<p>The contributions of this thesis are in the area of control of systems with nonlinear dynamics. The thesis is divided into three parts. The two first parts are similar in the sense that they both consider output feedback of rather general classes of nonlinear systems, and both approaches are based on mathematical programming (although in quite different ways). The third part contains a state feedback approach for a specific system class, and is more application oriented.</p><p>The first part treats control of systems described by nonlinear difference equations, possibly with uncertain terms. The system dynamics are represented by piecewise affine difference inclusions, and for this system class, piecewise affine controller structures are suggested. Controller synthesis inequalities for such controller structures are given in the form of Bilinear Matrix Inequalities (BMIs). A solver for the BMIs is developed. The main contribution is to the output feedback case, where an observer-based controller structure is proposed. The theory is exemplified through two examples.</p><p>In the second part the output feedback problem is examined in the setting of Nonlinear Model Predictive Control (NMPC). The state space formulation of NMPC is inherently a state feedback approach, since the state is needed as initial condition for the prediction in the controller. Consequently, for output feedback it is natural to use observers to obtain estimates of the state. A high gain observer is applied for this purpose. It is shown that for several existing NMPC schemes, the state feedback stability properties ``semiglobally'' hold in the output feedback case. The theory is illuminated with a simple example.</p><p>Finally, a state feedback controller for a class of positive systems is proposed. Convergence of the state to a certain subset of the first orthant, corresponding to a constant ``total mass'' (interpreting states as masses) is obtained. Conditions are given under which convergence to this set implies asymptotic stability of an equilibrium. Simple examples illustrate some properties of the controller. Furthermore, the control strategy is applied to the stabilization of a gas-lifted oil well, and simulations on a rigorous multi-phase dynamic simulator of such a well demonstrate the controller performance.</p>

The contributions of this thesis are in the area of control of systems with nonlinear dynamics. The thesis is divided into three parts. The two first parts are similar in the sense that they both consider output feedback of rather general classes of nonlinear systems, and both approaches are based on mathematical programming (although in quite different ways). The third part contains a state feedback approach for a specific system class, and is more application oriented. The first part treats control of systems described by nonlinear difference equations, possibly with uncertain terms. The system dynamics are represented by piecewise affine difference inclusions, and for this system class, piecewise affine controller structures are suggested. Controller synthesis inequalities for such controller structures are given in the form of Bilinear Matrix Inequalities (BMIs). A solver for the BMIs is developed. The main contribution is to the output feedback case, where an observer-based controller structure is proposed. The theory is exemplified through two examples. In the second part the output feedback problem is examined in the setting of Nonlinear Model Predictive Control (NMPC). The state space formulation of NMPC is inherently a state feedback approach, since the state is needed as initial condition for the prediction in the controller. Consequently, for output feedback it is natural to use observers to obtain estimates of the state. A high gain observer is applied for this purpose. It is shown that for several existing NMPC schemes, the state feedback stability properties ``semiglobally'' hold in the output feedback case. The theory is illuminated with a simple example. Finally, a state feedback controller for a class of positive systems is proposed. Convergence of the state to a certain subset of the first orthant, corresponding to a constant ``total mass'' (interpreting states as masses) is obtained. Conditions are given under which convergence to this set implies asymptotic stability of an equilibrium. Simple examples illustrate some properties of the controller. Furthermore, the control strategy is applied to the stabilization of a gas-lifted oil well, and simulations on a rigorous multi-phase dynamic simulator of such a well demonstrate the controller performance.

