By Mathukumalli Vidyasagar
This publication introduces the so-called "stable factorization procedure" to the synthesis of suggestions controllers for linear keep an eye on structures. the most important to this technique is to view the multi-input, multi-output (MIMO) plant for which one needs to layout a controller as a matrix over the fraction box F linked to a commutative ring with identification, denoted by way of R, which additionally has no divisors of 0. during this environment, the set of single-input, single-output (SISO) sturdy regulate structures is exactly the hoop R, whereas the set of solid MIMO keep an eye on structures is the set of matrices whose parts all belong to R. The set of volatile, that means no longer inevitably good, keep watch over structures is then taken to be the sphere of fractions F linked to R within the SISO case, and the set of matrices with parts in F within the MIMO case. The imperative inspiration brought within the booklet is that, in so much occasions of sensible curiosity, each matrix P whose components belong to F could be "factored" as a "ratio" of 2 matrices N,D whose components belong to R, in this sort of method that N,D are coprime. within the commonly used case the place the hoop R corresponds to the set of bounded-input, bounded-output (BIBO)-stable rational move features, coprimeness is resembling capabilities no longer having any universal zeros within the closed correct half-plane together with infinity. despite the fact that, the concept of coprimeness extends quite simply to discrete-time structures, distributed-parameter structures in either the continual- in addition to discrete-time domain names, and to multi-dimensional structures. therefore the good factorization procedure permits one to trap most of these occasions inside of a standard framework. the most important lead to the strong factorization technique is the parametrization of all controllers that stabilize a given plant. it truly is proven that the set of all stabilizing controllers might be parametrized through a unmarried parameter R, whose parts all belong to R. additionally, each move matrix within the closed-loop procedure is an affine functionality of the layout parameter R. hence difficulties of trustworthy stabilization, disturbance rejection, powerful stabilization and so on. can all be formulated by way of deciding upon a suitable R. it is a reprint of the ebook keep an eye on approach Synthesis: A Factorization process initially released through M.I.T. Press in 1985. desk of Contents: advent / right sturdy Rational features / Scalar structures: An creation / Matrix jewelry / Stabilization
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Additional info for Control System Synthesis - A Factorization Approach, Part I (Synthesis Lectures on Control and Mechatronics)
A study of strong stabilizability is important for at least two reasons: (i) Suppose p is strongly stabilizable. Let c ∈ S stabilize p and let p1 = p/(1 + pc) denote the resulting plant transfer function. Since c is stable, a factorization of c is given by nc = c, dc = 1. 5) that p1 = np where np is the numerator of a factorization of p. In other words, if p is stabilized using a stable compensator, then the resulting plant transfer function p1 = p/(1 + pc) has the same C+e -zeros as p and no others.
9) can be satisfied by a unit u(·) ∈ S that has all of its poles at s = −1 (or any other prespecified point on the negative real axis). 3 for constructing u also leads to the same conclusion. 5, u(j ) = 1 + j 2, u(∞) = 2 . 17) Let f (z) = u((1 + z)/(1 − z)). Then f must satisfy f (0) = 1, f (0) = −1, f (j ) = 1 + j 2, f (1) = 2 . 18) Note that f (0) is computed using the chain rule: f (0) = du ds |s=1 · |z=0 . 3, an irrational unit h is constructed that satisfies h(0) = 1, h (0) = −1, h(j ) = 1 + j 2, h(1) = 2 .
14) shows that either F −1 (∞) 2 = F −1 , or else F −1 (j ω) 2 = F −1 for some finite ω. Case (i) F −1 (∞) 2 = F −1 . Since F −1 (∞) is a real matrix, there exists a real vector v such that v 2 = 1, and F −1 (∞)v 2 = F −1 (∞) 2 . For brevity let c denote the constant F −1 (∞) 2 = F −1 , and let u denote the vector F −1 (∞)v. Then routine calculations show that H = 1/c. However, [F (∞) − H ]u = 0, which shows that |(F − H )(∞)| = 0. Thus G = F − H is not a unit. Case (ii) F −1 (0) 2 = F −1 . Since F −1 (0) is a real matrix, the above proof applies.
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