Fractional-Order Discrete-Time Laguerre Filters: A New Tool for Modeling and Stability Analysis of Fractional-Order LTI SISO Systems

This paper presents new results on modeling and analysis of dynamics of fractional-order discrete-time linear time-invariant single-input single-output (LTI SISO) systems by means of new, two-layer, “fractional-order discrete-time Laguerre filters.” It is interesting that the fractionality of the filters at the upper system dynamics layer is directly projected from the lower Laguerre-based approximation layer for the Grünwald-Letnikov difference. A new stability criterion for discrete-time fractional-order Laguerre-based LTI SISO systems is introduced and supplemented with a stability preservation analysis. Both the stability criterion and the stability preservation analysis bring up rather surprising results, which is illustrated with simulation examples

Constrained optimization for hierarchical control system design

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1999.Includes bibliographical references (p. 115-116).by Michael Brian Jamoom.S.M

A Data-Driven Frequency-Domain Approach for Robust Controller Design via Convex Optimization

The objective of this dissertation is to develop data-driven frequency-domain methods for designing robust controllers through the use of convex optimization algorithms. Many of today's industrial processes are becoming more complex, and modeling accurate physical models for these plants using first principles may be impossible. With the increased developments in the computing world, large amounts of measured data can be easily collected and stored for processing purposes. Data can also be collected and used in an on-line fashion. Thus it would be very sensible to make full use of this data for controller design, performance evaluation, and stability analysis. The design methods imposed in this work ensure that the dynamics of a system are captured in an experiment and avoids the problem of unmodeled dynamics associated with parametric models. The devised methods consider robust designs for both linear-time-invariant (LTI) single-input-single-output (SISO) systems and certain classes of nonlinear systems. In this dissertation, a data-driven approach using the frequency response function of a system is proposed for designing robust controllers with H∞ performance. Necessary and sufficient conditions are derived for obtaining H∞ performance while guaranteeing the closed-loop stability of a system. A convex optimization algorithm is implemented to obtain the controller parameters which ensure system robustness; the controller is robust with respect to the frequency-dependent uncertainties of the frequency response function. For a certain class of nonlinearities, the proposed method can be used to obtain a best-linear-approximation with an associated frequency dependent uncertainty to guarantee the stability and performance for the underlying linear system that is subject to nonlinear distortions. The concepts behind these design methods are then used to devise necessary and sufficient conditions for ensuring the closed-loop stability of systems with sector-bounded nonlinearities. The conditions are simple convex feasibility constraints which can be used to stabilize systems with multi-model uncertainty. Additionally, a method is proposed for obtaining H∞ performance for an approximate model (i.e., describing function) of a sector-bounded nonlinearity. This work also proposes several data-driven methods for designing robust fixed-structure controllers with H∞ performance. One method considers the solution to a non-convex problem, while another method convexifies the problem and implements an iterative algorithm to obtain the local solution (which can also consider H2 performance). The effectiveness of the proposed method(s) is illustrated by considering several case studies that require robust controllers for achieving the desired performance. The main applicative work in this dissertation is with respect to a power converter control system at the European Organization for Nuclear Research (CERN) (which is used to control the current in a magnet to produce the desired field in controlling particle trajectories in accelerators). The proposed design methods are implemented in order to satisfy the challenging performance specifications set by the application while guaranteeing the system stability and robustness using data-driven design strategies

Constrained H̳₂ design via convex optimization with applications

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1998.In title on t.p., double-underscored "H" appears in script.Includes bibliographical references (p. 133-138).A convex optimization controller design method is presented which minimizes the closed-loop H2 norm, subject to constraints on the magnitude of closed-loop transfer functions and transient responses due to specified inputs. This method uses direct parameter optimization of the closed-loop Youla or Q-parameter where the variables are the coefficients of a stable orthogonal basis. The basis is constructed using the recently rediscovered Generalized Orthonormal Basis Functions (GOBF) that have found application in system identification. It is proposed that many typical control specifications including robustness to modeling error and gain and phase margins can be posed with two simple constraints in the frequency and time domain. With some approximation, this formulation allows the controller design problem to be cast as a quadratic program. Two example applications demonstrate the practical utility of this method for real systems. First this method is applied to the roll axis of the EOS-AM1 spacecraft attitude control system, with a set of performance and robustness specifications. The constrained H2 controller simultaneously meets the specifications where previous model-based control studies failed. Then a constrained H2 controller is designed for an active vibration isolation system for a spaceborne optical technology demonstration test stand. Mixed specifications are successfully incorporated into the design and the results are verified with experimental frequency data.by Beau V. Lintereur.S.M

On Finite-Dimensional Transformations of Anisochronic Controllers Designed by Algebraic Means: A User Interface

