Institute for Systems Research

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    Efficient Implementation of Controllers for Large Scale Linear Systems via Wavelet Packet Transforms
    (1998) Kantor, George A.; Krishnaprasad, Perinkulam S.; ISR; CDCSS
    In this paper we present a method of efficiently implementing controllers for linear systems with large numbers of sensors and actuators. It is well known that singular value decomposition can be used to diagonalize any real matrix. Here, we use orthogonal transforms from the wavelet packet to "approximate" SVD of the plant matrix. This yields alternatebases for the input and output vector which allow for feedback control using local information. This fact allows for the efficient computation of a feedback control law in the alternate bases. Since the wavelet packet transforms are also computationally efficient,this method provides a good alternative to direct implementation of a controller matrix for large systems.

    This paper was presented at the 32nd CISS, March 18-21, 1998.

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    Mixed H2 /H∞ Optimization: A BMI Solution
    (1996) Yen, S.D.; ISR
    The mixed H2 /H∞ problem arises as one means to achieve robust good performance for a controlled linear time-invariant system. The idea is to achieve H2 optimal performance subject to an H∞ bound as robustness. It has been difficult to find good algorithms for solving this problem. In this study the problem was transformed into a bilinear matrix inequality problem (BMI). Solving the BMI by the method of centers was shown to be complicated by discontinuities resulting from the unobservability of the closed-loop system. Transforming the BMI problem into a lattice of BMI subproblems makes it possible to avoid the discontinuity and solve the original problem. A robust flight control system for the F-14 is included as an example of the algorithm.
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    Simultaneous and Robust Stabilization of Nonlinear Systems by Means of Continuous and Time-Varying Feedback
    (1996) Ho-Mock-Qai, Bertina; Dayawansa, W.P.; ISR
    In this dissertation, we address the stabilization of uncertain systems described by finite, countably infinite or uncountable families of systems. We adopt an approach that enables us to consider control systems with merely continuous dynamics as well as continuous time-invariant and time-varying feedback laws.

    We show that for any countable family of asymptotically stabilizable systems, there exists a continuous nonlinear time-invariant controller that simultaneously stabilizes (not asymptotically) the family. Although these controllers do not achieve simultaneous asymptotic stabilization in the general case, we manage to modify our construction in order to design continuous time-invariant feedback laws that simultaneously asymptotically stabilize certain pairs of systems in the plane.

    By introducing continuous time-varying feedback laws, we then prove that an finite family of linear time-invariant (LTI) systems is simultaneously asymptotically stabilizable by means of continuous nonlinear time-varying feedback if each system of the family is asymptotically stabilizable by a LTI controller. We also provide sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of LTI systems, by means of continuous time-varying feedback.

    We then obtain sufficient conditions for the existence of a continuous time- varying feedback law that simultaneously asymptotically stabilizes a finite family on nonlinear systems. We illustrate these results by establishing the simultaneous asymptotic stabilizability of the elements of a class of pairs of homogeneous nonlinear systems We finally consider a class of parameterized families of systems in the plane [where the parameter lies in an uncountable set] that are not robustly asymptotically stabilizable by means of C1 feedback. We solve their robust asymptotic stabilization by means of continuous feedback, through a new approach where a robust asymptotic stabilizer is considered as a feedback law that simultaneously robustly asymptotically stabilizes two sub-families of the family under consideration.

