Fractional interval observers and initialization of fractional systems

International audienceIn this paper an interval observer is synthesized for fractional linear systems with additive noise and disturbances. The contribution of system whole past to future output is taken into account as an initialization function. Provided the initialization function is upper and lower bounded, it is shown in this paper that the fractional interval observer (FIO) allows to bound pseudo-state free responses by an upper and a lower trajectory. In case interval observers cannot be synthesized straightforwardly, so as to obtain a stable and non-negative estimation error, it is shown that a change of coordinates allows to overcome this problem. The proposed methodology allows to bound fractional systems trajectories when the whole past is unknown but can be bounded. Finally, a numerical example is given to show the effectiveness of the proposed methods on the initialization of fractional linear systems

Design of Interval Observers for Estimation and Stabilization of Discrete-Time LPV Systems

International audienceThis work is devoted to interval observers design for discrete-time Linear Parameter-Varying (LPV) systems under the assumption that the vector of scheduling parameters is not available for measurements. Two problems are considered: a pure estimation problem and an output stabilizing feedback design problem where the stability conditions are expressed in terms of Linear Matrix Inequalities (LMIs). The efficiency of the proposed approach is demonstrated through computer simulations

다층레이어 네트워크 구조를 가지는 선형 시불변 다개체 시스템의 상태일치

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 전기·정보공학부, 2021. 2. 심형보.전통적으로 다 개체 시스템의 상태 일치 문제는 한 가지의 네트워크 상에서 한 가지의 정보를 주고받는 경우에 대해서 주로 연구가 되었다. 하지만 최근에는 이러한 가정은 보다 복잡한 상호작용을 나타내는 데 한계가 있기 때문에 새로운 접근법이 필요한 상황이다. 본 논문에서는 각 개체가 서로 다른 정보를 서로 다른 네트워크 상에서 주고받는 경우를 고려한다. 이러한 관계를 표현하기 위해 다층레이어 네트워크 (multilayer network)라는 개념을 도입하였다. 이때 동적인 제어기로 방향성이 없는 네트워크에서 상태 일치를 이루는 새로운 필요충분조건을 제시한다. 특히 제시한 조건은 그래프 이론적인 조건과 시스템 이론적인 조건을 결합하였으며, 통신 네트워크와 주고받는 정보의 상호작용을 강조한다. 더 나아가 제시한 조건을 사용하여 방향성이 없는 네트워크상에서 상태 일치를 이루는 관측기 기반 동적 제어기를 제시한다. 주요 결과는 방향성이 있는 네트워크 상에서 출력 일치를 이루는 문제로 확장한다. 아쉽게도 이 상황에서는 제시한 조건은 더 이상 필요충분조건이 되지 못하며 이런 어려움들을 다양한 예제를 통해서 설명한다. 그럼에도 불구하고, 개체의 동역학에 추가적인 조건을 가함으로써 방향성이 없는 네트워크에서 필요충분조건을 회복한다. 또한 방향성이 있는 네트워크에서 출력 일치 문제를 푸는 충분조건을 제시하고 이를 이루는 제어기를 제안한다. 본 논문의 효용성은 여러 가지 적용 예제를 통해 보인다. 첫 번째로 분산 관측 문제를 다층 레이어 네트워크 상의 상태 일치 문제로 표현한다. 제시된 방법을 사용하면 주변 개체와의 통신량을 기존 결과들 보다 줄이는 새로운 분산 관측기를 제시한다. 두번째로 논문의 결과를 사용해 편대 제어 문제를 푼다. 특히, 원하는 편대의 모양이 개체의 상대적인 위치와 상대적인 각도로 주어진 경우를 고려한다. 제시한 방법을 사용하여 원하는 편대를 이루는 동적 제어기를 제시하였고, 편대의 크기를 유기적으로 조절하는 알고리즘을 제시한다. 마지막으로 다층 레이어 네트워크를 분산 최적화 문제에 적용을 한다. 이를 통해 매시간 결정 변수의 일부분만을 통신하는 통신적으로 더 효율적인 알고리즘을 제시한다.Traditionally, the consensus of multi-agent systems is often studied by assuming that there is a single network consisting of a single type of interaction. Recently, such an assumption is being challenged due to its limitation in representing more complex interactions. In this thesis, we consider the case where each agent is interacting using multiple, different types of output information. In order to model such interactions, the concept of a multilayer graph is employed. A novel necessary and sufficient condition is proposed for the existence of a dynamic coupling law to achieve state consensus for a multi-agent system over an undirected network. Specifically, the proposed condition couples graph theoretic conditions with system theoretic conditions and highlights the interplay between the communication network and information exchange between agents. Furthermore, based on the proposed condition, an observer-based dynamic controller is designed to achieve state consensus over an undirected network. The main results are then extended to output consensus problem over a directed network. Unfortunately, the proposed conditions are no longer necessary and sufficient and the challenge is illustrated through various examples. Nevertheless, additional assumptions are made on the dynamics of the agent to recover the equivalence for output consensus over the undirected