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Thesis (MS)?Naval Postgraduate School, 1967

The present work addresses a finite-horizon linear-quadratic optimal control problem for uncertain systems driven by piecewise constant controls. The precise values of the system parameters are unknown, but assumed to belong to a finite set (i.e., there exist only finitely many possible models for the plant). Uncertainty is dealt with using a min-max approach (i.e., we seek the best control for the worst possible plant). The optimal control is derived using a multi-model version of Lagrange's multipliers method, which specifies the control in terms of a discrete-time Riccati equation and an optimization problem over a simplex. A numerical algorithm for computing the optimal control is proposed and tested by simulation.

We develop an indirect-adaptive model predictive control algorithm for uncertain linear systems subject to constraints. The system is modeled as a polytopic linear parameter varying system where the convex combination vector is constant but unknown. Robust constraint satisfaction is obtained by constraints enforcing a robust control invariant. The terminal cost and set are constructed from a parameter-dependent Lyapunov function and the associated control law. The proposed design ensures robust constraint satisfaction and recursive feasibility, is input-to-state stable with respect to the parameter estimation error and it only requires the online solution of quadratic programs.

We develop an indirect-adaptive model predictive control algorithm for uncertain linear systems subject to constraints. The system is modeled as a polytopic linear parameter varying system where the convex combination vector is constant but unknown. Robust constraint satisfaction is obtained by constraints enforcing a robust control invariant. The terminal cost and set are constructed from a parameter-dependent Lyapunov function and the associated control law. The proposed design ensures robust constraint satisfaction and recursive feasibility, is input-to-state stable with respect to the parameter estimation error and it only requires the online solution of quadratic programs.

We generalize the classical theory on algebraic Riccati equations and optimization to infinite-dimensional well-posed linear systems, thus completing the work of George Weiss, Olof Staffans and others. We show that the optimal control is given by the stabilizing solution of an integral Riccati equation. If the input operator is not maximally unbounded, then this integral Riccati equation is equivalent to the algebraic Riccati equation. Using the integral Riccati equation, we show that for (nonsingular) minimization problems the optimal state-feedback loop is always well-posed. In particular, the optimal state-feedback operator is admissible also for the original semigroup, not only for the closed-loop semigroup (as has been known in some cases); moreover, both settings are well-posed with respect to an external input. This leads to the positive solution of several central, previously open questions on exponential, output and dynamic (aka. "internal") stabilization and on coprime factorization of transfer functions. Our theory covers all quadratic (possibly indefinite) cost functions, but the optimal state feedback need not be well-posed (admissible) unless the cost function is uniformly positive or the system is sufficiently regular.

In the first part of this paper we introduced an algorithm that uses reachable set approximation to approximate the minimum time function of linear control problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e. we study cases in which the minimum time function is H\"older continuous or even discontinuous.

In the first part of this paper we introduced an algorithm that uses reachable set approximation to approximate the minimum time function of linear control problems. To illustrate the error estimates and to demonstrate differences to other numerical approaches we provide a collection of numerical examples which either allow higher order of convergence with respect to time discretization or where the continuity of the minimum time function cannot be sufficiently granted, i.e. we study cases in which the minimum time function is H\"older continuous or even discontinuous.

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Book Source: Digital Library of India Item 2015.234922 dc.contributor.author: Y S Apte dc.date.accessioned: 2015-07-10T19:11:41Z dc.date.available: 2015-07-10T19:11:41Z dc.date.digitalpublicationdate: 2005-09-27 dc.identifier.barcode: 5990010124756 dc.identifier.origpath: /rawdataupload1/upload/0124/759 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/234922 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 257 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: I I T Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Electrical Engineering dc.title: Linear Multivariable Control Systems

Book Source: Digital Library of India Item 2015.234922 dc.contributor.author: Y S Apte dc.date.accessioned: 2015-07-10T19:11:41Z dc.date.available: 2015-07-10T19:11:41Z dc.date.digitalpublicationdate: 2005-09-27 dc.identifier.barcode: 5990010124756 dc.identifier.origpath: /rawdataupload1/upload/0124/759 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/234922 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 257 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: I I T Kanpur dc.rights: Out_of_copyright dc.source.library: I I T Kanpur dc.subject.classification: Technology dc.subject.classification: Engineering. Technology In General dc.subject.classification: Electrical Engineering dc.title: Linear Multivariable Control Systems

Discrete control of linear distributed systems with application to telescope primary mirror of orbiting astronomical observatory

The closely-related problems of designing reliable feedback stabilization strategy and coordinating decentralized feedbacks are considered. Two approaches are taken. A geometric characterization of the structure of control interaction (and its dual) was first attempted and a concept of structural homomorphism developed based on the idea of 'similarity' of interaction pattern. The idea of finding classes of individual feedback maps that do not 'interfere' with the stabilizing action of each other was developed by identifying the structural properties of nondestabilizing and LQ-optimal feedback maps. Some known stability properties of LQ-feedback were generalized and some partial solutions were provided to the reliable stabilization and decentralized feedback coordination problems. A concept of coordination parametrization was introduced, and a scheme for classifying different modes of decentralization (information, control law computation, on-line control implementation) in control systems was developed.

