By benjamin; okamoto, Hideaki; lodge, Goerge C. Roberts
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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.