Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00002631/00001
## Material Information- Title:
- Robustness of extended least squares parameter estimation in the presence of unmodeled dynamics
- Creator:
- Sierzchula, David Raymond
- Publication Date:
- 1993
- Language:
- English
- Physical Description:
- v, 100 leaves : illustrations ; 29 cm
## Subjects- Subjects / Keywords:
- Least squares ( lcsh )
Parameter estimation ( lcsh ) Least squares ( fast ) Parameter estimation ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Bibliography:
- Includes bibliographical references (leaves 99-100).
- General Note:
- Submitted in partial fulfillment of the requirements for the degree, Master of Science, Electrical Engineering.
- General Note:
- Department of Electrical Engineering
- Statement of Responsibility:
- by David Raymond Sierzchula.
## Record Information- Source Institution:
- University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 30674322 ( OCLC )
ocm30674322 - Classification:
- LD1190.E54 1993m .S54 ( lcc )
## Auraria Membership |

Full Text |

ROBUSTNESS OF EXTENDED LEAST SQUARES PARAMETER ESTIMATION IN THE PRESENCE OF UNMODELED DYNAMICS by David Raymond Sierzchula B.S., Rensselaer Polytechnic Institute, 1985 A thesis submitted to the Faculty of the Graduate School of the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Electrical Engineering This thesis for the Master of Science degree by David Raymond Sierzchula has been approved for the Department of Electrical Engineering by ixh7ff3 Date Sierzchula, David Raymond (M.S., Electrical Engineering) Robustness of Extended Least Squares Parameter Estimation in the Presence of Unmodeled Dynamics Thesis directed by Assistant Professor Miloje S. Radenkovic ABSTRACT Robustness of least squares parameter estimation to unmodeled dynamics without using parameter projection is an unsolved problem. It is examined here, in the context of a discrete-time SISO self-tuning regulator, using computer simulation experiments to supplement ongoing research. Convergence behavior of least squares estimates is first examined by simulating an optimal predictor applied to a simple ARMA system. Numerical examples demonstrate stability of the self-tuning controller when the intensities of unmodeled dynamics and unstructured external disturbances are bounded appropriately. In these examples, parameter estimates converge without using parameter projection. Behavior of the scalar quantity p(t), the time-varying term in the denominator of the covariance matrix calculation, demonstrates the idea of a self- excitation mechanism in the adaptive loop. This abstract accurately represents the content of the candidate's thesis. The form and content of this abstract are approved. I recommend its publication. Signed Miloje S. Radenkovic in ACKNOWLEDGMENTS First and foremost, the author wishes to express deep appreciation for the help, patience and understanding of Tracy L. Godfrey, who provided incalculable emotional support, technical help, and a fine personal computing system. Professor Miloje S. Radenkovic provided the topic and served as the thesis advisor, and his research results served as a point-of-departure for investigation. He also provided positive feedback to the author at a time when it was sorely needed. Martin Marietta Astronautics financially supported the authors studies through its Study Under Company Auspices program. Special gratitude is also due to the authors supervisor, Thomas C. Redfield, as well as the authors department and coworkers, for their patience and encouragement when time away from work was needed to perform research. Professors Jan T. Bialasiewicz and Douglas A. Ross were kind enough to serve as committee members, and the author greatly appreciates their efforts in reading and commenting on this work and attending its defense. Finally, the author thanks God for providing the strength to complete this thesis. IV CONTENTS Chapter 1. Introduction.......................................................1 1.1. Early Background of Adaptive Control..............................2 1.2. Robustness........................................................3 1.3. Least Squares and the Self-Tuning Controller......................5 1.4. Overview of Analysis..............................................7 2. Parameter Estimation Using Extended Least Squares..................9 2.1. The Recursive Extended Least Squares Algorithm....................9 2.2. The Optimal Predictor.............................................12 2.2.1. Algorithm Development...........................................13 2.2.2. Simulations.....................................................18 3. The Self-Tuning Controller.........................................27 3.1. Development of Algorithm..........................................27 3.2. Review of Stability Results.......................................32 4. Adaptive Control Simulations.......................................37 4.1. Simple Nu merical Examples........................................37 4.1.1. Case 1..........................................................37 4.1.2. Case II.........................................................53 4.2. ATS-6 Satellite.................................................61 5. Conclusions........................................................75 APPENDIX A: MATLAB SCRIPTS.............................................77 BIBLIOGRAPHY...........................................................99 v 1. INTRODUCTION Robustness of an important class of parameter estimation and adaptive control algorithms is examined here; the primary tool used in this inquiry is simulation by digital computer. Adaptive control algorithms have great potential for practical application, but the current immaturity of adaptive control theory requires safety nets and modifications to standard algorithms in order to assure stable and safe performance for systems in use. These safety nets and modifications serve to complicate the realization of adaptive control systems by imposing hardware requirements that increase cost; robustness without such modifications would ease implementation. Adaptive control theory is still maturing; global stability and robustness results are widely scattered in the literature, and unification is not yet comprehensive. Thus, research continues whose objective is to verify the feasibility of a variety of adaptive control algorithms in realistic situations. The following sections in this chapter present a historical perspective of adaptive control from early attempts to the present investigation in robustness, illustrate the motivation for robustness research, and review results for one particularly important combination of adaptive control algorithm (direct self-tuning) and parameter estimator (extended least squares). This lays the groundwork for computer simulations of the considered system, which will build upon analytical stability results recently generated and demonstrate simplification of the conditions necessary for its stability. 