Yes / In this technical note, stochastic coupling attenuation is investigated for a class of multi-variable bilinear stochastic systems and a novel output feedback m-block backstepping controller with linear estimator is designed, where gradient descent optimization is used to tune the design parameters of the controller. It has been shown that the trajectories of the closed-loop stochastic systems are bounded in probability sense and the stochastic coupling of the system outputs can be effectively attenuated by the proposed control algorithm. Moreover, the stability of the stochastic systems is analyzed and the effectiveness of the proposed method has been demonstrated using a simulated example.

A compact model representing the dynamics between piezoelectric voltage inputs and cantilever probe positioning, including nonlinear surface interaction forces, for atomic force microscopes (AFM) is considered. By considering a relatively large cantilever stiffness, singular perturbation methods reduce complexity in the model and allows for faster responses to Van der Waals interaction forces experienced by the cantilever's tip and measurement sample. In this study, we outline a nonlinear near-optimal feedback control approach for non-contact mode imaging designed to move the cantilever tip laterally about a desired trajectory and maintain the tip vertically about the equilibrium point of the attraction and repulsion forces. We also consider the universal instance when the tip-sample interaction force is unknown, and we construct cascaded high-gain observers to estimate these forces and multiple AFM dynamics for the purpose of output feedback control. Our proposed output feedback controller is used to accomplish the outlined control objective with only the piezotube position available for state feedback. / Master of Science / In this thesis, the idea of an atomic force microscope (AFM), specifically the applications of the non-contact mode, will be discussed. An atomic force microscope (AFM) is a tool that measures the surface height of nanometer sized samples. To improve the speed and precision of the machine under a non-contact mode objective, a controller is designed based on optimality and is applied to the system. The system contains a series of equations designed to steer the system towards a desired trajectory and minimal vibrations. Given the complexity of the system, resulting from nonlinearities, we will apply singular perturbation principles on the system's stiffness property to separate the larger problem into two smaller ones. These two problems are inserted into a near-optimal controller and a series of simulations are conducted to demonstrate performance. Alongside this, we will outline an observer to estimate the unknown dynamics of the system. These estimates are then applied to our controller to demonstrate that only the AFM's piezotube position is to be known in order to estimate and control the remaining dynamics of the system.

