Observability conservation by output feedback and observability Gramian bounds

International audienceThough it is a trivial fact that the observability of a linear state space system is conserved by output feedback, it requires a rigorous proof to generalize this result to uniform complete observability, which is defined with the observability Gramian. The purpose of this paper is to present such a proof. Some issues in existing results are also discussed. The uniform complete observability of closed loop systems is useful for the analysis of some adaptive systems and of the Kalman filter

Control and Estimation Oriented Model Order Reduction for Linear and Nonlinear Systems

Optimization based controls are advantageous in meeting stringent performance requirements and accommodating constraints. Although computers are becoming more powerful, solving optimization problems in real-time remains an obstacle because of associated computational complexity. Research efforts to address real-time optimization with limited computational power have intensified over the last decade, and one direction that has shown some success is model order reduction. This dissertation contains a collection of results relating to open- and closed-loop reduction techniques for large scale unconstrained linear descriptor systems, constrained linear systems, and nonlinear systems. For unconstrained linear descriptor systems, this dissertation develops novel gramian and Riccati solution approximation techniques. The gramian approximation is used for an open-loop reduction technique following that of balanced truncation proposed by (Moore, 1981) for ordinary linear systems and (Stykel, 2004) for linear descriptor systems. The Riccati solution is used to generalize the Linear Quadratic Gaussian balanced truncation (LQGBT) of (Verriest, 1981) and (Jonckheere and Silverman, 1983). These are applied to an electric machine model to reduce the number of states from $>$100000 to 8 while improving accuracy over the state-of-the-art modal truncation of (Zhou, 2015) for the purpose of condition monitoring. Furthermore, a link between unconstrained model predictive control (MPC) with a terminal penalty and LQG of a linear system is noted, suggesting an LQGBT reduced model as a natural model for reduced MPC design. The efficacy of such a reduced controller is demonstrated by the real-time control of a diesel airpath. Model reduction generally introduces modeling errors, and controlling a constrained plant subject to modeling errors falls squarely into robust control. A standard assumption of robust control is that inputs/states/outputs are constrained by convex sets, and these sets are ``tightened'' for robust constraint satisfaction. However, robust control is often overly conservative, and resulting control strategies cannot take advantage of the true admissible sets. A new reduction problem is proposed that considers the reduced order model accuracy and constraint conservativeness. A constant tube methodology for reduced order constrained MPC is presented, and the proposed reduced order model is found to decrease the constraint conservativeness of the reduced order MPC law compared to reduced order models obtained by gramian and LQG reductions. For nonlinear systems, a reformulation of the empirical gramians of (Lall et al., 1999) and (Hahn et al., 2003) into simpler, yet more general forms is provided. The modified definitions are used in the balanced truncation of a nonlinear diesel airpath model, and the reduced order model is used to design a reduced MPC law for tracking control. Further exploiting the link between the gramian and Riccati solution for linear systems, the new empirical gramian formulation is extended to obtain empirical Riccati covariance matrices used for closed-loop model order reduction of a nonlinear system. Balanced truncation using the empirical Riccati covariance matrices is demonstrated to result in a closer-to-optimal nonlinear compensator than the previous balanced truncation techniques discussed in the dissertation.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/140839/1/riboch_1.pd

On Modeling and Nonlinear Model Reduction in Automotive Systems

The current control design development process in automotive industry and elsewhere involves many expensive experiments and hand-tuning of control parameters. Model based control design is a promising approach to reduce costs and development time. In this process low complexity models are essential and model reduction methods are very useful tools. This thesis combines the areas of modeling and model reduction with applications in automotive systems. A model reduction case study is performed on an engine air path. The heuristic method commonly used when modeling engine dynamics is compared with a more systematic approach based on the balanced truncation method.The main contribution of this thesis is a method for model reduction of nonlinear systems. The procedure is focused on reducing the number of states using information obtained by linearization around trajectories. The methodology is closely tied to existing theory on error bounds and good results are shown in form of examples such as a controller used in real-world cars. Also, a model of the exhaust gas oxygen sensor, used for air-fuel ratio control in automotive spark-ignition engines, is developed and successfully validated

