Convex Formulation of Controller Synthesis for Piecewise-Affine Systems

This thesis is divided into three main parts. The contribution of the first part is to present a controller synthesis method to stabilize piecewise-affine (PWA) slab systems based on invariant sets. Inspired by the theory of sliding modes, sufficient stabilization conditions are cast as a set of Linear Matrix Inequalities (LMIs) by proper choice of an invariant set which is a target sliding surface. The method has two steps: the design of the attractive sliding surface and the design of the controller parameters. While previous approaches to PWA controller synthesis are cast as Bilinear Matrix Inequalities (BMIs) that can, in some cases, be relaxed to LMIs at the cost of adding conservatism, the proposed method leads naturally to a convex formulation. Furthermore, the LMIs obtained in this work have lower dimension when compared to other methods because the dimension of the closed-loop state space is reduced. In the second part of the thesis, it is further shown that the proposed approach is less conservative than other approaches. In other words, it will be shown that for every solution of the LMIs resulting from previous approaches, there exists a solution for the LMIs obtained from the proposed method. Furthermore, it will be shown that while previous convex controller synthesis methods have no solutions to their LMIs for some examples of PWA systems, the approach proposed in this thesis yields a solution for these examples. The contribution of the last part of this thesis is to formulate the PWA time-delay synthesis problem as a set of LMIs. In order to do so, we first define a sliding surface, then control laws are designed to approach the specified sliding surface and ensure that the trajectories will remain on that surface. Then, using Lyapunov-Krasovskii functionals, sufficient conditions for exponential stability of the resulting reduced order system will be obtained. Several applications such as pitch damping of a helicopter (2nd order system), rover path following example (3rd order system) and active flutter suppression (4th order system) along with some other numerical examples are included to demonstrate the effectiveness of the approaches

Stability analysis and controller synthesis for a class of piecewise smooth systems

This thesis deals with the analysis and synthesis of piecewise smooth (PWS) systems. In general, PWS systems are nonsmooth systems, which means their vector fields are discontinuous functions of the state vector. Dynamic behavior of nonsmooth systems is richer than smooth systems. For example, there are phenomena such as sliding modes that occur only in nonsmooth systems. In this thesis, a Lyapunov stability theorem is proved to provide the theoretical framework for the stability analysis of PWS systems. Piecewise affine (PWA) and piecewise polynomial (PWP) systems are then introduced as important subclasses of PWS systems. The objective of this thesis is to propose efficient computational controller synthesis methods for PWA and PWP systems. Three synthesis methods are presented in this thesis. The first method extends linear controllers for uncertain nonlinear systems to PWA controllers. The result is a PWA controller that maintains the performance of the linear controller while extending its region of convergence. However, the synthesis problem for the first method is formulated as a set of bilinear matrix inequalities (BMIs), which are not easy to solve. Two controller synthesis methods are then presented to formulate PWA and PWP controller synthesis as convex problems, which are numerically tractable. Finally, to address practical implementation issues, a time-delay approach to stability analysis of sampled-data PWA systems is presented. The proposed method calculates the maximum sampling time for a sampled-data PWA system consisting of a continuous-time plant and a discrete-time emulation of a continuous-time PWA state feedback controller

Direct data‐driven design of switching controllers

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Fault tolerant control for bimodal piecewise affine systems

This thesis addresses the design of fault-tolerant controllers and a fault identification technique for bimodal piecewise affine systems. A new fault-tolerant control methodology is presented. Fault-tolerant, state feedback controllers are synthesized for piecewise-affine (PWA) systems while minimizing an upper bound on the expected value of a quadratic cost function. The controllers are designed to deal with partial loss of control authority in the closed loop PWA system. The proposed controller design technique stabilizes and satisfies performance bounds for both the nominal and faulty systems. Another contribution is the development of a fault identification technique for bimodal piecewise affine (PWA) systems. A Luenberger-based observer structure is applied to estimate partial loss of control authority in PWA systems. More specifically, the unknown value of the fault parameter is estimated by an observer equation obtained from a Lyapunov function. The design procedure is formulated as a set of linear matrix inequalities (LMIs) and guarantees asymptotic stability of the estimation error, provided the norm of the input is upper and lower bounded by positive constants. The new PWA identification method is illustrated in a numerical example. Asa third contribution, an active fault-tolerant controller structure is proposed for bimodal PWA systems. The new active fault-tolerant controller structure is illustrated in a numerical example

Green Scheduling of Control Systems

Electricity usage under peak load conditions can cause issues such as reduced power quality and power outages. For this reason, commercial electricity customers are often subject to demand-based pricing, which charges very high prices for peak electricity demand. Consequently, reducing peaks in electricity demand is desirable for both economic and reliability reasons. In this thesis, we investigate the peak demand reduction problem from the perspective of safe scheduling of control systems under resource constraint. To this end, we propose Green Scheduling as an approach to schedule multiple interacting control systems within a constrained peak demand envelope while ensuring that safety and operational conditions are facilitated. The peak demand envelope is formulated as a constraint on the number of binary control inputs that can be activated simultaneously. Using two different approaches, we establish a range of sufficient and necessary schedulability conditions for various classes of affine dynamical systems. The schedulability analysis methods are shown to be scalable for large-scale systems consisting of up to 1000 subsystems. We then develop several scheduling algorithms for the Green Scheduling problem. First, we develop a periodic scheduling synthesis method, which is simple and scalable in computation but does not take into account the influence of disturbances. We then improve the method to be robust to small disturbances while preserving the simplicity and scalability of periodic scheduling. However the improved algorithm usually result in fast switching of the control inputs. Therefore, event-triggered and self-triggered techniques are used to alleviate this issue. Next, using a feedback control approach based on attracting sets and robust control Lyapunov functions, we develop event-triggered and self-triggered scheduling algorithms that can handle large disturbances affecting the system. These algorithms can also exploit prediction of the disturbances to improve their performance. Finally, a scheduling method for discrete-time systems is developed based on backward reachability analysis. The effectiveness of the proposed approach is demonstrated by an application to scheduling of radiant heating and cooling systems in buildings. Green Scheduling is able to significantly reduce the peak electricity demand and the total electricity consumption of the radiant systems, while maintaining thermal comfort for occupants

