Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares

In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H_infinity control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series

Parameterized macromodeling of passive and active dynamical systems

L'abstract è presente nell'allegato / the abstract is in the attachmen

Dual-Blind Deconvolution for Overlaid Radar-Communications Systems

The increasingly crowded spectrum has spurred the design of joint radar-communications systems that share hardware resources and efficiently use the radio frequency spectrum. We study a general spectral coexistence scenario, wherein the channels and transmit signals of both radar and communications systems are unknown at the receiver. In this dual-blind deconvolution (DBD) problem, a common receiver admits a multi-carrier wireless communications signal that is overlaid with the radar signal reflected off multiple targets. The communications and radar channels are represented by continuous-valued range-time and Doppler velocities of multiple transmission paths and multiple targets. We exploit the sparsity of both channels to solve the highly ill-posed DBD problem by casting it into a sum of multivariate atomic norms (SoMAN) minimization. We devise a semidefinite program to estimate the unknown target and communications parameters using the theories of positive-hyperoctant trigonometric polynomials (PhTP). Our theoretical analyses show that the minimum number of samples required for near-perfect recovery is dependent on the logarithm of the maximum of number of radar targets and communications paths rather than their sum. We show that our SoMAN method and PhTP formulations are also applicable to more general scenarios such as unsynchronized transmission, the presence of noise, and multiple emitters. Numerical experiments demonstrate great performance enhancements during parameter recovery under different scenarios.Comment: 26 pages, 13 figures, 1 tabl

Data-driven extraction of uniformly stable and passive parameterized macromodels

A Robust algorithm for the extraction of reduced-order behavioral models from sampled frequency responses is proposed. The system under investigation can be any Linear and Time Invariant structure, although the main emphasis is on devices that are relevant for Signal and Power Integrity and RF design, such as electrical interconnects and integrated passive components. We assume that the device under modeling is parameterized by one or more design variables, which can be related to geometry or materials. Therefore, we seek for multivariate macromodels that reproduce the dynamic behavior over a predefined frequency band, with an explicit embedded dependence of the model equations on these external parameters. Such parameterized macromodels may be used to construct component libraries and prove very useful in fast system-level numerical simulations in time or frequency domain, including optimization, what-if, and sensitivity analysis. The main novel contribution is the formulation of a finite set of convex constraints that are applied during model identification, which provide sufficient conditions for uniform model stability and passivity throughout the parameter space. Such constraints are characterized by an explicit control allowing for a trade-off between model accuracy and runtime, thanks to some special properties of Bernstein polynomials. In summary, we solve the longstanding problem of multivariate stability and passivity enforcement in data-driven model order reduction, which insofar has been tackled only via either overconservative or heuristic and possibly unreliable methods

Systems Structure and Control

The title of the book System, Structure and Control encompasses broad field of theory and applications of many different control approaches applied on different classes of dynamic systems. Output and state feedback control include among others robust control, optimal control or intelligent control methods such as fuzzy or neural network approach, dynamic systems are e.g. linear or nonlinear with or without time delay, fixed or uncertain, onedimensional or multidimensional. The applications cover all branches of human activities including any kind of industry, economics, biology, social sciences etc

Fast Simulation of Analog Circuit Blocks under Nonstationary Operating Conditions

This paper proposes a black-box behavioral modeling framework for analog circuit blocks operating under small-signal conditions around non-stationary operating points. Such variations may be induced either by changes in the loading conditions or by event-driven updates of the operating point for system performance optimization, e.g., to reduce power consumption. An extension of existing data-driven parameterized reduced-order modeling techniques is proposed that considers the time-varying bias components of the port signals as non-stationary parameters. These components are extracted at runtime by a lowpass filter and used to instantaneously update the matrices of the reduced-order state-space model realized as a SPICE netlist. Our main result is a formal proof of quadratic stability of such Linear Parameter Varying (LPV) models, enabled by imposing a specific model structure and representing the transfer function in a basis of positive functions whose elements constitute a partition of unity. The proposed quadratic stability conditions are easily enforced through a finite set of small-size Linear Matrix Inequalities (LMI), used as constraints during model construction. Numerical results on various circuit blocks including voltage regulators confirm that our approach not only ensures the model stability, but also provides speedup in runtime up to 2 orders of magnitude with respect to full transistor-level circuits

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 [...

Regeneratively and Passively Constrained Control of Vibratory Networks

This dissertation is focused on the control of vibratory networks. Mechanical examples of vibratory systems include a civil structure, automobile, and a cantilever beam. These systems are excited by external disturbances such as earthquakes, wind, or uneven road elevations. Both passive and active control laws can be utilized to suppress vibrations in these networks. Each type of control law possesses inherent advantages and drawbacks. Active control provides the highest performance but is expensive, relies on an external power source, and is complicated to implement and maintain. Passive control devices (composed of springs, inertial elements, dashpots) represent the cheapest option and provide energy-autonomy, but have inferior performance when compared to an active control device. Due to their reliability and low cost, passive control technologies set the baseline for comparison for other, more sophisticated technologies. On the other hand, although it yields superior performance, active control presumes availability of unlimited energy, which may be an impractical or unreliable assumption. This dissertation examines a new class of control technologies, called regenerative control systems. A regenerative control system theoretically possesses energy-autonomy, but does so with better performance when compared to a passive control system. However, regenerative control devices are more expensive than passive and therefore the improved performance they attain must warrant utilization. A regenerative control device is assumed to be connected to a large energy storage device (battery, supercapacitor, etc). At times, the control device will draw energy from the energy storage device in order to actuate the network. At other times, the control device converts mechanical energy from the network into electrical energy and replenishes the energy in the storage device. The regenerative controller is constrained such that, on average, it generates more energy than it expends. This constraint, which is a relaxation of a passive control law constraint, ensures the local energy storage device never completely depletes. One of the main focuses of this research is to develop theory which can can solve for optimal regenerative and passive control laws. Optimizing control laws for both types of technology, in the context of the same problem, allows for a fair comparison.The regenerative control design problem can be formulated as a convex optimization and therefore can be solved easily with many commercial solvers. Passively constrained control design is a nonconvex problem and a new technique, Iterative Convex Over-Bounding (ICO) is proposed and developed to solve this nonconvex optimization. We show that optimal regenerative control outperforms optimal passive control if parasitic losses are sufficiently small. We also propose a technique to quantify how large the parasitics can be for a regenerative controller to still outperform a passive controller for a given problem.PHDCivil EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138455/1/erwarner_1.pd