Discretized Feedback Systems

Abstract

As described in Chap. 1, since discretized/quantized feedback systems become nonlinear, the analysis and design of those types of systems has not been elucidated.

In this chapter, the allowable sector for discretized nonlinear characteristics is defined, and some lemmas for norm inequalities in an ℓ ₂ space are given as a premise for the robust stability analysis of discrete feedback systems in Chap. 3. In the analysis of a response of a discrete control system, the sign of a sum of trapezoidal areas should be checked. In the mathematical development, an assumption with respect to the time responses is necessary. The validity of the assumption will be verified in the numerical examples with respect to the relationship between the steepness of responses and the resolution values and sampling period.

Appendices

Appendix A: Norms and Inner Products of L _p and ℓ _p Spaces

In this appendix, inner products and norms in an ℓ ₂ space are explained for discrete-time systems. In general, norms of L _p and ℓ _p spaces are defined as follows. For a continuous-time signal x: $\mathbb{R}_{+}\to \mathbb{R}$,

$$\begin{aligned} &\|x(t)\|_p:=\left(\int_0^\infty |x(t)|^p dt\right)^{1/p}, ~~~1\le p<\infty, \end{aligned}$$

(2.47)

$$\begin{aligned} &\|x(t)\|_\infty:=\mathrm{ess}\sup_{t\in [0,\infty)} |x(t)|, \end{aligned}$$

(2.48)

and for a discrete-time signal x: $\mathbb{Z}_{+}\to\mathbb{R}$ (or $\mathbb{Z}$),

$$\begin{aligned} &\|x(k)\|_p:=\left(\sum_{k=1}^\infty |x(k)|^p \right)^{1/p}, ~~~1\le p<\infty, \end{aligned}$$

(2.49)

$$\begin{aligned} &\|x(k)\|_\infty:=\sup_{k\ge 1} |x(k)|. \end{aligned}$$

(2.50)

In the ℓ ₂ space, the norm is defined as

$$ \|x(k)\|_2:=\left(\sum_{k=1}^\infty |x(k)|^2\right)^{1/2}, $$

(2.51)

and the inner product is given by

$$ \big\langle x(k),y(k) \big\rangle :=\sum_{k=1}^\infty x(k)y(k). $$

(2.52)

The preceding definitions for finite time series x(k)(k=0,1,2,⋯,N) are written as follows:

$$\begin{aligned} &\|x(k)\|_{2,N}:=\left(\sum_{k=1}^N |x(k)|^2\right)^{1/2}, \end{aligned}$$

(2.53)

$$\begin{aligned} &\big\langle x(k),y(k) \big\rangle _N:=\sum_{k=1}^N x(k)y(k). \end{aligned}$$

(2.54)

When N→∞, these definitions are written as:

$$\begin{aligned} &\|x(k)\|_2:=\lim_{N\to\infty}\|x(k)\|_{2,N}, \end{aligned}$$

(2.55)

$$\begin{aligned} &\big\langle x(k),y(k) \big\rangle := \lim_{N\to\infty}\big\langle x(k),y(k) \big\rangle _N. \end{aligned}$$

(2.56)

Appendix B: Hölder and Schwarz Inequalities

(1) In an L _p space, the following inequality holds:

$$ \int_0^\infty |x(t)y(t)|\mathrm{d}t \le\left(\int_0^\infty|x(t)|^p\mathrm{d}t\right)^{1/p} \left(\int_0^\infty|y(t)|^q\mathrm{d}t\right)^{1/q}, ~~~~\frac{1}{p}+\frac{1}{q}=1. $$

(2.57)

As for discrete signals, the following inequality holds in an ℓ _p space:

$$ \sum_{k=1}^\infty|x(k)y(k)|\le\left(\sum_{k=1}^\infty|x(k)|^p\right)^{1/p} \left(\sum_{k=1}^\infty|y(k)|^q\right)^{1/q},~~~~ \frac{1}{p}+\frac{1}{q}=1. $$

(2.58)

These are called Hölder’s inequalities [4, 8, 10]. The proof is given for 1≤p≤∞ (i.e., 1≤q≤∞) in, e.g., [8].

(2) An important special case of (2.58) for p=q=2 is given as

$$ \sum_{k=1}^\infty|x(k)y(k)|\le\left(\sum_{k=1}^\infty|x(k)|^2\right)^{1/2} \left(\sum_{k=1}^\infty|y(k)|^2\right)^{1/2}. $$

(2.59)

Equation (2.59) is called Schwarz’s inequality. The easier proof of (2.59) is as follows. For finite sums of N steps, Schwarz’s inequality (2.59) is rewritten as

$$\begin{aligned} &\left(\sum_{k=1}^N|x(k)y(k)|\right)^2\le \left(\sum_{k=1}^N|x(k)|^2\right) \left(\sum_{k=1}^N|y(k)|^2\right) \end{aligned}$$

(2.60)

$$\begin{aligned} &~~~~~~~~~~\sum_{k=1}^N|x(k)y(k)|^2+ 2\sum_{k,l=1,k\neq l}^N|x(k)y(k)|\cdot|x(l)y(l)| \\ &~~~~~~~~~~\le\sum_{k=1}^N|x(k)|^2|y(k)|^2 +\sum_{k,l=1,k\neq l}^N|x(k)|^2|y(l)|^2. \end{aligned}$$

(2.61)

In (2.61), the following sum must be non-negative:

$$ \sum_{k,l=1,k\neq l}^N|x(k)|^2|y(l)|^2 -2\sum_{k,l=1,k\neq l}^N|x(k)y(k)|\cdot|x(l)y(l)| =\left(\sum_{k,l=1,k\neq l}^N|x(k)y(l)|\right)^2, $$

(2.62)

and it must hold for N→∞. Thus, inequality (2.60) has been proved.

