Sanjoy K. Mitter

Stabilization of linear systems with limited information

We show that the coarsest, or least dense, quantizer that quadratically stabilizes a single input linear discrete time invariant system is logarithmic, and can be computed by solving a special linear quadratic regulator problem. We provide a closed form for the optimal logarithmic base exclusively in terms of the unstable eigenvalues of the system. We show how to design quantized state-feedback controllers, and quantized state estimators. This leads to the design of hybrid output feedback controllers. The theory is then extended to sampling and quantization of continuous time linear systems sampled at constant time intervals. We generalize the definition of density of quantization to the density of sampling and quantization in a natural way, and search for the coarsest sampling and quantization scheme that ensures stability. Finally, by relaxing the definition of quadratic stability, we show how to construct logarithmic quantizers with only finite number of quantization levels and still achieve practical stability of the closed-loop system.

Representation and control of infinite dimensional systems (vol. 1)

There is disclosed a charged particle beam apparatus capable of adjusting the convergence angle of the charged particle probe independent of the current in the probe. The apparatus comprises a charged particle gun producing a charged particle beam, a first condenser lens, a second condenser lens, an objective lens, an aperture, and an auxiliary condenser lens. The first condenser lens, the second condenser lens, and the objective lens which are arranged in this order act to sharply focus the beam to form an charged particle probe impinging on a specimen. The aperture is installed between the second condenser lens and the objective lens. The auxiliary lens is located between the objective lens and the aperture to control the convergence angle of the probe. The passage of the charged particle beam which extends from the aperture to the specimen is wide enough to prevent the current in the charged particle probe striking the specimen from varying if the strength of the auxiliary lens and the strength of the objective lens are varied.

Probabilistic Solution of Ill-Posed Problems in Computational Vision

We formulate several problems in early vision as inverse problems. Among the solution methods we review standard regularization theory, discuss its limitations, and present new stochastic (in particular, Bayesian) techniques based on Markov Random Field models for their solution. We derive efficient algorithms and describe parallel implementations on digital parallel SIMD architectures, as well as a new class of parallel hybrid computers that mix digital with analog components.

The conjugate gradient method for optimal control problems

This paper extends the conjugate gradient minimization method of Fletcher and Reeves to optimal control problems. The technique is directly applicable only to unconstrained problems; if terminal conditions and inequality constraints are present, the problem must be converted to an unconstrained form; e.g., by penalty functions. Only the gradient trajectory, its norm, and one additional trajectory, the actual direction of search, need be stored. These search directions are generated from past and present values of the objective and its gradient. Successive points are determined by linear minimization down these directions, which are always directions of descent. Thus, the method tends to converge, even from poor approximations to the minimum. Since, near its minimum, a general nonlinear problem can be approximated by one with a linear system and quadratic objective, the rate of convergence is studied by considering this case. Here, the directions of search are conjugate and hence the objective is minimized over an expanding sequence of sets. Also, the distance from the current point to the miminum is reduced at each step. Three examples are presented to compare the method with the method of steepest descent. Convergence of the proposed method is much more rapid in all cases. A comparison with a second variational technique is also given in Example 3.

Control over noisy channels

Communication is an important component of distributed and networked controls systems. In our companion paper, we presented a framework for studying control problems with a digital noiseless communication channel connecting the sensor to the controller. Here, we generalize that framework by applying traditional information theoretic tools of source coding and channel coding to the problem. We present a general necessary condition for observability and stabilizability for a large class of communication channels. Then, we study sufficiency conditions for Internet-like channels that suffer erasures.

Stochastic linear control over a communication channel

We examine linear stochastic control systems when there is a communication channel connecting the sensor to the controller. The problem consists of designing the channel encoder and decoder as well as the controller to satisfy some given control objectives. In particular, we examine the role communication has on the classical linear quadratic Gaussian problem. We give conditions under which the classical separation property between estimation and control holds and the certainty equivalent control law is optimal. We then present the sequential rate distortion framework. We present bounds on the achievable performance and show the inherent tradeoffs between control and communication costs. In particular, we show that optimal quadratic cost decomposes into two terms: A full knowledge cost and a sequential rate distortion cost.

The Capacity of Channels With Feedback

In this paper, we introduce a general framework for treating channels with memory and feedback. First, we prove a general feedback channel coding theorem based on Masseys concept of directed information. Second, we present coding results for Markov channels. This requires determining appropriate sufficient statistics at the encoder and decoder. We give a recursive characterization of these sufficient statistics. Third, a dynamic programming framework for computing the capacity of Markov channels is presented. Fourth, it is shown that the average cost optimality equation (ACOE) can be viewed as an implicit single-letter characterization of the capacity. Fifth, scenarios with simple sufficient statistics are described. Sixth, error exponents for channels with feedback are presented.

Volatility of Power Grids Under Real-Time Pricing

The paper proposes a framework for modeling and analysis of the dynamics of supply, demand, and clearing prices in power systems with real-time retail pricing and information asymmetry. Characterized by passing on the real-time wholesale electricity prices to the end consumers, real-time pricing creates a closed-loop feedback system between the physical layer and the market layer of the system. In the absence of a carefully designed control law, such direct feedback can increase sensitivity and lower the systems robustness to uncertainty in demand and generation. It is shown that price volatility can be characterized in terms of the systems maximal relative price elasticity, defined as the maximal ratio of the generalized price-elasticity of consumers to that of the producers. As this ratio increases, the system may become more volatile. Since new demand response technologies increase the price-elasticity of demand, and since increased penetration of distributed generation can also increase the uncertainty in price-based demand response, the theoretical findings suggest that the architecture under examination can potentially lead to increased volatility. This study highlights the need for assessing architecture systematically and in advance, in order to optimally strike the trade-offs between volatility/robustness and performance metrics such as economic efficiency and environmental efficiency.

A Theory of Modal Control

Although considerable progress has been made in various aspects of control theory, there still appears to be no adequate theory for the control of large-scale linear time-invariant multivariable systems. If the engineering specifications required of the controlled system can be effectively summarized in a quadratic performance measure, then linear optimal control theory, in principle, provides a linear feedback controller which would perform the required task. Even under these circumstances the computational problems may be insurmountable. In an effort to circumvent these difficulties Rosenbrock suggested the use of modal control as a design aid. Modal control may be defined as control which changes the modes (i.e., the eigenvalues of the system matrix) to achieve the desired control objectives. This paper presents a complete and rigorous theory of modal control as well as recursive algorithms which permit modal control to be realized.

CONTROLLABILITY, OBSERVABILITY AND OPTIMAL FEEDBACK CONTROL OF AFFINE HEREDITARY DIFFERENTIAL SYSTEMS*

This paper is concerned with two aspects of the control of affine hereditary differential systems. They are (i) the theory of various types of controllability and observability for such systems and (ii) the problem of optimal feedback control with a quadratic cost. The study is undertaken within the framework of hereditary differential systems with initial data in the space