In this thesis, we consider problems connected to navigation and tracking for autonomousrobots under the assumption of constraints on sensors and kinematics. We study formation controlas well as techniques for filtering and smoothing of noise contaminated input. The scientific contributions of the thesis comprise five papers.In Paper A, we propose three cascaded, stabilizing formation controls for multi-agent systems.We consider platforms with non-holonomic kinematic constraints and directional rangesensors. The resulting formation is a leader-follower system, where each follower agent tracksits leader agent at a specified angle and distance. No inter-agent communication is required toexecute the controls. A switching Kalman filter is introduced for active sensing, and robustnessis demonstrated in experiments and simulations with Khepera II robots.In Paper B, an optimization-based adaptive Kalman filteringmethod is proposed. The methodproduces an estimate of the process noise covariance matrix Q by solving an optimization problemover a short window of data. The algorithm recovers the observations h(x) from a system˙ x = f (x), y = h(x)+v without a priori knowledge of system dynamics. The algorithm is evaluatedin simulations and a tracking example is included, for a target with coupled and nonlinearkinematics. In Paper C, we consider the problem of estimating a closed curve in R2 based on noisecontaminated samples. A recursive control theoretic smoothing spline approach is proposed, thatyields an initial estimate of the curve and subsequently computes refinements of the estimateiteratively. Periodic splines are generated by minimizing a cost function subject to constraintsimposed by a linear control system. The optimal control problem is shown to be proper, andsufficient optimality conditions are derived for a special case of the problem using Hamilton-Jacobi-Bellman theory.Paper D continues the study of recursive control theoretic smoothing splines. A discretizationof the problem is derived, yielding an unconstrained quadratic programming problem. Aproof of convexity for the discretized problem is provided, and the recursive algorithm is evaluatedin simulations and experiments using a SICK laser scanner mounted on a PowerBot from ActivMedia Robotics. Finally, in Paper E we explore the issue of optimal smoothing for control theoretic smoothingsplines. The output of the control theoretic smoothing spline problem is essentially a tradeoff between faithfulness to measurement data and smoothness. This tradeoff is regulated by the socalled smoothing parameter. In Paper E, a method is developed for estimating the optimal valueof this smoothing parameter. The procedure is based on general cross validation and requires noa priori information about the underlying curve or level of noise in the measurements. / QC 20100722

This thesis concerns the analysis of dynamical systems suitable to be modelled by piecewise-smooth differential equations. In such systems the continuous-in-time dynamics is interrupted by discrete-in-time jumps in the state or governing equations of motion. Not only can this framework be used to describe existing systems with strong nonlinear behaviour such as impacts and friction, but the non-smooth properties can be exploited to design new mechanical devices. As suggested in this work it opens up the possibility of, for example, fast limit switches and energy transfer mechanisms. Particularly, the dynamics at the onset of low-velocity impacts in systems with recurrent dynamics, so called grazing bifurcations in impact-oscillators, are investigated. As previous work has shown, low-velocity impacts is a strong source of instability to the dynamics, and efforts to control the behaviour is of importance. This problem is approached in two ways in this work. One is to investigate the influence of parameter variations on the dynamic behaviour of the system. The other is to implement low-cost control strategies to regulate the dynamics at the grazing bifurcation. The control inputs are of impulsive nature, and utilizes the natural dynamics of the system to the greatest extent. The scientific contributions of this work is collected in five appended papers. The first paper consists of an experimental verification of a map that captures the correction to the smooth dynamics induced by an impact, known in the literature as the discontinuity map. It is shown that the lowest order expansion of the map accurately captures the transient growth rate of impact velocities. The second paper presents a constructive proof of a control algorithm for a rather large class of impact oscillators. The proof is constructive in the sense that it gives control parameters which stabilizes the dynamics at the onset of low-velocity impacts. In the third paper a piecewise-smooth quarter-car model is derived, and the control strategy is implemented to reduce impact velocities in the suspension system. In the fourth and fifth papers the grazing bifurcation of an impact oscillator with dry friction type damping is investigated. It turns out that the bifurcation is triggered by the disappearance of an interval of stable stick solutions. A condition on the parameters of the system is derived which differentiates between stable and unstable types of bifurcation scenarios. Additionally, a low-cost control strategy is proposed, similar to the one previously mentioned, to regulate the bifurcation scenario. / QC 20100811