1.1. Early Background of Adaptive Control The definition of adaptive control4' has long been a matter of debate, but there seems to be at least partial agreement [1]. Generally, it encompasses any regulator that can modify its own behavior or adjust its own parameters in response to learning of, or changes in, the dynamics of the process it controls and the disturbances associated with that process; it excludes classical constant-gain feedback systems. Descriptions often applied to adaptive control systems are learning and self-organizing. Some in the adaptive control community consider such systems to have two categories of states, which change at two different rates; the slower rate governs the parameters of the system. An adaptive approach to control was first researched in the 1940s and 1950s in conjunction with high-performance aircraft [1], [2]. Such aircraft were intended to perform in a wide range of changing operating conditions requiring different control dynamics. Researchers investigated adaptive control strategies as a way of adjusting the aircraft controller mechanisms as the operating conditions varied. Flight test disasters occurred, and analog hardware and poor theoretical insight combined to restrict design solutions, consequently reducing adaptive control research until the burst of control theory contributions of the 1960s. The development at that time of such theoretical notions as state space, stochastic control, dynamic programming, recursive schemes, system identification, and parameter estimation, coupled with advances in stability theory, provided new insight into adaptive control. 2 1.2. Robustness An ideal system model for the purpose of studying adaptive control algorithms is characterized by linearity, time-invariance, known relative degree, and an upper bound on its order; it may also have well-modeled disturbances [3], [4], A sizable body of theory exists to describe the behavior of such systems; many important stability results and proofs, usually based on highly restrictive assumptions, appeared in the late 1970s and early 1980s. However, the realistic situation of actual physical systems found in industrial applications introduces perturbations affecting the ideal model: all physical systems possess dynamic behavior which cannot be modeled accurately, resulting in reduced-order modeling of the system being controlled which can lead to problems'such as systematic bias errors [5]; external disturbances influence many physical systems; noise is present in measurements of the states and outputs of physical systems. Adaptive control researchers noted that stable behavior in the ideal situation of algorithms being proposed and studied did not guarantee such behavior in the presence of influences found in realistic situations. They also found drift of parameter estimates to occur. For example, [6] questioned robustness of adaptive control algorithms to unmodeled dynamics and external disturbances; in this instance, maintaining stability requires application of dominantly rich reference inputs providing persistent excitation of appropriate order [7]. In order to realize adaptive control in real-world applications, it is clearly necessary to show robustness how adaptive algorithms can achieve control 3 objectives and maintain stability in the presence of realistic conditions. The focus on adaptive control robustness research began in the early-1980s. Meanwhile, by the same time frame, digital computer technology had developed so far that adaptive controllers began to appear on the commercial market; these could realize adaptive control algorithms, often much more complex than non-adaptive controllers, for practical applications. From this sequence of events, one concludes that the maturity of the theory lags the hardware technology required for physical realization in many applications; indeed, adaptive controllers in use today require safety nets, often elaborate, for safe operation [1]. This motivates the proliferation of robustness research today. Many of the issues and strategies identified in research to date are reviewed in [8], [4]. There are several classes of adaptive control algorithms: self-tuners, model reference, pole placement, dual control, and variations on each. Likewise, different methods exist of estimating parameters: least squares, gradient, stochastic averaging, and variations of these. A wide range of algorithm possibilities results, and specific examples of robustness research within this range include: investigation of estimated plant model admissibility through restriction of parameter space and sufficient excitation for adaptive pole placement using several classes of least squares algorithms [4]; use of set membership and a priori knowledge of the plant model to provide robustness [9], [10]; development of a Robust Ultimate Boundedness Theorem for global stability analysis and design of perturbed adaptive systems with gradient and normalization 4 estimation algorithms [11]; the results shows parameter projection is required for global stability, but persistent excitation is not; examination of least squares estimation robustness as applied to self-tuning regulators, which is addressed in the following. 