multilayer network. A sufficient condition is also given for output consensus problem over the directed network and the corresponding controller design is presented. The effectiveness of the work is shown by a series of applications of the main results. First, the distributed state estimation problem is formulated into a consensus problem over a multilayer network. The proposed approach allowed us to develop a novel design for a distributed observer that communicates less information with its neighbors compared to existing designs. Secondly, the main results are applied to the formation control problem. Specifically, we consider the case when the desired formation is given by a combination of relative positional constraint and bearing constraint. Using the proposed approach, a dynamic controller is designed to achieve the desired formation while organically scaling the overall size of the formation. Finally, a multilayer network is also applied to the distributed optimization problem. Through multilayer networks, a communication-efficient algorithm is proposed which only communicates a part of the decision vector at each time instant.ABSTRACT i List of Figures ix List of Tables ix Notation and Symbols xi 1 Introduction 1 1.1 Research Background 1 1.2 Contributions and Outline of Dissertation 7 2 Preliminaries on Graph Theory and Convex Optimization 13 2.1 Graph Theory and Consensus Problem 13 2.1.1 Basic Definitions 13 2.1.2 Connectedness of the Graph 14 2.1.3 Laplacian Matrix and Its Properties 17 2.2 Multilayer Graph Theory 22 2.3 Convex Optimization 24 2.3.1 Convex Functions and Useful Properties 24 2.3.2 Optimization Algorithms 28 3 Consensus Problem over the Multilayer Network 41 3.1 Problem Formulation 41 3.2 A Necessary and Sufficient Condition for State Consensus 45 3.3 Proof of Necessity 51 3.4 Proof of Sufficiency 58 3.4.1 Additional Considerations for the Controllers 63 4 Extension to Output Consensus over Directed Network 67 4.1 Necessary Conditions for the Output Consensus Problem 67 4.2 Challenges for the Output Consensus over Directed Networks 71 4.3 Controller Design for the Output Consensus Problem 74 4.3.1 Controller Design under System Theoretic Constraint 74 4.3.2 Controller Design under Information Structural Constraint 82 4.4 Static Output Diffusive Coupling 84 4.5 Summary of Results 86 4.5.1 Comparison with Single-layer Consensus Problem 86 4.5.2 Relation between Necessary and Sufficient Conditions 87 5 Application to the Distributed State Estimation Problem 89 5.1 Problem Formulation 89 5.2 Distributed State Estimation over Static Network 92 5.2.1 Design Procedures 100 5.3 Distributed State Estimation over Switching Network 103 5.4 Simulation Results 111 6 Application to the Formation Control Problem 115 6.1 Problem Formulation 115 6.2 Formation Control Problem using Multilayer Network 117 6.3 Simulation Results 119 6.3.1 Achieving a Static Formation 119 6.3.2 Scaling Formation via Multilayer Network 123 7 Application to the Distributed Optimization Problem 127 7.1 Problem Formulation 127 7.2 Distributed PI Algorithm 129 7.2.1 Distributed PI Algorithm under Static Network 129 7.2.2 State Transformation for Analysis 132 7.3 Convergence Analysis for the PI Algorithm 136 7.3.1 Convergence with Weak Coupling 136 7.3.2 Convergence with Strong Coupling 139 7.3.3 Convergence under Fast Switching 153 7.4 Construction of Distributed Algorithms 158 7.4.1 Distributed Gradient Descent Method 158 7.4.2 Distributed Heavy-ball Method 160 7.4.3 Distributed Heavy-ball Method with Cyclic Coordinate Descent 166 7.5 Numerical Experiments 170 7.5.1 Distributed PI Algorithm 170 7.5.2 Distributed Heavy-ball Algorithm 172 7.6 Remark on the Study of Continuous-time Algorithms 175 8 Conclusions and Further Issues 177 APPENDIX 183 A.1 Technical Lemmas 183 A.2 Comparisons with Existing Consensus Problems 185 A.2.1 Consensus Problem of Homogeneous Agents over Singlelayer Network 186 A.2.2 Consensus Problem of Heterogeneous Agents over Singlelayer Network 188 A.2.3 Consensus Problem over Matrix-weighted Network 190 A.3 Detectability Interpretation of the Necessary Conditions 191 BIBLIOGRAPHY 195 국문초록 209Docto