This dissertation addresses the control problem for the general class of control non-affine, non-standard singularly perturbed continuous-time systems. The problem of control for nonlinear multiple time scale systems is addressed here for the first time in a systematic manner. Toward this end, this dissertation develops the theory of feedback passivation for non-affine systems. This is done by generalizing the Kalman-Yakubovich-Popov lemma for non-affine systems. This generalization is used to identify conditions under which non-affine systems can be rendered passive. Asymptotic stabilization for non-affine systems is guaranteed by using these conditions along with well-known passivity-based control methods. Unlike previous non-affine control approaches, the constructive static compensation technique derived here does not make any assumptions regarding the control influence on the nonlinear dynamical model. Along with these control laws, this dissertation presents novel hierarchical control design procedures to address the two major difficulties in control of multiple time scale systems: lack of an explicit small parameter that models the time scale separation and the complexity of constructing the slow manifold. These research issues are addressed by using insights from geometric singular perturbation theory and control laws are designed without making any assumptions regarding the construction of the slow manifold. The control schemes synthesized accomplish asymptotic slow state tracking for multiple time scale systems and simultaneous slow and fast state trajectory tracking for two time scale systems. The control laws are independent of the scalar perturbation parameter and an upper bound for it is determined such that closed-loop system stability is guaranteed. Performance of these methods is validated in simulation for several problems from science and engineering including the continuously stirred tank reactor, magnetic levitation, six degrees-of-freedom F-18/A Hornet model, non-minimum phase helicopter and conventional take-off and landing aircraft models. Results show that the proposed technique applies both to standard and non-standard forms of singularly perturbed systems and provides asymptotic tracking irrespective of the reference trajectory. This dissertation also shows that some benchmark non-minimum phase aerospace control problems can be posed as slow state tracking for multiple time scale systems and techniques developed here provide an alternate method for exact output tracking.

In this thesis we study a number of nonlinear control problems motivated by their appearance in flight control. The results are presented in a general framework and can also be applied to other areas. The two main topics are backstepping and control allocation. Backstepping is a nonlinear control design method that provides an alternative to feedback linearization. Here, backstepping is used to derive robust linear control laws for two nonlinear systems, related to angle of attack control and flight path angle control, respectively. The resulting control laws require less modeling information than corresponding designs based on feedback linearization, and achieve global stability in cases where feedback linearization can only be performed locally. Further, a method for backstepping control of a rigid body is developed, based on a vector description of the dynamics. We also discuss how to augment an existing nonlinear controller to suppress constant input disturbances. Two methods, based on adaptive backstepping and nonlinear observer design, are proposed. Control allocation deals with actuator utilization for overactuated systems. In this thesis we pose the control allocation problem as a constrained least squares problem to account for actuator position and rate constraints. Efficient solvers based on active set methods are developed with similar complexity to existing, approximate, pseudoinverse methods. A method for dynamic control allocation is also proposed which enables a frequency dependent control distribution among the actuators to be designed. Further, the relationship between control allocation and linear quadratic control is investigated. It is shown that under certain circumstances, the two techniques give the same freedom in distributing the control effort among the actuators. An advantage of control allocation, however, is that since the actuator constraints are considered, the control capabilities of the actuator suite can be fully exploited.

This thesis focuses on decentralized deadbeat output regulation of discrete-time nonlinear plants that are composed of multiple agents. These agents interact, via scalar-valued signals, in a known structured way represented with a graph. This work is motivated by applications where it is infeasible and/or undesirable to introduce control action within each plant agent; instead, control agents are introduced to interact with certain plant agents, where each control agent focuses on regulating a specific plant agent, called its target. Then, two analyses are carried out to determine if regulation is achieved: targeting analysis is used to determine if control laws can be found to regulate all target agents, then growing analysis is used to determine the effect of those control laws on non-target plant agents. The strength of this novel approach is the intuitively-appealing notion of each control agent focusing on the regulation of just one plant agent. This work goes beyond previous research by generalizing the class of allowable plant dynamics, considering not only arbitrary propagation times through plant agents, but also allowing for non-symmetrical influence between the agents. Moreover, new necessary and sufficient algebraic conditions are derived to determine when targeting succeeds. The main contribution of this work, however, is the development of new easily-verifiable conditions necessary for targeting and/or growing to succeed. These new conditions are valuable due to their simplicity and scalability to large systems. They concern the positioning of control agents and targets as well as the propagation time of signals through the plant, and they help significantly with design decisions. Various graph structures (such as queues, grids, spiders, rings, etc.) are considered and for each, these conditions are used to develop a control scheme with the minimum number of control agents needed.

