By Rosario Toscano
Structured Controllers for doubtful platforms makes a speciality of the improvement of easy-to-use layout recommendations for powerful low-order or fixed-structure controllers (particularly the industrially ubiquitous PID controller). those innovations are in keeping with a recently-developed stochastic optimization strategy termed the "Heuristic Kalman set of rules" (HKA) using which ends up in a simplified method that allows the answer of the dependent keep an eye on challenge and not using a great quantity of user-defined parameters. an outline of the most stochastic tools employable within the context of constant non-convex optimization difficulties can be supplied and diverse optimization standards for the layout of a dependent controller are thought of; H
∞, H2, and combined H2/H∞ each one advantages a bankruptcy to itself. Time-domain-performance necessities might be simply integrated within the design.
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These different concepts are detailed hereafter. 1 Signal Norms We consider real valued signals1 that are piecewise continuous functions of time t ∈ [0, ∞). In this section we introduce some different norms for these signals. 1) φ(v + w) ≤ φ(v) + φ(w) for all α ∈ R and v, w ∈ V. A norm is defined on a vector space. To apply this concept to the case of signals, it is necessary to define sets of signals that are vector spaces. This is the case of the signal spaces described below. 1 In the case of a stochastic signal, we will always assume that it is modeled as an ergodic stationary stochastic process.
3 System Norms Given an LTI-system an important issue is to characterize, in some sense, the amplification (or attenuation) introduced by the system for a given input signal. To emphasize the importance of this issue, consider the control problem shown in Fig. 2 where K(s) is the controller to be designed and G(s) is the transfer matrix of the system to be controlled. e. the control signal must be admissible by the system) despite the external influences r, d, and n. One way to evaluate the performance of the closed-loop system is to measure the gain provided by the system T , between the inputs (r, d and n) and the outputs e 36 2 Signal and System Norms and u: ⎡ ⎤ r e = T (s) ⎣d ⎦ u n Good performance is then obtained if the transfer matrix T (s) is small or, more specifically, if the gain of T (s) is small.
For better reading, the parts of the book dedicated to these issues are also given. 1 The Uncertain System G The design of a controller requires a mathematical description, called a model, of the system to be controlled. This model is always obtained at the price of approximations and idealizations and so we have an approximate description of the system behavior. 8 However, linear models often constitute a gross oversimplification of the real system, and so we have to take into account the uncertainties resulting from this simplification.