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Appendix
Positive Matrices
This chapter concerns positive matrices which are matrices with elements of the positive real numbers. The motivations for the inclusion of the algebraic structure of positive matrices are the problems (1) of stability of the system of probability measures of the Markov process o
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Appendix
Stochastic Processes
Specialized topics on the theory of stochastic processes are described which are used in the body of this book. Defined are a filtration and stochastic processes relative to a filtration. Elementary martingale theory is discussed. Stopping times and a stochastic process indexed b
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Appendix
Control and System Theory of Deterministic Systems
Concepts and theorems of the system theory of deterministic linear systems are summarized. Controllability, observability, and a realization are formulated. Realization theory includes necessary and sufficient conditions for the existence of a realization, a characterization of t
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Appendix
State-Variance Matrices
Concepts and results of the geometric structure of the set of state-variance matrices of a time-invariant Gaussian system are provided in this chapter. With respect to a condition, the set is convex with a minimal and a maximal element. In case the noise variance matrix satisfies
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Appendix
Mathematics
The reader finds in this short appendix concepts and results of various topics of mathematics. These topics are used in the body of the book but are not part of control theory. Topics covered are: algebra of set theory; a canonical form; algebraic structures including monoids, gr
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Appendix
Covariance Functions and Dissipative Systems
The concept of a dissipative system is defined for a deterministic linear system and is satisfied if there exists a storage function and a supply rate such that the dissipation inequality holds. It is proven that a deterministic linear system is dissipative if and only if a relat
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Appendix
Matrix Equations
The Lyapunov equation and the algebraic Riccati equation are treated in depth. The Lyapunov equation arises as the equation for the asymptotic covariance matrix of the state of a stationary Gaussian system. The algebraic Riccati equation arises in the Kalman filter, in stochastic
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Appendix
Probability
Concepts and results of probability theory are presented in this chapter which complement those of Chapter 2. Concepts covered in detail include the canonical variable decomposition of a tuple of Gaussian random variables, a set of stable probability distribution functions, condi
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Power-imbalance allocation control of power systems
A frequency bound for time-varying loads
For secondary frequency control of power systems with lossless networks, we proposed a method named Power-Imbalance Allocation Control (PIAC) to restore the nominal frequency after a disturbance. PIAC has shown a good transient performance where under a constant power imbalance,
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The synchronization of power generators is an important condition for the proper functioning of a power system, in which the fluctuations in frequency and the phase angle differences between the generators are sufficiently small when subjected to stochastic disturbances. Serious
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Positive systems with positive inputs and positive outputs are used in several branches of engineering, biochemistry, and economics. Both control theory and system theory require the concept of reachability of a time-invariant discrete-time linear positive system. The subset of t
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The synchronization stability of a complex network system of coupled phase oscillators is discussed. In case the network is affected by disturbances, a stochastic linearized system of the coupled phase oscillators may be used to determine the fluctuations of phase differences in
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A consensus-control-based multi-level control law named Multi-Level Power-Imbalance Allocation Control (MLPIAC) is presented for a large-scale power system partitioned into two or more groups. Centralized control is implemented in each group while distributed control is implement
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The problem considered is to construct all solutions of an equation for a tuple of languages. The tuple in synchronous composition should equal a considered language. Of special interest are the maximal solutions with respect to a partial order relation on the set of solutions. T
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The traditional secondary frequency control of power systems restores nominal frequency by steering Area Control Errors (ACEs) to zero. Existing methods are a form of integral control with the characteristic that large control gain coefficients introduce an overshoot and small on
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To balance the power supply and demand with optimized control cost and nominal synchronized frequency, we propose a secondary frequency control approach, named Power-Imbalance Allocation Control (PIAC), for power systems with lossless networks, consisting of synchronous machines,
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The study of control of stochastic systems requires knowledge of probability and of stochastic processes. Probability is summarized in this chapter in a way which is sufficient for studying the control and system theory of the subsequent chapters. Additional concepts and results
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Optimal stochastic control problems are considered for a time-invariant stochastic control system with partial observations on an infinite horizon. Such problems can be solved by a dynamic programming method for partial observations. Both the average cost and the discounted cost
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The filter problem is to derive an expression for the conditional distribution of the state of a stochastic system conditioned on the past outputs of the considered system and a recursion of the parameters of that conditional distribution. In this chapter the filter problem for a
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The electrical power grid is a fundamental infrastructure in today’s society. The synchronization of the frequency to nominal frequency over all the network is essential for the proper functioning of the power grid. The current transition to a more distributed generation by weath
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