This chapter intended to propound the reader a methodology for algebraic controller design for systems with internal delays, followed by a comparison of several easy-handling techniques for rational (i.e. finite-dimensional) approximation of anisochronic (i.e. infinite-dimensional) controllers – or their transfer functions, more precisely. Matlab with Simulink was a very useful assistant here. The authors programmed a simple user interface which enables the user to enter a nominal transfer function and select approximation methods to be used and their orders. As a result, the program returns the accuracies in both text and graphical forms. Simulation experiments with the program were made. Control of a simple stable TDS, control of unstable TDS of a skater on the swaying bow and control of a laboratory circuits heating plant were benchmark examples. The results were very interesting and startling because the habitual Padé approximation proved to be very good and, moreover, the higher order approximation did not automatically mean the better result for systems with internal delays.P(ED2.1.00/03.0089

Identification of nonlinear interconnected systems

This thesis was submitted for the degree of Master of Philosophy and awarded by Brunel University.In this work we address the problem of identifying a discrete-time nonlinear system composed of a linear dynamical system connected to a static nonlinear component. We use linear fractional representation to provide a united framework for the identification of two classes of such systems. The first class consists of discrete-time systems consists of a linear time invariant system connected to a continuous nonlinear static component. The identification problem of estimating the unknown parameters of the linear system and simultaneously fitting a math order spline to the nonlinear data is addressed. A simple and tractable algorithm based on the separable least squares method is proposed for estimating the parameters of the linear and the nonlinear components. We also provide a sufficient condition on data for consistency of the identification algorithm. Numerical examples illustrate the performance of the algorithm. Further, we examine a second class of systems that may involve a nonlinear static element of a more complex structure. The nonlinearity may not be continuous and is approximated by piecewise a±ne maps defined on different convex polyhedra, which are defined by linear combinations of lagged inputs and outputs. An iterative identification procedure is proposed, which alternates the estimation of the linear and the nonlinear subsystems. Standard identification techniques are applied to the linear subsystem, whereas recently developed piecewise affine system identification techniques are employed for the estimation of the nonlinear component. Numerical examples show that the proposed procedure is able to successfully profit from the knowledge of the interconnection structure, in comparison with a direct black box identification of the piecewise a±ne system.Funding was obtained as a Marie Curie Early Stage Researcher Training fellowship, under the NET-ACE project (MEST-CT-2004-6724)

From Fixed-Order Gain-Scheduling to Fixed-Structure LPV Controller Design

This thesis focuses on the development of some fixed-order controller design methods in the gain-scheduling/Linear Parameter Varying (LPV) framework. Gain-scheduled controllers designed using frequency-domain Single Input Single Output (SISO) models are considered first, followed by LPV controller design in the SISO transfer function setting and, finally, by Multiple Input Multiple Output (MIMO) LPV controller design in the state-space setting. In addition to the guarantee of closed-loop stability, each of the methods optimizes some classical performance measure, such as the $\mathscr{H}_\infty$ or $\mathscr{H}_2$ performance metrics. In the LPV state-space setting, the practical assumption of bounded scheduling parameter variations is taken into account in order to allow a higher performance level to be achieved. The fixed-order gain-scheduled controller design method is based on frequency-domain models dependent on the scheduling parameters. Based on the linearly parameterized gain-scheduled controllers and desired open-loop transfer functions, the $\mathscr{H}_\infty$ performance of the weighted closed-loop transfer functions is presented in the Nyquist diagram as a set of convex constraints. No a posteriori interpolation is needed, so the stability and performance level are guaranteed for all values of scheduling parameters considered in the design. Controllers designed with this method are successfully applied to the international benchmark in adaptive regulation. These low-order controllers ensure good rejection of the multisinusoidal disturbance with time-varying frequencies on the active suspension testbed. One issue related to the gain-scheduled controller design using the frequency response model is the computational burden due to the constraint sampling in the frequency domain. The other is a guarantee of stability and performance for all the values of scheduling parameters, not just those treated in design. To overcome these issues, a method for the design of fixed-order LPV controllers with the transfer function representation is proposed. The LPV controller parameterization considered in this approach leads to design variables in both the numerator and denominator of the controller. Stability and $\mathscr{H}_\infty$ performance conditions for all fixed values of scheduling parameters are presented in terms of Linear Matrix Inequalities (LMIs). With a problem of rejection of a multisinusoidal disturbance with time-varying frequencies in mind, LPV controller is designed for an LTI plant with a transfer function model. The extension of these methods from SISO to MIMO systems is far from trivial. The state-space setting is used for this reason, as there the transition from SISO to MIMO systems is natural. A method for fixed-order output-feedback LPV controller design for continuous-time state-space LPV plants with affine dependence on scheduling parameters is proposed. Bounds on the scheduling parameters and their variation rates are exploited in design through the use of affine Parameter Dependent Lyapunov Functions (PDLFs). The exponential decay rate, induced $\mathscr{L}_2$-norm and $\mathscr{H}_2$ performance constraints are expressed through a set of LMIs. The proposed method is applied to the 2DOF gyroscope experimental setup. In practice control is performed using digital computers, so some effort needs to be put into the LPV controller discretization. If the discrete-time LPV model of the system is available [...