<p>his thesis proposes two motion synchronization approaches to coordinate the motion of a follower to a leader within the Euler-Lagrange system framework. The information requirements from the leader are that of position and orientation only, i.e. the mathematical model with its parameters and the velocity and acceleration of the leader are considered unknown and unmeasured.</p><p>The follower is responsible for the control action necessary to coordinate the systems, and the leader system is free to manoeuvre independently of the follower. There is no off-line synchronization of the systems through predefined paths or trajectories. %The parameters of the dynamic model of the leader are unknown, and its unmeasured system states (velocity and acceleration) must be estimated in order to be utilized in the coordination controller of the follower.</p><p>The concept of motion control of multiple objects is discussed in terms of the different forms of synchronization; cooperation (where all objects contribute equally) and coordination (where one object governs the motion of the others). Motivating examples and literature provide the motivation for the definition of two motion coordination problems. The output reference state feedback synchronization problem is defined by utilizing only output feedback from the desired motion reference, while assuming state feedback for the follower in the coordination control law. Furthermore, to increase the usefulness of the proposed control schemes and to provide robustness towards loss or poor quality of velocity measurements, the requirements of state information for the follower are alleviated in the definition of the output reference output feedback synchronization problem utilizing only output information of both the leader and the follower in the synchronization design. Furthermore, the necessary tools of stability are presented to prove that the proposed coordination schemes are uniformly ultimately bounded or practically asymptotically stable closed-loop systems.</p><p>In order to solve the output reference state feedback and the output reference output feedback synchronization problems, an observer-controller scheme is proposed that estimates the unknown states of the leader indirectly through a nonlinear model-based error observer. The observer-controller approach makes the follower system a physical observer of the leader system through the coupled observer and controller error-dynamics. A second nonlinear model-based observer is introduced for the follower to remove the state feedback assumption. The observer-controller scheme is proven to be uniformly globally ultimately bounded when utilizing state feedback of the follower in the coordination control law, and to be uniformly semiglobally ultimately bounded when utilizing only output feedback of the follower in the coordination control law. The observer-controller approach to motion coordination is studied through simulations and experiments, and a back-to-back comparison between ideal simulations and practical experiments is presented to allow for a discussion on the performance of the scheme under modelling errors, measurement noise and external disturbances. The observer-controller scheme is demonstrated to be suitable for practical applications.</p><p>Furthermore, a virtual vehicle scheme is proposed to solve the output reference state/ output feedback synchronization problems through a cascaded approach. The virtual vehicle approach is based on a two-level control structure to decouple the estimation and coordination error dynamics in the stability analysis and the tuning process. The virtual vehicle scheme estimates the unknown states of the leader through a virtual kinematic vehicle stabilized to the output of the leader system. A stable first-order velocity filter is introduced for the follower to remove the state feedback assumption. The virtual vehicle scheme is proven to be uniformly globally practically asymptotically stable when utilizing state feedback of the follower in the coordination control law, and to be uniformly semiglobally practically asymptotically stable when utilizing only output feedback of the follower in the coordination control law. Application of the virtual vehicle scheme to both vehicle coordination and robot manipulator coordination is presented, and the virtual vehicle approach to motion coordination is studied through simulations and experiments. The virtual vehicle scheme is demonstrated to be suitable for practical applications. In addition, an extension to a dynamic synchronization scheme is proposed to impose a smooth behaviour on the follower during a change of relative position.</p><p>The proposed coordination schemes are compared in terms of estimation principle, performance and robustness. Simulation studies compare the performance of the proposed schemes in terms of gain tuning and bounds on the closed-loop errors, and in terms of impact from external disturbances, modelling errors and measurement noise. The two coordination schemes are distinguished by concept rather than by performance, and both of the proposed schemes are believed to be suitable for practical implementation in coordination applications.</p>

<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.

his thesis proposes two motion synchronization approaches to coordinate the motion of a follower to a leader within the Euler-Lagrange system framework. The information requirements from the leader are that of position and orientation only, i.e. the mathematical model with its parameters and the velocity and acceleration of the leader are considered unknown and unmeasured. The follower is responsible for the control action necessary to coordinate the systems, and the leader system is free to manoeuvre independently of the follower. There is no off-line synchronization of the systems through predefined paths or trajectories. %The parameters of the dynamic model of the leader are unknown, and its unmeasured system states (velocity and acceleration) must be estimated in order to be utilized in the coordination controller of the follower. The concept of motion control of multiple objects is discussed in terms of the different forms of synchronization; cooperation (where all objects contribute equally) and coordination (where one object governs the motion of the others). Motivating examples and literature provide the motivation for the definition of two motion coordination problems. The output reference state feedback synchronization problem is defined by utilizing only output feedback from the desired motion reference, while assuming state feedback for the follower in the coordination control law. Furthermore, to increase the usefulness of the proposed control schemes and to provide robustness towards loss or poor quality of velocity measurements, the requirements of state information for the follower are alleviated in the definition of the output reference output feedback synchronization problem utilizing only output information of both the leader and the follower in the synchronization design. Furthermore, the necessary tools of stability are presented to prove that the proposed coordination schemes are uniformly ultimately bounded or practically asymptotically stable closed-loop systems. In order to solve the output reference state feedback and the output reference output feedback synchronization problems, an observer-controller scheme is proposed that estimates the unknown states of the leader indirectly through a nonlinear model-based error observer. The observer-controller approach makes the follower system a physical observer of the leader system through the coupled observer and controller error-dynamics. A second nonlinear model-based observer is introduced for the follower to remove the state feedback assumption. The observer-controller scheme is proven to be uniformly globally ultimately bounded when utilizing state feedback of the follower in the coordination control law, and to be uniformly semiglobally ultimately bounded when utilizing only output feedback of the follower in the coordination control law. The observer-controller approach to motion coordination is studied through simulations and experiments, and a back-to-back comparison between ideal simulations and practical experiments is presented to allow for a discussion on the performance of the scheme under modelling errors, measurement noise and external disturbances. The observer-controller scheme is demonstrated to be suitable for practical applications. Furthermore, a virtual vehicle scheme is proposed to solve the output reference state/ output feedback synchronization problems through a cascaded approach. The virtual vehicle approach is based on a two-level control structure to decouple the estimation and coordination error dynamics in the stability analysis and the tuning process. The virtual vehicle scheme estimates the unknown states of the leader through a virtual kinematic vehicle stabilized to the output of the leader system. A stable first-order velocity filter is introduced for the follower to remove the state feedback assumption. The virtual vehicle scheme is proven to be uniformly globally practically asymptotically stable when utilizing state feedback of the follower in the coordination control law, and to be uniformly semiglobally practically asymptotically stable when utilizing only output feedback of the follower in the coordination control law. Application of the virtual vehicle scheme to both vehicle coordination and robot manipulator coordination is presented, and the virtual vehicle approach to motion coordination is studied through simulations and experiments. The virtual vehicle scheme is demonstrated to be suitable for practical applications. In addition, an extension to a dynamic synchronization scheme is proposed to impose a smooth behaviour on the follower during a change of relative position. The proposed coordination schemes are compared in terms of estimation principle, performance and robustness. Simulation studies compare the performance of the proposed schemes in terms of gain tuning and bounds on the closed-loop errors, and in terms of impact from external disturbances, modelling errors and measurement noise. The two coordination schemes are distinguished by concept rather than by performance, and both of the proposed schemes are believed to be suitable for practical implementation in coordination applications.

