An Overview of Integral Quadratic Constraints for Delayed Nonlinear and Parameter-Varying Systems

A general framework is presented for analyzing the stability and performance of nonlinear and linear parameter varying (LPV) time delayed systems. First, the input/output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints (IQCs). A constant delay is a linear, time-invariant system and this leads to a simple, intuitive interpretation for these frequency domain constraints. This simple interpretation is used to derive new IQCs for both constant and varying delays. Second, the performance of nonlinear and LPV delayed systems is bounded using dissipation inequalities that incorporate IQCs. This step makes use of recent results that show, under mild technical conditions, that an IQC has an equivalent representation as a finite-horizon time-domain constraint. Numerical examples are provided to demonstrate the effectiveness of the method for both class of systems

Robust nonlinear control of vectored thrust aircraft

An interdisciplinary program in robust control for nonlinear systems with applications to a variety of engineering problems is outlined. Major emphasis will be placed on flight control, with both experimental and analytical studies. This program builds on recent new results in control theory for stability, stabilization, robust stability, robust performance, synthesis, and model reduction in a unified framework using Linear Fractional Transformations (LFT's), Linear Matrix Inequalities (LMI's), and the structured singular value micron. Most of these new advances have been accomplished by the Caltech controls group independently or in collaboration with researchers in other institutions. These recent results offer a new and remarkably unified framework for all aspects of robust control, but what is particularly important for this program is that they also have important implications for system identification and control of nonlinear systems. This combines well with Caltech's expertise in nonlinear control theory, both in geometric methods and methods for systems with constraints and saturations

A new solution approach to polynomial LPV system analysis and synthesis

Based on sum-of-squares (SOS) decomposition, we propose a new solution approach for polynomial LPV system analysis and control synthesis problems. Instead of solving matrix variables over a positive definite cone, the SOS approach tries to find a suitable decomposition to verify the positiveness of given polynomials. The complexity of the SOS-based numerical method is polynomial of the problem size. This approach also leads to more accurate solutions to LPV systems than most existing relaxation methods. Several examples have been used to demonstrate benefits of the SOS-based solution approach

Robust Performance Analysis for Gust Loads Computation

In the design process of modern aircraft, a comprehensive analysis of worst case structural gust loads is imperative. Because this analysis requires to consider millions of cases, the examination is extremely time consuming. To solve this problem, a new approach based on robust performance analysis is introduced: the worst case energy-to-peak gain is used to efficiently determine worst case loads of nominal, uncertain, and linear parameter varying gust loads models

Parameter Varying Mode Decoupling for LPV systems

The paper presents the design of parameter varying input and output transformations for Linear Parameter Varying systems, which make possible the control of a selected subsystem. In order to achieve the desired decoupling the inputs and outputs of the plant are blended together, and so the MIMO control problem is reduced to a SISO one. The new input of the blended system will only interact with the selected subsystem, while the response of the undesired dynamical part is suppressed in the single output. Decoupling is achieved over the whole parameter range, and no further dynamics are introduced. Linear Matrix Inequality methods form the basis of the proposed approach, where the minimum sensitivity (denoted by the H − index) is maximized for the subsystem to be controlled, while the H∞ norm of the subsystem to be decoupled is minimized. The method is evaluated on a flexible wing aircraft model

Control, stability analysis and grid integration of wind turbines.

In Chapters 2 and 3 of the thesis we propose a self-scheduled control method for a doublyfed induction generator driven by a wind turbine (DFIGWT), whose rotor is connected to the power grid via two back-to-back PWM power converters. We design a controller for this system using the linear matrix inequality based approach to linear parameter varying (LPV) systems, which takes into account the nonlinear dynamics of the system. We propose a two-loop hierarchical control structure. The inner-loop current controller, which considers the synchronous speed and the generator rotor speed as a parameter vector, achieves robust tracking of the rotor current reference signals. The outer-loop electrical torque controller aims for wind energy capture maximization, grid frequency support and generates the reference rotor current. We perform a controller reduction for the inner-loop LPV controller, which is not doable by conventional model-reduction techniques, because the controller is parameter-dependent. In simulation, the reduced order controller has been tested on a nonlinear 4th order DFIG model with a two-mass model for the drive-train. Stability and high performances have been achieved over the entire operating range of the DFIGWT. More importantly, simulation results have demonstrated the capability and contribution of the proposed two-loop control systems to grid frequency support. In Chapter 4 we investigate the integral input-to-state stability (iISS) property for passive nonlinear systems. We show that under mild assumptions, a passive nonlinear system which is globally asymptotically stable is also iISS. Moreover, the integral term from the definition of the iISS property has a very simple form (like an L1 norm). These theoretical results will be useful for our stability analysis of wind turbine systems in Chapter 5. In Chapter 5 we investigate the stability of a variable-speed wind turbine operating under low to medium wind speed. The turbine is controlled to capture as much wind energy as possible. We concentrate on the mechanical level of the turbine system, more precisely on the drive-train with the standard quadratic generator torque controller. We consider both the one-mass and the two-mass models for the drive-train, with the inputs being the deviation of the active torque from an arbitrary positive nominal value and the tracking error of the generator torque. We show that the turbine system is input-to-state stable for the one-mass model and iISS for the two-mass model. Using our abstract results from Chapter 4, we identify the iISS gain of this system. We also propose an adaptive search algorithm for the optimal gain of the quadratic torque controller

Optimal control and robust estimation for ocean wave energy converters

This thesis deals with the optimal control of wave energy converters and some associated observer design problems. The first part of the thesis will investigate model predictive control of an ocean wave energy converter to maximize extracted power. A generic heaving converter that can have both linear dampers and active elements as a power take-off system is considered and an efficient optimal control algorithm is developed for use within a receding horizon control framework. The optimal control is also characterized analytically. A direct transcription of the optimal control problem is also considered as a general nonlinear program. A variation of the projected gradient optimization scheme is formulated and shown to be feasible and computationally inexpensive compared to a standard nonlinear program solver. Since the system model is bilinear and the cost function is not convex quadratic, the resulting optimization problem is shown not to be a quadratic program. Results are compared with other methods like optimal latching to demonstrate the improvement in absorbed power under irregular sea condition simulations. In the second part, robust estimation of the radiation forces and states inherent in the optimal control of wave energy converters is considered. Motivated by this, low order H∞ observer design for bilinear systems with input constraints is investigated and numerically tractable methods for design are developed. A bilinear Luenberger type observer is formulated and the resulting synthesis problem reformulated as that for a linear parameter varying system. A bilinear matrix inequality problem is then solved to find nominal and robust quadratically stable observers. The performance of these observers is compared with that of an extended Kalman filter. The robustness of the observers to parameter uncertainty and to variation in the radiation subsystem model order is also investigated. This thesis also explores the numerical integration of bilinear control systems with zero-order hold on the control inputs. Making use of exponential integrators, exact to high accuracy integration is proposed for such systems. New a priori bounds are derived on the computational complexity of integrating bilinear systems with a given error tolerance. Employing our new bounds on computational complexity, we propose a direct exponential integrator to solve bilinear ODEs via the solution of sparse linear systems of equations. Based on this, a novel sparse direct collocation of bilinear systems for optimal control is proposed. These integration schemes are also used within the indirect optimal control method discussed in the first part.Open Acces