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    Residue Calculus and Effective Nullstellensatz
    (1996) Berenstein, Carlos A.; Yger, A.; ISR
    We provide new tools to compute multidimensional residues for rational functions, even over fields of positive characteristic. As a corollary one obtains solutions of the Betout equation for polynomials over a ring with a site that have almost optimal estimates for degree and size.
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    Non-Smooth Simultaneous Stabilization of Nonlinear Systems: Interpolation of Feedback Laws
    (1996) Ho-Mock-Qai, Bertina; Dayawansa, Wijesuriya P.; ISR
    In this paper, we introduce a method that enables us to construct a continuous simultaneous stabilizer for pairs of systems in the plane that cannot be simultaneously stabilized by smooth feedback. We extend this method to higher dimensional systems and prove that any pair of asymptotically stabilizable nonlinear systems can be simultaneously stabilized (not asymptotically) by means of continuous feedback. The resulting simultaneous stabilizer depends on a partition of unity and we show how to circumvent the computation of this partition of unity by constructing an explicit simultaneous stabilizer.
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    ALADDIN: A Computational Toolkit for Interactive Engineering Matrix and Finite Element Analysis
    (1995) Austin, Mark; Chen, Xiaoguang; Lin, Wane-Jang; ISR
    This report describes Version 1.0 of ALADDIN, an interaction computational toolkit for the matrix and finite element analysis of engineering systems. The ALADDIN package is designed around a language specification that includes quantities with physical units, branching constructs and looping constructs. The basic language functionality is enhanced with external libraries of matrix and finite element functions.
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    A New Framework for Supervisory Control of Discrete Event Systems
    (1995) Shayman, M.A.; Kumar, Ratnesh; ISR
    We propose a new framework for supervisory control design for discrete event systems. Some of the features of the proposed approach are: (i) By associating control and observation capabilities and limitations with the plant as well as the supervisor, it models reactive systems, and also treats plant and supervisory processes in a symmetric way. (ii) By introducing a single general interconnection operation, called masked composition, it permits open-loop as well as closed-loop control. (iii) By viewing the uncontrollability of events as corresponding to a projection-type control mask, and considering more general nonprojection-type control as well as observation masks, it treats the controllability and observability of events in a unified way. (iv) It applies to both deterministic and nondeterministic plant models and supervisory design. The sublanguages of a given language that are realizable under control are closed under union. Hence, the supremal realizable sublanguage always exists. In addition, it yields conditions under which existence of a non-deterministic supervisor implies existence of a deterministic supervisor. (v) By encapsulating control and observation masks with process logic to form process objects, and using a single type of interconnection operator to build complex process objects out of simpler component process objects, it provides a foundation for an object-oriented approach to discrete event control.
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    Nonlinear HControl with Delayed Measurements
    (1995) Baras, John S.; Patel, N.S.; ISR
    This paper considers the nonlinear Hcontrol problem for systems subject to delayed measurements. Necessary and sufficient conditions for the solvability of the problem are presented. We employ the concept of an information state to achieve separation between estimation and control. In particular, the information state derived is no longer the ﲷorst case cost to come function. We also briefly discuss certainty equivalence for systems with delayed measurements.
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    Modeling and Control of Dynamical Effects due to Impact on Flexible Structures
    (1994) Wei, Q.; Dayawansa, W.P.; ISR
    In the first part of this dissertation, we consider modeling and approximation of impact dynamics on flexible structures. A nonlinear model is developed through Hertz law of impact in conjunction with the dynamic equation of the flexible structure. We have analyzed this nonlinear model and established the existence and uniqueness of solutions of the nonlinear equation. A numerical method is developed based on the contraction mapping principle. By utilizing the fact that impact interval is very short in general, one may approximate the transfer functions of the systems to which the impacting bodies belong by Taylor polynomials of low order. We have developed the first and second order approximations. The first order approximation yields a special function which can be used for analytical and computational purposes. The second order approximation leads to a two-parameter family of ordinary differential equations of which the solutions exhibit universal features of impact problems. Simulation results of various examples have demonstrated the usefulness of the developed numerical method and approximation methods.

    The second part of this dissertation is devoted to control of impact dynamics. We have formulated and studied a control problem where a linear system is subjected to a series of impact forces. The impact forces are treated as disturbances to the system and modeled as finite duration events using the theory developed in part one. A reasonable control objective is to design a feedback controller to minimize the energy transferred from the disturbances to the controlled outputs in the L2 norm sense. Under the assumption that the disturbance information is known a priori, a (sub) optimal control strategy is derived based on dynamic game theory. We shown that, by taking advantage of the fact that the duration of each impact force is very short in general, we can derive a series of approximate solutions of the nonlinear problem. The higher order terms may be negligible for the disturbance attenuation problem in some applications. Hence, the approximation with the leading term renders a linear one. A (sub) Hcontroller is derived and a procedure to compute such a controller is given. The (sub) optimal solution is naturally associated with the existence of a stabilizing periodic solution of coupled Raccati equations. Hamiltonian theory is employed to analyze the coupled Raccati equations. Finally, we investigate the digital implementation of this control algorithm by using a sampled-data controller. We have shown that under a certain sampling condition, the controller structure could become simpler than the continuous version.

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    Robust H-infinity Output Feedback Control for Nonlinear Systems
    (1994) Teolis, C.A.; J.S.Baras; ISR
    The study of robust nonlinear control has attracted increasing interest over the last few years. Progress has been aided by the recent extension of the linear quadratic results which links the theories of L2 gain control (nonlinear H∞ control), differential games, and stochastic risk sensitive control. In fact, significant advances in both linear and nonlinear H∞ theory have drawn upon results from the theories of differential games and stochastic risk sensitive control. Despite these advances in H∞ control theory, practical controllers for complex nonlinear systems which operate on basic H∞ principles have not been realized to date. Issues of importance to the design of a practical controller include (i) computational complexity, (ii) operation solely with observable quantities, and (iii) implementability in finite time. In this dissertation we offer a design procedure which yields, practical and implementable H∞ controllers and meets the, mandate of the above issues for general nonlinear systems. In particular, we develop a well defined and realistically implementable procedure for designing robust output feedback controllers for a large class of nonlinear systems. We analyze this problem in both continuous time and discrete time settings. The robust output feedback control problem is formulated as a dynamic game problem. The solution to the game is obtained by transforming the problem into an equivalent full state feedback problem where the new state is called the information state. The information state method provides a separated control policy which involves the solution of a forward and a backward dynamic programming equation. Obtained from the forward equation is the information state, and from the backward equation is the value function of the game and the optimal information state control. The computer implementation of the information state controller is addressed and several approximations are introduced. The approximations are designed to decrease the online computational complexity of controller.