We propose a very general framework that systematizes the notion of a hybrid system, combining differential equations and automata, governed by a hybrid controller that issues continuous-variable commands and makes logical decisions. We first identify the phenomena that arise in real-world hybrid systems. Then, we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems, evolving on continuous-variable state spaces and subject to continuous controls and discrete transitions. The model captures the identified phenomena, subsumes previous models, yet retains enough structure to pose and solve meaningful control problems. We develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework. In particular, we demonstrate the existence of optimal (relaxed) and near-optimal (precise) controls and derive "generalized quasi-variational inequalities" that the associated value function satisfies. We summarize algorithms for solving these inequalities based on a generalized Bellman equation, impulse control, and linear programming.

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A unified framework for hybrid control: model and optimal control theory

The placement of a minimal set of phasor measurement units (PMUs) so as to make the system measurement model observable, and thereby linear, is investigated. A PMU placed at a bus measures the voltage as well as all the current phasors at that bus, requiring the extension of the topological observability theory. In particular, the concept of spanning tree is extended to that of spanning measurement subgraph with an actual or a pseudomeasurement assigned to each of its branches. The minimal PMU set is found through a dual search algorithm which uses both a modified bisecting search and a simulated-annealing-based method. The former fixes the number of PMUs while the latter looks for a placement set that leads to an observable network for a fixed number of PMUs. In order to accelerate the procedure, an initial PMU placement is provided by a graph-theoretic procedure which builds a spanning measurement subgraph according to a depth-first search. From computer simulation results for various test systems it appears that only one fourth to one third of the system buses need to be provided with PMUs in order to make the system observable. >

Power system observability with minimal phasor measurement placement

This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity equations, the boundary and continuity conditions, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper.

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Optimal control of switched systems based on parameterization of the switching instants

Controlling mechanical systems with backlash-a survey

: Complex systems typically possess a hierarchical structure, characterized by continuous-variable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today perform computer-coded checks and issue logical as well as continuous-variable control commands. Such are 'hybrid' systems. Traditionally, the hybrid nature of these systems is suppressed by converting them into either purely discrete or continuous entities. Motivated by real-world problems, we introduce 'hybrid systems' as interacting collections of dynamical systems, evolving on continuous-variable state spaces, and subject to continuous controls and discrete phenomena. We identify the discrete phenomena that arise in hybrid systems and review previously proposed models. We propose a hybrid control model, coupling differential equations and automata, that encompasses them. Our unified model is natural for posing and solving hybrid analysis and control problems. We discuss topological issues that arise in hybrid systems analysis. Then we compare the computational capabilities of analog, digital, and hybrid machines by proposing intuitive notions of analog machines simulating digital ones. We show that simple continuous systems possess the power of universal computation. Hybrid systems have further simulation capabilities. For instance, we settle the famous asynchronous arbiter problem in both continuous and hybrid settings. Further, we develop analysis tools for limit cycle existence, perturbation robustness, and stability. We analyze a hybrid control system, typically used in aircraft, that logically switches between two conventional controllers. Stability of such systems has previously only been tested using extensive simulation; we prove global asymptotic stability for a realistic set of cases. Our tools demonstrate robustness of this stability with respect

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Studies in hybrid systems: modeling, analysis, and control

Electric utilities are becoming increasingly interested in using synchronized phasor measurements from around power systems to enhance their protection and remedial action control strategies. Accordingly, the task of predicting future behavior of the power system before it actually occurs has become an important area of research. This paper presents and analyses several approaches for solving the real-time prediction problem. In order to solve power systems with detailed load models fast enough for real-time prediction, the authors present a new piecewise constant current load model approximation technique that can solve a model as complex as the New England 39 bus system with composite voltage dependent loads much faster than in real-time. If the reduced order model is too large for real-time solution, then a pattern recognition tool such as decision trees can be trained off line to associate the post-fault phasor measurements with the outcome of future behavior. In this case also, the piecewise constant current technique would be needed to perform the offline training set generation with sufficient speed and accuracy. >

/pdf/predicting-future-behavior-of-transient-events-rapidly-7jkqsa6h73.pdf

Predicting future behavior of transient events rapidly enough to evaluate remedial control options in real-time

In this paper, the concepts of transiently chaotic swings and windowed Lyapunov exponents in power system dynamics are described. An efficient computer method to detect a transiently chaotic swing from a set of real-time phasor measurements is presented. The method estimates the largest windowed Lyapunov exponent, /spl lambda//spl circ//sub [N/spl Delta/T]//sup 1/ as a detection index. In simulation results, it is shown that the proposed method has the potential to detect a transiently chaotic swing and the estimated /spl lambda//spl circ//sub [N/spl Delta/T]//sup 1/ can be used as an online stability index to predict multi-swing transient instability. >

Detection of transiently chaotic swings in power systems using real-time phasor measurements

This paper proposes methods for calculating saddle-node bifurcation points of power system power flow equations. The first method calculates a saddle-node bifurcation point along a given ray in the parameter space of power flow equations. By exploiting the special structure of power flow equations, the method calculates a saddle-node bifurcation point along a given ray as a solution to a constrained optimization problem. The constrained optimization can be solved efficiently with standard optimization methods. The second method calculates a locally closest saddle-node bifurcation point with respect to the operating point. This method uses an iterative process of computing a saddle-node bifurcation point along a ray, and then updating the direction of the ray for calculating a closer saddle-node bifurcation point. The method is a quasi-Newton method that updates the direction of a ray based on the first-order derivatives and the approximations to the second-order derivatives of the distance between saddle-node bifurcation points and the operating point. The paper compares the proposed methods with other methods on two test examples. >

http://ntur.lib.ntu.edu.tw/bitstream/246246/151755/1/04.pdf

New methods for computing a saddle-node bifurcation point for voltage stability analysis

We consider optimal control design for systems that involve continuous and discrete states. The discontinuous nature of such a system results in the inconsistency of the derivatives of the objective function used by gradient-based numerical optimization algorithms. To avoid this difficulty, we propose a novel algorithm using constrained differential dynamic programming to solve the optimal control problem. >

http://ieeexplore.ieee.org/iel2/1070/7738/00325606.pdf

Optimal control of systems with continuous and discrete states

The problem of controller placements in large-scale linear systems is considered. Specifically, given a large-scale linear system it is desired to find the optimal number of linear controllers, the optimal locations in the system to place these controllers and the optimal feedback gain of each controller so that an objective function is optimised, and also so that other desirable properties of the controlled system, such as eigenstructure assignment, can occur. A formulation of the controller placement problem is presented. A general solution algorithm based on simulated annealing is developed for the controller placement problem. The solution algorithm can arrive at the globally optimal solution. The solution algorithm has been applied to an eleven-bay structure comprising a 132-dimension linear system with very promising results.

Jin Lu

Papers

Predicting future behavior of transient events rapidly enough to evaluate remedial control options in real-time

Detection of transiently chaotic swings in power systems using real-time phasor measurements

New methods for computing a saddle-node bifurcation point for voltage stability analysis

Optimal control of systems with continuous and discrete states

Optimal controller placements in large-scale linear systems