva plant and the ACC Benchmark. Both systems are used to demonstrate the efficacy of integral action in estimating the disturbance of unknown inputs, plant perturbations and fault actuators. In all cases the integral action allowed the PI observer-based controller to out perform a P-based controller, and in some cases even outperform a LQR using full state feedback.

The universal appeal of the ACC Benchmark is it’s simple mass-spring-plant. It provides an ideal vehicle for benchmarking the QRC methodology because all plant parameters have qualitative, linguistic interpretation. Extensive comparisons were made with existing solution for the Benchmark. These simulations validates the superior performance of our QRC controller with regards to disturbances caused by unknown inputs and plant perturbations.

The new technologies described in this thesis can be useful over a wide range of applications because disturbances are prevalent in real world systems. One application shown by this thesis is the use of integral action to estimating and actively accommodating for the disturbances caused by actuator nonlinearities . This capability has a two fold benefit. First, less precise actuators with known non-linearities can be incorporated into a control system. This results in a great cost reduction and allows for greater flexibility in selecting appropriate components. Also, the less precise actuator is often of sturdier design and can require significantly less maintenance. Second, integral action can be used to monitor the gradual degradation of an actuator. Preventative maintenance can then be scheduled before the accommodating capabilities of the integral action is overwhelmed, and a catastrophic failure occurs.

Another application demonstrated by this thesis is the rejection of vibrations in flexible structures, which include large mechanical structures, such as cranes , and space structures, such as the new space station Endeavor. The QRC fuzzy controller developed in this thesis for the flexible structure of the 1992 ACC Robust Control Benchmark is especially well suited for controlling space structures. Space structures use thrusters for maneuvering and attitude control. These thrusters are limited in their effectiveness because they are essentially binary actuators; the thrusters are either on or off. The QRC controller also uses discrete pulses to dampen vibrations of the structure.

This dissertation continues with a more detailed description of the material covered in this introduction. First, the pertinent mathematical background for my research is described in Chapter 2. Next, the validations benchmarks are described in Chapter 4. The theoretical results are given in Chapter 5 and 6, while Chapter 7 and 8 give detailed simulation results that validate the theoretical results. Chapter 9 provides concluding remarks.

Stephen Paul Linder

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Stephen Paul Linder

Ph.D. Thesis Introduction