Observability and observer design for switched linear systems

Hybrid vehicles, HVAC systems in new/old buildings, power networks, and the like require safe, robust control that includes switching the mode of operation to meet environmental and performance objectives. Such switched systems consist of a set of continuous-time dynamical behaviors whose sequence of operational modes is driven by an underlying decision process. This thesis investigates feasibility conditions and a methodology for state and mode reconstruction given input-output measurements (not including mode sequence). An application herein considers insulation failures in permanent magnet synchronous machines (PMSMs) used in heavy hybrid vehicles. Leveraging the feasibility literature for switched linear time-invariant systems, this thesis introduces two additional feasibility results: 1) detecting switches from safe modes into failure modes and 2) state and mode estimation for switched linear time-varying systems. This thesis also addresses the robust observability problem of computing the smallest structured perturbations to system matrices that causes observer infeasibility (with respect to the Frobenius norm). This robustness framework is sufficiently general to solve related robustness problems including controllability, stabilizability, and detectability. Having established feasibility, real-time observer reconstruction of the state and mode sequence becomes possible. We propose the embedded moving horizon observer (EMHO), which re-poses the reconstruction as an optimization using an embedded state model which relaxes the range of the mode sequence estimates into a continuous space. Optimal state and mode estimates minimize an L2-norm between the measured output and estimated output of the associated embedded state model. Necessary conditions for observer convergence are developed. The EMHO is adapted to solve the surface PMSM fault detection problem

WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction

WavePacket is an open-source program package for numeric simulations in quantum dynamics. It can solve time-independent or time-dependent linear Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows, e.g., to simulate molecular quantum dynamics beyond the Born-Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semi-classical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. Being highly versatile and offering visualization of quantum dynamics 'on the fly', WavePacket is well suited for teaching or research projects in atomic, molecular and optical physics as well as in physical or theoretical chemistry. Building on the previous Part I which dealt with closed quantum systems and discrete variable representations, the present Part II focuses on the dynamics of open quantum systems, with Lindblad operators modeling dissipation and dephasing. This part also describes the WavePacket function for optimal control of quantum dynamics, building on rapid monotonically convergent iteration methods. Furthermore, two different approaches to dimension reduction implemented in WavePacket are documented here. In the first one, a balancing transformation based on the concepts of controllability and observability Gramians is used to identify states that are neither well controllable nor well observable. Those states are either truncated or averaged out. In the other approach, the H2-error for a given reduced dimensionality is minimized by H2 optimal model reduction techniques, utilizing a bilinear iterative rational Krylov algorithm

Dynamic temperature estimation of power electronics systems

This thesis proposes a method for accurate temperature estimation of thermally-aware power electronics systems. The duality between electrical systems and thermal systems was considered for thermal modeling. High dimensional thermal models present a challenge for online estimation. RC (resistor-capacitor) circuits that create a tradeoff between accuracy and complexity were used to simulate the dynamic thermal behavior of power electronics. The complexity of the thermal network was further reduced by applying a structure-preserving model order reduction technique. The reduced order thermal model was an RC circuit with fewer capacitors. Preserving the physical correspondence between the reduced order model and the physical system allows the user to use the reduced order thermal model in the sensor placement optimization process. The accuracy of the thermal estimates can be easily increased by increasing the number of sensors in the system. However, a large number of sensors increases the cost and complexity of the system. It might also interfere with the circuit design and create packaging problems. An optimal number and optimal placement of temperature sensors was found. The optimal sensor placement problem was solved by maximizing the trace of observability Gramian. The optimal number of temperature sensors was based on the state estimation error obtained from a Kalman filter. The dynamic thermal behavior of the power electronics systems was represented by a linear state space model by applying the conservation of energy principle. Therefore, assuming Gaussian noise, it is well-known that a Kalman filter is an optimal estimator for such systems. A continuous-discrete Kalman filter was used to estimate the dynamic thermal behavior of power electronics systems using an optimal number of temperature sensors placed at optimal locations. The proposed method was applied on 2-D and 3-D power electronics systems. Theoretical results were validated experimentally using IR thermal imaging and thermocouples. It was shown that the proposed method can accurately reconstruct the dynamic temperature profile of power electronics systems using a small number of temperature sensors

Spacecraft trajectory tracking and parameter estimation in the presence of a splitting contact binary asteroid

Increasing interest in asteroid mining and in-situ resource utilization will lead to an increase in asteroid surface operations. The composition of asteroids is often unknown and potentially unstable, many of which are bound together predominantly by gravitational forces. Surface operations such as mining may significantly alter the asteroid’s structure or, in the case of contact binary asteroids, cause the asteroid to split. Conditions on the evolution of the contact binary system after splitting are derived. Observability issues causing inconclusive results in the previous work were addressed here by identifying an observable set of constant parameters which describes the splitting system. The coupled problem of estimating unknown parameters of a newly-formed binary system and controlling a spacecraft’s trajectory in the system’s vicinity is investigated. An indirect adaptive control scheme is utilized to simultaneously and accurately meet both objectives. Finally, a simpler control scheme, based only on 2-body dynamics is derived to show significant improvement in performance when using the proposed adaptive control scheme over using the simple 2-body controller. For larger contact binary asteroids, the adaptive control scheme has shown up to a 20% reduction in control cost over the 2-body controller