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This paper proposes a new approach for the adaptive functional estimation of second order infinite dimensional systems with structured perturbations. First, the proposed observer is formulated in the natural second order setting thus ensuring the time derivative of the estimated position is the estimated velocity, and therefore called natural adaptive observer. Assuming that the system does not yield a positive real system when placed in first order form, then the next step in deriving parameter adaptive laws is to assume a form of input-output collocation. Finally, to estimate structured perturbations taking the form of functions of the position and/or velocity outputs, the parameter space is not identified by a finite dimensional Euclidean space but instead is considered in a Reproducing Kernel Hilbert Space. Such a setting allows one not to be restricted by a priori assumptions on the dimension of the parameter spaces. Convergence of the position and velocity errors in their respective norms is established via the use of a parameter-dependent Lyapunov function, specifically formulated for second order infinite dimensional systems that include appropriately defined norms of the functional errors in the reproducing kernel Hilbert spaces. Boundedness of the functional estimates immediately follow and via an appropriate definition of a persistence of excitation condition for functional estimation, a functional convergence follows. When the system is governed by vector second order dynamics, all abstract spaces for the state evolution collapse to a Euclidean space and the natural adaptive observer results simplify. Numerical results of a second order PDE and a multi-degree of freedom finite dimensional mechanical system are presented. 

This paper considers a class of distributed parameter systems that can be controlled by an actuator onboard a mobile platform. In order to avoid computational costs and control architecture complexity associated with a joint optimization of actuator guidance and control law, a suboptimal policy is proposed that significantly reduces the computational costs. By utilizing a continuous-discrete optimal control design, a mobile actuator moves to a new position at the beginning of a new time interval and resides for a prescribed time. Using the cost to go with variable lower limit, the optimization simplifies to solving algebraic Riccati equations instead of differential Riccati equations. Adding a hardware feature whereby the mobile sensors are constrained to stay within the proximity of the mobile actuator, a feedback kernel decomposition scheme is proposed to approximate a full state feedback controller by the weighted sum of sensor measurements. 

The problem of sensor placement for second order infinite dimensional systems is examined within the context of a disturbance-decoupling observer. Such an observer takes advantage of the knowledge of the spatial distribution of disturbances to ensure that the resulting estimation error dynamics are not affected by the temporal component of the disturbances. When such an observer is formulated in a second order setting, it results in a natural observer. Further, when the natural observer is combined with a disturbance decoupling observer, the necessary operator identities needed to ensure the well-posedness of the observer, are expressed in terms of the stiffness, damping, input and output operators. A further extension addresses the question of where to place sensors so that the resulting natural disturbance decoupling observer is optimal with respect to an appropriately selected performance measure. This paper proposes this performance measure which is linked to the mechanical energy of second order infinite dimensional systems. The proposed sensor optimization is demonstrated by a representative PDE in a second order setting. 

This paper presents a new formulation of consensus filters for parabolic PDEs. Using modal decompositions, the information a given distributed filter transmits to and receives from the remaining networked filters depends on the modal information needed. If a given distributed filter can completely reconstruct a specific mode or modes of the PDE, then it does not need any information from any of the networked filters. Similarly, if a distributed filter cannot adequately reconstruct a given mode, then it receives information from the filter that can completely reconstruct that specific mode. This then presents a connectivity which is based on the information needed. This consensus protocol which is dictated by the information a filter does not have but needs, is essentially a projection of information needed onto the unobservable space. This is demonstrated for a diffusion PDE in 1D and subsequently its abstraction is formulated for Riesz-spectral systems. Numerical studies demonstrate the proposed modal consensus filters. 

This paper presents an adaptive functional estimation scheme for the fault detection and diagnosis of nonlinear faults in positive real infinite dimensional systems. The system is assumed to satisfy a positive realness condition and the fault, taking the form of a nonlinear function of the output, is assumed to enter the system at an unknown time. The proposed detection and diagnostic observer utilizes a Reproducing Kernel Hilbert Space as the parameter space and via a Lyapunov redesign approach, the learning scheme for the unknown functional is used for the detection of the fault occurrence, the diagnosis of the fault and finally its accommodation via an adaptive control reconfiguration. Results on parabolic PDEs with either boundary or in-domain actuation and sensing are included. 

The economic aspects as a new factor in the selection of sensors for improved filtering of dynamical systems are introduced. By using the price of a single sensor, reflected by high values of the associated covariance, an economic aspect of the sensor optimization for optimal filtering is introduced. Both the unit price and the total price of a network of inexpensive noisy sensors are used as an alternative to the performance of a single expensive and highly accurate sensor. Algorithms for the integrated sensor optimization for both finite and infinite dimensional systems are presented and examples are provided to demonstrate these effects. 

An alternative to regression-based estimation of spatial fields is adaptive-based estimation. Harnessing a widely used assumption on the series expansion of an unknown spatial field, the on-line estimation of the spatial field enables the integration of the real-time estimation of the field with any other tasks required of sensing agents. Parameter convergence in the adaptive estimation case requires the property of persistence of excitation. This condition reduces to imposing the integral of the outer product of a regressor vector be uniformly positive definite. With a single sensor measurement this is impossible to achieve unless the measurements are mobile. In this work, it is shown that in the adaptive estimation of a spatial field, a single mobile sensor is capable of inducing persistence of excitation and hence provide the sought after parameter convergence. Thus, the motion of a single sensor is a necessary condition for parameter convergence. It is shown with the appropriate control design for the platform carrying onboard the sensor, it also is a sufficient condition for persistence of excitation. Numerical results examining the time-variation of the regressor vector to induce a persistence of excitation along with user-defined guidance for the adaptive estimation of spatial fields are included to demonstrate the effects of mobile sensors in inducing persistence of excitation. 

This work is concerned with the use of mobile sensors to approximate and replace the full state feedback controller by static output feedback controllers for a class of PDEs. Assuming the feedback operator associated with the full-state feedback controller admits a kernel representation, the proposed optimization aims to approximate the inner product of the kernel and the full state by a finite sum of weighted scalar outputs provided by the mobile sensors. When the full state feedback operator is time-dependent thus rendering its associated kernel time-varying, the approximation results in moving sensors with time-varying static gains. To calculate the velocity of the mobile sensors within the spatial domain the time-varying kernel is set equal to the sensor density and thus the solution to an associated advection PDE reveals the velocity field of the sensor network. To obtain the speed of the finite number of sensors, a domain decomposition based on a modification of the Centroidal Voronoi Tessellations (µ-CVT) is used to decompose the kernel into a finite number of cells, each of which contains a single sensor. A subsequent application of the µ-CVT on the velocity field provides the individual sensor speeds. The nature of this µ-CVT ensures collision avoidance by the very structure of the kernel decomposition into non-intersecting cells. Numerical simulations are provided to highlight the proposed sensor guidance.