Control of linear, discrete time, stochastic systems with unknown control gain parameters is discussed. Two suboptimal adaptive control schemes are derived: one is based on underestimating future control and the other is based on overestimating future control. Both schemes require little on-line computation and incorporate in their control laws some information on estimation errors. The performance of these laws is studied by Monte Carlo simulations on a computer. Two single input, third order systems are considered, one stable and the other unstable, and the performance of the two adaptive control schemes is compared with that of the scheme based on enforced certainty equivalence and the scheme where the control gain parameters are known.

A neighboring extremal guidance law for systems with piecewise linear control using multiple optimization along the suboptimal trajectory is developed. The guidance law computes control gains that relate the difference in the perturbed and nominal trajectory with a control update. Previous research in neighboring extremal control has concentrated on calculating the control gains using a single optimization from the initial conditions to the final constraint manifold. The purpose of this study is to develop a more optimal method of computing the control gains at each node through multiple optimization. Multiple optimization reoptimizes the trajectory from each control node on the suboptimal trajectory. The suboptimal control vectors for each subtrajectory are used to compute a more optimal set of control gains. An example optimization problem, the lunar launch problem, is used to compare the performance of both the single optimization and multiple optimization control laws. Overall, the multiple optimization gains returned results comparable to those achieved with single optimization. For perturbations of + or - 5% in either gravity or thrust acceleration, endpoint conditions for multiple optimization gains never exceeded more than 50 feet in position or 4.3 ft/sec in vertical velocity from the desired boundary conditions. Direct comparisons of performance for specific model perturbations result in mixed conclusions. For thrust perturbations, single optimization gains deliver smaller errors in meeting end point conditions, while for gravity perturbations, multiple gains give better results.

Hybird method for optimal linear control of nonlinear systems

Journal of Research of the National Bureau of Standards

Journal of Research of the National Bureau of Standards

Thesis (M.S. in Aeroelectronics)--U.S. Naval Postgraduate School, 1967

Thesis (M.S. in Aeroelectronics)--U.S. Naval Postgraduate School, 1967

Thesis (M.S. in Aeroelectronics)--U.S. Naval Postgraduate School, 1967

This paper describes a General Theory of filtering and control problems for linear systems, arising in particular from distributed parameter systems. (Author)

The relationship between numerical methods for realizations of E(t)x'(t) + F(t)x(t)=f(t) based on derivative arrays and geometric control realization procedures based on Lie derivatives is examined.

In this paper, a unified approach to sequence-based control and estimation of linear networked systems with multiple sensors is proposed. Time delays and data losses in the controller-actuator-channel are compensated by sending sequences of control inputs. The sequence-based design paradigm is further extended to the sensor-controller-channels without increasing the load of the network. In this context, we present a recursive solution based on the Hypothesizing Distributed Kalman Filter (HKF) that is included in the overall sequence-based controller design.

In this paper, a unified approach to sequence-based control and estimation of linear networked systems with multiple sensors is proposed. Time delays and data losses in the controller-actuator-channel are compensated by sending sequences of control inputs. The sequence-based design paradigm is further extended to the sensor-controller-channels without increasing the load of the network. In this context, we present a recursive solution based on the Hypothesizing Distributed Kalman Filter (HKF) that is included in the overall sequence-based controller design.

In this paper, a unified approach to sequence-based control and estimation of linear networked systems with multiple sensors is proposed. Time delays and data losses in the controller-actuator-channel are compensated by sending sequences of control inputs. The sequence-based design paradigm is further extended to the sensor-controller-channels without increasing the load of the network. In this context, we present a recursive solution based on the Hypothesizing Distributed Kalman Filter (HKF) that is included in the overall sequence-based controller design.

In this paper, a unified approach to sequence-based control and estimation of linear networked systems with multiple sensors is proposed. Time delays and data losses in the controller-actuator-channel are compensated by sending sequences of control inputs. The sequence-based design paradigm is further extended to the sensor-controller-channels without increasing the load of the network. In this context, we present a recursive solution based on the Hypothesizing Distributed Kalman Filter (HKF) that is included in the overall sequence-based controller design.

