By Chang-Hee Won, Cheryl B. Schrader, Anthony N. Michel
This volume—dedicated to Michael ok. Sain at the party of his 70th birthday—is a suite of chapters masking fresh advances in stochastic optimum keep watch over conception and algebraic platforms conception. Written through specialists of their respective fields, the chapters are thematically geared up into 4 parts:
* half I makes a speciality of statistical keep an eye on theory, the place the price functionality is seen as a random variable and function is formed via rate cumulants. during this appreciate, statistical keep an eye on generalizes linear-quadratic-Gaussian and H-infinity control.
* half II addresses algebraic platforms theory, reviewing using algebraic platforms over semirings, modules of zeros for linear multivariable platforms, and zeros in linear time-delay systems.
* half III discusses advances in dynamical platforms characteristics. The chapters specialize in the steadiness of a discontinuous dynamical procedure, approximate decentralized fastened modes, direct optimum adaptive regulate, and balance of nonlinear structures with restricted information.
* half IV covers engineering education and features a distinct bankruptcy on theology and engineering, considered one of Sain's most up-to-date learn interests.
The e-book can be an invaluable reference for researchers and graduate scholars in platforms and regulate, algebraic platforms thought, and utilized arithmetic. Requiring simply wisdom of undergraduate-level keep an eye on and structures idea, the paintings can be utilized as a supplementary textbook in a graduate direction on optimum regulate or algebraic structures theory.
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Extra resources for Advances in Statistical Control, Algebraic Systems Theory, and Dynamic Systems Characteristics: A Tribute to Michael K. Sain
The utility function can be determined experimentally, or we can use one of the known utility functions. Most utility functions are convex, which implies risk-aversive behavior of a decision maker. In classical LQG optimal control we can view the cost function as having a quadratic utility function. Now, the notion of risk can be defined precisely using a utility or a disutility function (a disutility function being the negative of a utility function) as Bertsekas [Ber76] or Whittle [Whi90] did in their books.
The loss function is given by L( j, θ ( j), k( j − 1)) = θ ( j), R( j − 1)θ + k( j − 1), P( j − 1)k( j − 1) , j ∈ I, (26) where < ·, · >: Rn × Rn → R is the Euclidean inner product and θ ( j) is the unique solution of equation (23) satisfying the initial condition x(n0 ) = x0 . R( j) and P( j) are positive definite,5 bounded, and symmetric for all j, j ∈ I. Similarly, the mean value constraint is given by h(n0 , Z(n0 )) = m(n0 ) + θ (n0 ), M(n0 )θ (n0 ) , (27) where m(n0 ) ∈ R+ and the matrix M(n0 ) ∈ Sn×n must be bounded and positive definite.
The separation principle, Kalman–Bucy and Kalman filters, and the Kalman optimal control gain formulae, have become commonplace. The approach of Kalman to LQG problems was, of course, based upon minimizing the average cost. We remark that the average cost is the first entry in two famous sequences of random cost statistics. The first sequence is that of the cost moments; the second sequence is that of the cost cumulants. Without more information it would not be possible to surmise whether Kalman’s formulae derived their efficacy from the average cost being a moment or from the average cost being a cumulant.
Advances in Statistical Control, Algebraic Systems Theory, and Dynamic Systems Characteristics: A Tribute to Michael K. Sain by Chang-Hee Won, Cheryl B. Schrader, Anthony N. Michel