1.3. Least Squares and the Self-Tuning Controller One adaptive control scheme that has received significant attention over the years is the self-tuning controller proposed by Astrom and Wittenmark in 1973 [12]. This control scheme incorporates the certainty equivalence principle to formulate its control signal directly from the estimated parameters of the process. Other self-tuning regulators are often described as indirect adaptive control algorithms, in the sense that the controller design equation is recursively solved using the estimated system parameters; once the controller parameters are determined, the controller generates the signal which causes the desired dynamic response from the plant [1]. The parameter estimator of the Astrom and Wittenmark self-tuning controller, an extended least squares algorithm, is an attractive choice because it has a convergence rate superior to other algorithms, such as the normalized gradient. 5 Simple block diagram of direct self-tuning controller Process Parameters Figure 1.1 Researchers have studied least squares parameter estimation in self-tuning control for the case where unmodeled dynamics are not present in the system. For example, convergence and consistency of least squares estimates has been shown in [13], assuming additive noise in the system is Gaussian and white. The author of [13] also obtained system stability and optimality, which applied specifically to the Astrom and Wittenmark self-tuning regulator, as well as a wide variety of other adaptive control schemes. Convergence of self-tuning algorithms using least squares was also shown in [14]. 6 In the presence of unmodeled dynamics and unstructured external disturbances, it is useful to weaken as much as possible the assumptions for robustness, in order to simplify the algorithms. Research has recently obtained global stability of the Astrom and Wittenmark self-tuning controller, provided a parameter projection mechanism assures bounding of the parameter estimates [3]. It is of interest here to investigate elimination of the parameter projection mechanism while maintaining stability of the controller. The paper [3] also identifies and formalizes a so-called self-stabilization mechanism in the adaptation loop that causes the covariance matrix of the least squares process to become bounded. A deeper review of these results of global stability and self-excitation later in this thesis provides a point of departure for the computer simulations that follow. These simulations demonstrate, using example systems, that a self-tuning regulator exhibits stable behavior under realistic perturbation conditions, and that its self-excitation mechanism operates as postulated, without using parameter projection. 1.4. Overview of Analysis Chapter 2 examines the estimation aspect of the problem. A review of the least squares method precedes an extension of the algorithm for use in an optimal predictor, to study the convergence behavior of its estimates. Chapter 3 derives a self-tuning controller algorithm that uses least squares, and reviews important stability results for this controller when it uses parameter projection. Finally, Chapter 4 summarizes simulations of the controller without parameter projection in a perturbed environment Two simple numerical examples and one based on a real plant incorporate unstructured 7 external disturbances and unmodeled dynamics; stability of the output, behavior of the parameter estimates, and a measure of the self-stabilization mechanism are examined. 8 2. PARAMETER ESTIMATION USING EXTENDED LEAST SQUARES 2.1. The Recursive Extended Least Squares Algorithm According to [1], the principle of least squares was developed by Gauss to determine planetary orbits; he argued that the unknown parameters of a math model should be chosen such that the sum of the squares of the difference between the observed values and computed values times their respective precision factors should be minimized. As developed in [1], for the regression model y(f) = + with y(t) the observed variable, 0T = \dx d2...dn] the vector of unknown parameters, and (j>T(t) = [^ 02...0] the vector of regressors, each of which is paired with an observation from an experiment, it is desired to determine the parameters such that the outputs calculated from the model agree in the least squares sense as closely as possible with the observations. The loss function which expresses this is ne,t)=\t,{y(i)-fii)B)2=\\Y- where y( 1) Y(t) = y( 2) (2.3) 9 and m= (2.4) The loss function is quadratic in 0; provided T(f)(r) is nonsingular, (2.2) can berewritten and is minimized with respect to 0. The result for estimate 0 of the unknown parameter vector is 0 = (&T(byl (2.5) P(t) = (<&r(f)O(0)_1 = fl V >=1 (2.6) then (2.5) can be expressed as 0(0 = Mi)y(i) . =i (2.7) (00(0 nonsingular is called an excitation condition. If the estimate were to converge to its true value as the number of observations tended to infinity, the estimate would be said to possess the property of consistency. This estimation method obviously assumes all the data has been collected prior to calculating the estimate. It would be useful to calculate updated estimates recursively for use in real-time applications such as adaptive control algorithms. Development of the recursive computation begins with (2.6), from which follows = P~l (r -1) + ^(r)0r(r) (2.8) 10 -(2.7) can alternately be expressed as fi-1 0(0 = Pit) ^MOyiO + QiOyiO \i=1 Manipulation of the previous two equations gives (2.9) 2>(/)y( o=p~\t - -1)=p-\oht -1) mfioht d (2.10) i=l and using this result in (2.9) yields the following recursive relationship: ko=kt -1) Piomfiokt -1)+pmioyio = 9(t -1) + P(04>(0[y(0 0r(Okt -1)) (2.11) = 0(t-l) + P(0(K0e(0 To avoid repeated matrix inversions in calculating the covariance matrix P(t) recursively, the Matrix Inversion Lemma (A + BCDy1 = A'1 A_15(Cr1 + DA_1fl)_1 DA~X (2.12) is applied to (2.8) giving p(0 = (<&r(fW))-1 = (<*>'(*-irn -1)+4>(04>t(0)~' = (p-1(r-l) + ^(O0r(O)_1 (2.13) =p(t D p(t i m)(i+f(OP(t -1) mt This can be written for an SISO system as Pit) = Pit-\)- P(t-mt)T(0P(t-0 (2.14) l + piOPit-DIHO because the dimensions of
(2.11) and (2.14) summarize the algorithm. In words, these say that a correction
asymptotically stable linear dynamical system such that the output of the system {t)(y{t +1) y(f +1)) |