High Dynamic Range (HDR) Display Perception

Displays have undergone a huge development in the last several decades. From cathode-ray tube (CRT), liquid crystal display (LCD), to organic light-emitting diode (OLED), even Q-OLED, the new configurations of the display bring more and more functions into industry and daily life. In the recent several years, high dynamic range (HDR) displays become popular. HDR displays usually refer to that the black level of the display is darker and the peak being brighter compared with the standard dynamic range (SDR) display. Traditionally, the peak luminance level can be used as the white in characterization and calibration. However, for HDR displays, the peak luminance is higher than the traditional diffuse white level. Exploration of the perceptual diffuse white in HDR image when presented in displays is proposed, which can be beneficial to the characterizing and the optimizing the usage of the HDR display. Moreover, in addition to the ``diffuse white , 3D color gamut volume can be calculated in some specific color appearance models. Calculation and modeling of the 3D color gamut volume can be very useful for display design and better characterizing display color reproduction capability. Furthermore, the perceptional color gamut volume can be measured through psychophysical experiments. Comparison between the perceptional color gamut volume and the theoretical 3D gamut volume calculations will reveal some insights for optimizing the usage of HDR displays. Another advantage of the HDR display is its darker black compared with the SDR display. Compared with the real black object, what level of black is `perfect\u27 enough in displays? Experiments were proposed and conducted to evaluate that if the HDR display is capable of showing ``perfect black for different types of background images/patterns. A glare-based model was proposed to predict the visual ``perfect black. Additionally, the dynamic range of human vision system is very large. However, the simultaneous dynamic range of human vision system is much smaller and is important for the fine tuning usage of HDR displays. The simultaneous dynamic range was measured directly for different stimulus sizes. Also, it was found that the simultaneous dynamic range was peak luminance level dependent. A mathematical model was proposed based on the experimental data to predict the simultaneous dynamic range. Also the spatial frequency effect of the target pattern on the simultaneous dynamic range was measured and modeled. The four different assessments about HDR displays perception would provide experimental data and models for a better understanding of HDR perception and tuning of the HDR display

DIGITAL ENCODING OF TELEVISION SIGNALS USING THE PULSE WIDTH MODULATOR

An attempt is made to quantify the circuit complexity and mean circuit speed of linearly quantized straight PCM video encoding techniques. Any significant reduction in circuit complexity (i.e. the number of active and passive devices to be integrated) is considered important since this determines: chip area and yield if the encoder is to be fully integrated. Analysis indicates that the complexity of the more highly developed straight PCM video encoders can be reduced by typically a factor 3 using either non-programmed sequential encoding, pulse width modulator encoding or programmed sequential encoding (closed loop successive approximation). The encoder studied in this work is an 8-bit pulse width modulator video encoder using a 2-step production line technique and a detailed design procedure for a prototype encoder is given. This encoder is considered to achieve 7-bit resolution at a sampling rate of 13.3MHZ. A mathematical model of the encoder-decoder system is developed for numerical evaluation of the effect of encoder errors and white Gaussian noise upon a coded and decoded video signal. A triangular wave test is applied to examine the effect of encoder errors upon the statio transfer characteristic of the encoder. Dynamic errors are investigated by simulating colour subcarrier at the model input and observing the phase and gain errors at the filtered codec output. Using differential phase and gain, an attempt is made to determine a circuit design and alignment criterion such that most practical codecs will fall within specific bounds on these parameters (taken as ±6° and ±6% respective1y). In the absense of dither, Monte Carlo analysis indicates that the maximum voltage error incurred by each encoder error source should have a high probability (95%) of being less than a half quantum if 85 - 90% of codecs measured are to fall within the above bounds. If white Gaussian noise is used as a simple dither signal then the probability of a codec falling within the above bounds may increase to about 95%. Improvements to the encoder are discussed, including several automatic error correction techniques which combat instrumental errors and give a more robust PWM encoder. Also, by predetermining the most significant bit for each set of 4 coded bits it is possible to halve the encoder clock frequency (to 133MHZ) without significantly changing the encoder complexity

Inverse tone mapping

The introduction of High Dynamic Range Imaging in computer graphics has produced a novelty in Imaging that can be compared to the introduction of colour photography or even more. Light can now be captured, stored, processed, and finally visualised without losing information. Moreover, new applications that can exploit physical values of the light have been introduced such as re-lighting of synthetic/real objects, or enhanced visualisation of scenes. However, these new processing and visualisation techniques cannot be applied to movies and pictures that have been produced by photography and cinematography in more than one hundred years. This thesis introduces a general framework for expanding legacy content into High Dynamic Range content. The expansion is achieved avoiding artefacts, producing images suitable for visualisation and re-lighting of synthetic/real objects. Moreover, it is presented a methodology based on psychophysical experiments and computational metrics to measure performances of expansion algorithms. Finally, a compression scheme, inspired by the framework, for High Dynamic Range Textures, is proposed and evaluated