The thesis considers the problems of designing a dynamic output feedback controller to discrete time systems with polytopic uncertainty so that the closed-loop systems are DR stable with their transfer matrices having H2 norm and H¡Û norm bounded by a prescribed value ru. The formar part of the thesis provides less conservative LMI conditions for H2 and H¡Û analysis and the output feedback control of discrete system than those appeared in the current research. While the latter part of the thesis extend the current research to DR stable with H2 and H¡Û design. Finally, numerical examples are illustrated to show improvement of the propered result.

This dissertation presents novel adaptive control laws in both state feedback and output feedback forms. In the setting of state feedback adaptive control K-modification provides a tunable stiffness term that results in a frequency dependent filtering effect, smoother transient responses, and time delay robustness in an adaptive system. K-modification is combined with the recently developed Kalman filter (KF) based adaptive control and derivative-free (DF) adaptive control. K-modification and its combinations with KF adaptive control and DF adaptive control preserve the advantages of each of these methods and can also be combined with other modification methods such as - and e-modification. An adaptive output feedback control law based on a state observer is also developed. The main idea behind this approach is to apply a parameter dependent Riccati equation to output feedback adaptive control. The adaptive output feedback approach assumes that a state observer is employed in the nominal controller design. The observer design is modified and employed in the adaptive part of the design in place of a reference model. This is combined with a novel adaptive weight update law. The weight update law ensures that estimated states follow both the reference model states and the true states so that both state estimation errors and state tracking errors are bounded. Although the formulation is in the setting of model following adaptive control, the realization of the adaptive controller uses the observer of the nominal controller in place of the reference model to generate an error signal. Thus the only components that are added by the adaptive controller are the realizations of the basis functions and the weight adaptation law. The realization is even less complex than that of implementing a model reference adaptive controller in the case of state feedback. The design procedure of output feedback adaptive control is illustrated with two examples: a simple wingrock dynamics model and a more complex aeroelastic aircraft transport model.