This thesis examines a class of systems whose models are described by linear partial differential equations that depend on a small parameter epsilon. First, the spectral decomposition of the so-called stiff operators (using the terminology of 24) is investigated, including the convergence of clarifying their eigenvalue eigenvector pairs as epsilon approaches 0, with the objective of clarifying their singular behavior. Second, asymptotic approximation of the solution boundary value problems involving stiff operators are constructed, using the weak limits of their eigenvectors. This approach leads to a decomposition into regular approximation and internal layer approximation, which are found separately and then combined to provide an approximation to the original problem. This methodology is not complicated. Moreover, it alleviates the inherent stiffness when numerical algorithms are employed. Third, the same approach is applied to some control problems. In this case, similar results are obtained, provided additional requirements are satisfied, due to the type of control, which may drastically alter the system behavior. (Author)

The theory of high-gain error actuated feedback control, developed by Porter and Bradshaw, is applied to the design of various lateral-directional decoupling flight control systems for an advanced aircraft. The controllers developed in this report utilize output feedback with proportional plus integral control to produce desirable closed-loop responses with minimal interactions between outputs. Because of the structure of the system, measurement variables in addition to the outputs are necessary to apply this method. The report examines controller design robustness by varying the flight conditions or maneuver commands from the ones the controller is specifically designed for, and then judges system performance. The results show that the controller is robust with respect to varying flight conditions, but is not robust with respect to varying maneuver commands. This report also examines the effect of first-order actuator dynamics in the system model. Actuator dynamics are shown to significantly effect the control system response, indicating that a simplified model, without actuators, is not desireable in one's control design scheme. Also a computer program to determine transmission zeros and decoupling zeros is developed. (Author)

The design of a system or circuit in which stability is a key constraint frequently leads to an optimization problem over the space of functions analytic on the right half plane (R.H.P.) Mathematical techniques for solving such optimization problems for mean square error (L2 error) criteria have been widespread in engineering since the time of Wiener. Much of this research goes to developing techniques for handling worst case error (L infinity error) criteria. These occur naturally in design of control systems and amplifiers. Practically speaking there is evidence that frequency domain L infinity criteria control system designs have desirable robustness properties. The ultimate objective is to develop a new CAD approach to MIMO control design which has the flavor of classical control as well as a systematic approach to worst case frequency domain design as it occurs in many areas. The promise of this approach is sufficient to have attracted many investigators and it is currently the focus of much attention. This research addresses many aspects of the problem. They range from the development of computer algorithms of a radically different type to the discovery of theoretical methods for understanding computational design. Also considerable progress was made in extending existing H infinity control to nonlinear plants. Another major effort involves computer algebra for systems research. The objective is to treat (on a computer) systems formulas of the type an investigator would manipulate by hand. Considerable software was developed along these lines.

A new class of scalar and vector-state estimators and stochastic controllers for linear dynamic systems with additive Cauchy process and measurement noises has been developed. The Kalman filter and the linear-quadratic-Gaussian controller have been the main estimation and control paradigms in modern engineering. However, many practical system noises, such as radar glint, are better described by heavy tailed probability density functions (pdf). Although the Cauchy pdf has an infinite variance, the conditional density of a Cauchy random variable, given a linear measurement with an additive Cauchy noise, has a conditional mean and a finite conditional variance, both being functions of the measurement. Over the last three years, a theory of estimation and stochastic control has been developed for the vector state linear dynamic system. The methodology for scalar state systems entailed propagation of the conditional pdf, while the vector state case was addressed by developing a recursion for the analytic propagation of the character function of the unnormalized conditional pdf (ucpdf). Through a spectral transformation, the character function of the ucpdf is used explicitly in the development of stochastic controllers for vector-state systems.

A set of performance statistics known as cumulants show the way to new design criteria for stochastic linear control systems. The new criteria allow the designer to affect the probability distribution of performance measures which do away with the need for stochastic simulation. Controllers arising in this new format are linear-dynamical and exhibit the classical separation property. (Author)

New methods were developed for stabilizing a time-varying linear differential system. Three technical notes involving feedback stabilization of a discrete-time constant linear system, derivation new lower bounds on the solution matrix, to the algebraic matrix Riccati equation, and new sufficient conditions for the linear constant differential-difference system were published. Results have been obtained concerning a minimum energy regulator problem for linear time-invariant systems in which the control variable is subject to an average-power constraint on the response time interval. (Author)

A class of optimal linear time invariant tracking systems, both in continuous time and discrete time, of which the number of inputs (which are restricted to be step functions) is equal to the number of system outputs, is studied. Along with derivation of equations and design procedures, two discretization schemes are presented, constraining either the control or its time derivative, to be a constant over each sampling period. Descriptions are given for the linearized model of the F-8C aircraft longitudinal dynamics, and the C* handling qualities criterion, which then serve as an illustration of the applications of these linear tracking designs. A suboptimal reduced state design is also presented. Numerical results are given for both the continuous time and discrete time designs.