Model Predictive based load frequency control studies in a deregulated environment

A fundamental objective in power system operations is to ensure reliablity and quality supply, and one key action that aids the accomplishment of this objective is the load frequency control (LFC). Primarily, LFC is an automatic action that aims to restore system frequency and net tie line power between a control area (CA) and its neighbours to their scheduled values; these quantities deviate when there is an imbalance between active power demand and supply in a synchrononus interconnection. This thesis aims to investigate a model predictive control (MPC) technique for LFC problems in a deregulated power system environment which has become a challenging task. In deregulated power interconnections, generation companies (GenCos) and distribution companies (DisCos) exist in each CA, and a transmission system operator (TSO) in each area is responsible for grid reliability. Each TSO handles LFC in its CA and ensures that market participants (GenCos and DisCos) in other CAs have an unbiased and open access to its network. As a result, there has been a rise in cross-border transac- tions between GenCos and DisCos for bulk energy and load matching (LM) and consequently large frequency fluctuations recently. DisCos can participate in LFC by making bilateral LM contracts with GenCos. An extensive review of the LFC literature, in terms of strengths and weaknesses of different control techniques, is presented to identify the key gaps. The review reveals that MPC can bring some benefits in the deregulated environment but its strengths are underexploited. Beginning with a small-scale system to provide insights into deregulated system modelling and predictive control design, a centralised MPC (CMPC)-based LFC scheme is proposed for a 2-area deregulated power system with measured (contracted) and unmeasured (uncontracted) load changes, where the areas are assumed to equally rated. The 2-area deregulated system is developed by incorporating bilateral LM contracts in the well known traditional LFC model as a new set of information. It is assumed that DisCos handle contracted load changes via bilateral LM contracts with GenCos and a TSO handles any variations outside the LM con- tracts (uncontracted) via a supplementary control scheme which represents the CMPC. The CMPC algorithm is developed as a tracking one, with an observer to provide estimates of the system states and uncontracted load changes. Also, input and incremental state constraints, which depict limits on LFC control efforts and generation rate constraints (GRC) respectively, are considered. A simulation comparison of the proposed CMPC solution and optimal linear quadratic regulator (LQR) demonstrates the efficacy of CMPC. Developing deregulated LFC models for larger systems with complex topologies and a large number of CAs/market participants could be laborious. Therefore, a novel generalised modelling framework for deregulated LFC is further proposed. The key benefits of the generalised framework is that it provides a relatively easy and systematic procedure to develop deregulated LFC benchmark systems irrespective of the interconnection size, topology and number of market participants. It also offers the flexibility of accommodating LFC studies where CAs have either equal (often assumed) or unequal (more pragmatic) rated capacities. A 7-area deregulated benchmark model is developed from the generalised framework to illustrate its usage and significance, and the importance of incorporating area rated capacities is demonstrated via simulations. In addition, a 4-area benchmark model is developed to provide a reader with more insight into how the generalised formulation can be applied to develop LFC models for an arbitrary network. Furthermore, to demonstrate the scalability of an MPC design procedure, the CMPC proposed previously is extended to examine the LFC problem of the 7-area system. Key novelties here are CAs are assumed to have unequal rated capacities, some GenCos do not participate in supplementary control, and the control input to each GenCo is computed separately rather than a single lumped input for each CA which is the norm in previous deregulated LFC studies. The separate control inputs is to ensure that the input constraints of each GenCo is accounted for in the CMPC in addition to their GRCs and this is achieved by incorporating the area participation factors of the GenCos explicitly in the CMPC cost function. A test conducted on the 7-area benchmark confirms the benefits of this new approach. CMPC shows great potential for deregulated LFC in terms of multiple inputs coordination, effective disturbance rejection and constraints handling; however it is unrealistic for practical interconnections were CAs are operated by different organisations and have large geographical separations. This limitation is addressed by investigating a distributed MPC (DMPC) technique for rejecting incremental load changes, convenient for a finite number of control areas (subsystems), and therefore represents a more practical control architecture for LFC in multi-area systems. The proposed DMPC is non-cooperative and developed to operate using output feedback, where distributed observers using local measurements are developed to provide uncontracted load changes and subsystem states’ estimates to local MPCs. Moreover, the DMPC, unlike other non-cooperative schemes, is simple and devoid of extensive offline parameter tuning. Using the 4-area and the 7-area benchmarks models developed as test systems for the proposed DMPC, some comparisons of simulations results, regulation cost and discussions are provided between the proposed DMPC and alternative MPC schemes