This work considers formulation of new classes of adaptive controllers for double-integrator type systems where the underlying system parameters are uncertain and the complete state-vector is not available for feedback. Given the parameter uncertainty within the system model, a "separation principle" cannot generally be invoked towards an observer geared towards reconstruction of the full state vector using only measured variables. In this report, controllers are designed for some important sub-classes of Euler-Lagrange type mechanical systems, where states are physically interpreted as position and velocity variables, and only the position part of the state vector is available as measured output. The typical approach to obtain velocity estimates using position interpolation (also known as dirty differentiation), is known to be strongly susceptible to measurement noise and therefore does not usually represent a robust option for feedback control implementation. The proposed control scheme achieves global asymptotic stability for system dynamics subject to the condition that velocity states appear within the governing dynamics in a linear fashion. This arguably restrictive condition is loosened for the special case of scalar system with friction non-linearity as is typical within hardware implementations. The objective is to study prototypical mechanical systems with non-linearity appearing in the velocity rate equations with the eventual applications envisioned towards the attitude control problem accounting for the gyroscopic non-linearity in the Euler rotational dynamics. Based on classical certainty equivalence approaches for adaptive control, one cannot readily deal with cross terms associated with parameter estimates and unmeasured states in the Lyapunov function derivative in order to make the Lyapunov function negative definite or negative semi-definite. However, employing a new approach, this obstacle is shown in this report to be circumvented for scalar systems. In order to generalize the methodology for higher-order dynamics, a filtered state approach is used. Specifically, an auxiliary variable is introduced which plays an important role in determining restrictions on the control parameters and analyzing the stability. The proposed approach helps to overcome the uniform detectability obstacle. Additionally, this work can be applied to uncertain linear systems where independent control inputs are applied on each of the velocity state dynamics. Lastly, the solution for the scalar is applied to the rotor speed control system and is extended to the case where Coulomb friction is considered in addition to viscous friction. Since a sign function can be approximated as a hyperbolic tangent, the tanh model is used for the Coulomb friction. A controller is developed with the assumption that the coefficients of these frictions are unknown. The proposed control is then verified with Educational Control Product Model 750 Control Moment Gyroscope, and the simulation and actual test results are compared. / text

Numerical methods of designing control systems are currently an active area of research. Convex optimization with linear matrix inequalities (LMIs) is one such method. Control objectives like minimizing the H_2, H_infinity norms, limiting the actuating effort to avoid saturation, pole-placement constraints etc., are cast as LMIs and an optimal feedback controller is found by making use of efficient interior-point algorithms. A full-state feedback controller is designed and implemented in this thesis using this method which then forms the basis for designing a static output feedback (SOF) controller. A profile was generated that relates the change in the SOF control gain matrix required to keep the same value of the generalized H_2 norm of the transfer function from the road disturbance to the actuating effort with the change in the sprung mass of the quarter-car system. The quarter-car system makes use of a linear brushless permanent magnet motor (LBPMM) as an actuator, a linear variable differential transformer (LVDT) and two accelerometers as sensors for feedback control and forms a platform to test these control methodologies. For the full-state feedback controller a performance measure (H_2 norm of the transfer function from road disturbance to sprung mass acceleration) of 2.166*10^3 m/s^2 was achieved ensuring that actuator saturation did not occur and that all poles had a minimum damping ratio of 0.2. The SOF controller achieved a performance measure of 1.707*10^3 m/s^2 ensuring that actuator saturation does not occur. Experimental and simulation results are provided which demonstrate the effectiveness of the SOF controller for various values of the sprung mass. A reduction in the peak-to-peak velocity by 73 percent, 72 percent, and 71 percent was achieved for a sprung mass of 2.4 kg, 2.8 kg, and 3.4 kg, respectively. For the same values of the sprung mass, a modified lead-lag compensator achieved a reduction of 79 percent, 77 percent and, 69 percent, respectively. A reduction of 76 percent and 54 percent in the peak-to-peak velocity was achieved for a sprung mass of 6.0 kg in simulation by the SOF controller and the modified lead-lag compensator, respectively. The gain of the modified lead-lag compensator needs to be recomputed in order to achieve a similar attenuation as that of the SOF controller when the value of the sprung mass is changed. For a sprung mass of 3.4 kg and a suspension spring stiffness of 1640 N/m the peak-to-peak velocity of the sprung mass was attenuated by 42 percent.