The author develops a nonlinear longitudinal model of an aircraft modeled by rigid fuselage, tail, and wing, where the wing is attached to the fuselage with a torsional spring. The main focus of this research is to retain the full nonlinearities associated with the system and to perform gust load alleviation for the model by comparing the impact of a proportional-integral- lter nonzero setpoint linear controller with control rate weighting and a nonlinear Lyapunov-based controller. The four degree of freedom longitudinal system under consideration includes the traditional longitudinal three degree of freedom aircraft model and one additional degree of freedom due to the torsion from the wing attachment. Computational simulations are performed to show the aeroelastic response of the aircraft due to a gust load disturbance with and without control. Results presented in this thesis show that the linear model fails to capture the true nonlinear response of the system and the linear controller based on the linear model does not stabilize the nonlinear system. The results from the Lyapunov-based control demonstrate the ability to stabilize the nonlinear response, including the presence of an LCO, and emphasize the importance of examining the fully nonlinear system with a nonlinear controller.

Stability is a critical issue in gain-scheduled control problems in that the closed loop system may not be stable during the transitions between operating conditions despite guarantees that the gain-scheduled controller stabilizes the plant model at fixed values of the scheduling variable. For Linear Parameter Varying (LPV) model representations, a controller interpolation method using Youla parameterization that guarantees stability despite fast transitions in scheduling variables is proposed. By interconnecting an LPV plant model with a Local Controller Network (LCN), the proposed Youla parameterization based controller interpolation method allows the interpolation of controllers of different size and structure, and guarantees stability at fixed points over the entire operating region. Moreover, quadratic stability despite fast scheduling is also guaranteed by construction of a common Lyapunov function, while the characteristics of individual controllers designed a priori at fixed operating condition are recovered at the design points. The efficacy of the proposed approach is verified with both an illustrative simulation case study on variation of a classical MIMO control problem and an experimental implementation on a multi-evaporator vapor compression cycle system. The dynamics of vapor compression systems are highly nonlinear, thus the gain-scheduled control is the potential to achieve the desired stability and performance of the system. The proposed controller interpolation/switching method guarantees the nonlinear stability of the closed loop system during the arbitrarily fast transition and achieves the desired performance to subsequently improve thermal efficiency of the vapor compression system.

<p>For a successful design of an engineering system it is essential to pay careful attention to its dynamic response. This is particularly true, in the case of nonlinear systems, since they can exhibit very complex dynamic behaviour, including multiple co-existing stable solutions and chaotic motions, characterized by large sensitivity to initial conditions. In some systems nonlinear characteristics are desired and designed for, but in other cases they are unwanted and can cause fatigue and failure. A type of dynamical system which is highly nonlinear is discontinuous or non-smooth systems. In this work, systems with impacts are primarily investigated, and this is a typical example of a discontinuous system. To enhance or optimize the performance of dynamical systems, some kind of control can be implemented. This thesis concerns implementation of low-cost control strategies for discontinuous systems. Low-cost control means that a minimum amount of energy is used when performing the control actions, which is a desirable situation regardless of the application. The disadvantage of such a method is that the performance might be limited as compared with a control strategy with no restrictions on energy consumption. In this work, the control objective is to enforce a continuous or discontinuous grazing bifurcation of the system, whichever is desirable. In Paper A, the dynamic response and bifurcation behaviour of an impactoscillator with dry friction is investigated. For a one-degree-of-freedom model of the system, analytical solutions are found in separate regions of state space. These are then used to perform a perturbation analysis around a grazing trajectory. Through the analysis, a condition on the parameters of the system is derived, which assures a continuous grazing bifurcation. It is also shown that the result has bearing on the dynamic response of a two-degree-of-freedom model of the system. A low-cost active control strategy for a class of impact oscillators is proposed in Paper B. The idea of the control method is to introduce small adjustments in the position of the impact surface, at discrete moments in time, to assure a continuous bifurcation. A proof is given for what control parameters assures the stabilization. In Paper C, the proposed low-cost control method is implemented in a quarter-car model of a vehicle suspension, in order to minimize impact velocities with the bumpstop in case of high amplitude excitation. It is shown that the control method is effective for harmonic road excitation.</p>