The authors obtain results similar to those for LQG problems on the control system structure for optimal linear quadratic regulator problems with Poisson noise disturbances. If the coefficient matrices of the system dynamics and the performance index are constant, the optimal control of the finite time problem converges to the time-invariant control of the infinite time problem quasi-uniformly, almost surely. Both the long-term average cost criterion and the discounted cost criterion are investigated for infinite time problems.

We consider control design for positive compartmental systems in which each compartment's outflow rate is described by a concave function of the amount of material in the compartment.We address the problem of determining the routing of material between compartments to satisfy time-varying state constraints while ensuring that material reaches its intended destination over a finite time horizon. We give sufficient conditions for the existence of a time-varying state-dependent routing strategy which ensures that the closed-loop system satisfies basic network properties of positivity, conservation and interconnection while ensuring that capacity constraints are satisfied, when possible, or adjusted if a solution cannot be found. These conditions are formulated as a linear programming problem. Instances of this linear programming problem can be solved iteratively to generate a solution to the finite horizon routing problem. Results are given for the application of this control design method to an example problem. Key words: linear programming; control of networks; positive systems; controller constraints and structure.

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Distinguishability takes a crucial rule in studying observability of hybrid system such as switched system. Recently, for two linear systems, Lou and Si gave a condition not only necessary but also sufficient to the distinguishability of linear systems. However, the condition is not easy enough to verify. This paper will give a new equivalent condition which is relatively easy to verify.

Rocket booster control - time-optimal bounded phase coordinate control for systems modeled by linear recurrence equations

Rocket booster control - time-optimal bounded phase coordinate control for systems modeled by linear recurrence equations

Rocket booster control - time-optimal bounded phase coordinate control for systems modeled by linear recurrence equations

Rocket booster control - time-optimal bounded phase coordinate control for systems modeled by linear recurrence equations

Rocket booster control - time-optimal bounded phase coordinate control for systems modeled by linear recurrence equations

Rocket booster control - time-optimal bounded phase coordinate control for systems modeled by linear recurrence equations

Motivated by an open problem posed by J.P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the L2-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one.

Motivated by an open problem posed by J.P. Hespanha, we extend the notion of Barabanov norm and extremal trajectory to classes of switching signals that are not closed under concatenation. We use these tools to prove that the finiteness of the L2-gain is equivalent, for a large set of switched linear control systems, to the condition that the generalized spectral radius associated with any minimal realization of the original switched system is smaller than one.

The filtering problem for a partially observable stochastic system, with linear in observable states dynamics and non-Gaussian initial conditions is studied here. It is shown that the conditional expected value of the unobservable states, given the past observations, can be expressed in terms of a finite dimensional set of statistics. This result, which generalizes the conditionally Gaussian filter is used to derive a separation principle for a linear-quadratic control problem. (Author)

This thesis is the result of literary research on the subject of sensitivity in linear feedback control systems. It is a synopsis of information obtained from various technical publications and it designed to give the reader information on the theory of sensitivity and its application in design problems. Included are: (1) Definition of sensitivity function and root sensitivity for incremental variations of single parameter and their design application; (2) Definition of sensitivity function applied to the large parameter variations and its design application; (3) Special topics such as specification for sensitivity function, sensitivity integrals, and introduction to multi-parameter sensitivity.

It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variational theory. It allows us, in particular, to obtain efficient conditions via the system data supporting the sufficiency in Brockett's theorem and ensuring local exponential stabilization by means of continuous stationary feedback laws. Furthermore, we derive new necessary conditions for local exponential and asymptotic stabilization of continuous-time control systems by using both continuous and smooth stationary feedback laws and establish also some counterparts of the obtained sufficient conditions for local asymptotic stabilization by continuous stationary feedback laws in the case of nonlinear discrete-time control systems.

We consider the optimal control design problem for discrete-time LTI systems with state feedback, when the actuation signal is subject to unmeasurable switching propagation delays, due to e.g. the routing in a multi-hop communication network and/or jitter. In particular, we set up a constrained optimization problem where the cost function is the worst-case $\mathcal{L}_2$ norm for all admissible switching delays. We first show how to model these systems as pure switching linear systems, and as main contribution of the paper we provide an algorithm to compute a sub-optimal solution.