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

Pseudo Euler-Lagrange and Piecewise Affine Control Applied to Surge and Stall in Axial Compressors

This thesis addresses the control of the axial compressor surge and stall phenomena using Pseudo Euler-Lagrange and Piecewise Affine (PWA) controller synthesis techniques. These phenomena are considered as major gas turbine compressor instabilities that may result in failures such as the engine flame-out or severe mechanical damages caused by high blade vibration. The common approach towards the detection of the rotating stall and surge is to install various types of pressure sensors, hot wires and velocity probes. The inception of the rotating stall and surge is recognized by the presence of pressure fluctuation and velocity disturbances in the gas stream that are obtained through sensors. The necessary measure is then taken by applying proper stall and surge stabilizing control actions. The Lyapunov stability of pseudo Euler-Lagrange systems in the literature is extended to include additional nonlinear terms. Although Lyapunov stability theory is considered as the cornerstone of analysis of nonlinear systems, the generalization of this energy-based method poses a drawback that makes obtaining a Lyapunov function a difficult task. Therefore, proposing a method for generating a Lyapunov function for the control synthesis problem of a class of nonlinear systems is of potential importance. A systematic Lyapunov-based controller synthesis technique for a class of second order systems is addressed in this thesis. It is shown, in terms of stability characteristics, that the proposed technique provides a more robust solution to the compressor surge suppression problem as compared to the feedback linearization and the backstepping methods. The second contribution is a proposed new PWA approximation algorithm. Such an approximation is very important in reducing the complexity of nonlinear systems models while keeping the global validity of the models. The proposed method builds upon previous work on piecewise affine (PWA) approximation methods, which can be used to approximate continuous functions of n-variables by a PWA function. Having computed the PWA model of the stall and surge equations, the suppression problem is then solved by using PWA synthesis techniques. The proposed solution is shown to have higher damping characteristics as compared to the backstepping nonlinear method

Hybrid modeling and control of mechatronic systems using a piecewise affine dynamics approach

This thesis investigates the topic of modeling and control of PWA systems based on two experimental cases of an electrical and hydraulic nature with varying complexity that were also built, instrumented and evaluated. A full-order model has been created for both systems, including all dominant system dynamics and non-linearities. The unknown parameters and characteristics have been identi ed via an extensive parameter identi cation. In the following, the non-linear characteristics are linearized at several points, resulting in PWA models for each respective setup. Regarding the closed loop control of the generated models and corresponding experimental setups, a linear control structure comprised of integral error, feed-forward and state-feedback control has been used. Additionally, the hydraulic setup has been controlled in an autonomous hybrid position/force control mode, resulting in a switched system with each mode's dynamics being de ned by the previously derived PWA-based model in combination with the control structure and respective mode-dependent controller gains. The autonomous switch between control modes has been de ned by a switching event capable of consistently switching between modes in a deterministic manner despite the noise-a icted measurements. Several methods were used to obtain suitable controller gains, including optimization routines and pole placement. Validation of the system's fast and accurate response was obtained through simulations and experimental evaluation. The controlled system's local stability was proven for regions in state-space associated with operational points by using pole-zero analysis. The stability of the hybrid control approach was proven by using multiple Lyapunov functions for the investigated test scenarios.publishedVersio

Stability analysis and controller design for switched time-delay systems

In this thesis, the stability analysis and control synthesis for uncertain switched time-delay systems are investigated. It is known that a wide variety of real-world systems are subject to uncertainty and also time-delay in their dynamics. These characteristics, if not taken into consideration in analysis and synthesis, can lead to important problems such as performance degradation or instability in a control system. On the other hand, the switching phenomenon often appears in numerous applications, where abrupt change is inevitable in the system model. Switching behavior in this type of systems can be triggered either by time, or by the state of the system. A theoretical framework to study various features of switched systems in the presence of uncertainty and time-delay (both neutral and retarded) would be of particular interest in important applications such as network control systems, power systems and communication networks. To address the problem of robust stability for the class of uncertain switched systems with unknown time-varying delay discussed above, sufficient conditions in the form of linear matrix inequalities (LMI) are derived. An adaptive switching control algorithm is then proposed for the stabilization of uncertain discrete time-delay systems subject to disturbance. It is assumed that the discrete time-delay system is highly uncertain, such that a single fixed controller cannot stabilize it effectively. Sufficient conditions are provided subsequently for the stability of switched time-delay systems with polytopic-type uncertainties. Moreover, an adaptive control scheme is provided to stabilize the uncertain neutral time-delay systems when the upper bounds on the system uncertainties are not available a priori . Simulations are provided throughout the thesis to support the theoretical result