Appendix C: Minkowski Inequalities

• (1) In an L _p space, the following inequality holds:

  $$ \left(\int_0^\infty |x(t)+y(t)|^p\mathrm{d}t\right)^{1/p} \le\left(\int_0^\infty|x(t)|^p\mathrm{d}t\right)^{1/p} +\left(\int_0^\infty|y(t)|^p\mathrm{d}t\right)^{1/p}. $$

  (2.63)

  The norm expression based on (2.47) is given by

  $$ \|x(t)+y(t)\|_p\le\|x(t)\|_p+\|y(t)\|_p. $$

  (2.64)

  As for discrete signals, the following inequality holds in an ℓ _p space:

  $$ \left(\sum_{k=1}^\infty|x(k)+y(k)|^p\right)^{1/p} \le\left(\sum_{k=1}^\infty|x(k)|^p\right)^{1/p} +\left(\sum_{k=1}^\infty|y(k)|^p\right)^{1/p}. $$

  (2.65)

  The norm expression based on (2.49) is given by

  $$ \|x(k)+y(k)\|_p\le\|x(k)\|_p+\|y(k)\|_p. $$

  (2.66)

  These are called Minkowski’s inequalities [7, 8].

• (2) A special case of (2.65) for p=2 and finite sums of N steps is written as

  $$\begin{aligned} &\left(\sum_{k=1}^N|x(k)+y(k)|^2\right)^{1/2} \le\left(\sum_{k=1}^N|x(k)|^2\right)^{1/2}+ \left(\sum_{k=1}^N|y(k)|^2\right)^{1/2} \end{aligned}$$

  (2.67)
  $$\begin{aligned} &\|x(k)+y(k)\|_{2,N}\le\|x(k)\|_{2,N}+\|y(k)\|_{2,N}. \end{aligned}$$

  (2.68)

  In order to prove (2.65), consider the following equality:

  $$(|x(k)|+|y(k)|)^p=|x(k)|(|x(k)|+|y(k)|)^{p-1}+|y(k)|(|x(k)|+|y(k)|)^{p-1}. $$

  Hölder’s inequality gives

  $$\begin{aligned} &\sum_{k=1}^N|x(k)|(|x(k)|+|y(k)|)^{p-1} \\ &\le\left(\sum_{k=1}^N|x(k)|^p\right)^{1/p} \left(\sum_{k=1}^N(|x(k)|+|y(k)|)^{(p-1)q}\right)^{1/q} \end{aligned}$$

  (2.69)
  $$\begin{aligned} &\sum_{k=1}^N|y(k)|(|x(k)|+|y(k)|)^{p-1} \\ &\le\left(\sum_{k=1}^N|y(k)|^p\right)^{1/p} \left(\sum_{k=1}^N(|x(k)|+|y(k)|)^{(p-1)q}\right)^{1/q}. \end{aligned}$$

  (2.70)

  Since (p−1)q=p and 1/q=1−1/p, the addition of (2.70) and (2.69) gives

  $$\begin{aligned} &\sum_{k=1}^N(|x(k)|+|y(k)|)^p \le\left(\sum_{k=1}^N(|x(k)|+|y(k)|)^p\right)^{(1-1/p)} \\ &\cdot\left[\left(\sum_{k=1}^N|x(k)|^p\right)^{1/p}+ \left(\sum_{k=1}^N|y(k)|^p\right)^{1/p}\right]. \end{aligned}$$

  (2.71)

  Moreover, |x(k)+y(k)|≤|x(k)|+|y(k)|. Thus, Minkowski’s inequality (2.65) is obtained for N→∞.

  To provide a clear understanding, the following simple equality is considered here:

  $$(|x(k)|+|y(k)|)^2=|x(k)|(|x(k)|+|y(k)|) +|y(k)|(|x(k)|+|y(k)|). $$

  Schwarz’s inequality gives

  $$\begin{aligned} &|x(1)|(|x(1)|+|y(1)|)+|x(2)|(|x(2)|+|y(2)|)\\ &\le(|x(1)|^2+|x(2)|^2)^{1/2}[(|x(1)|+|y(1)|)^2 +(|x(2)|+|y(2)|)^2]^{1/2}\\ &|y(1)|(|x(1)|+|y(1)|)+|y(2)|(|x(2)|+|y(2)|)\\ &\le(|y(1)|^2+|y(2)|^2)^{1/2}[(|x(1)|+|y(1)|)^2 +(|x(2)|+|y(2)|)^2]^{1/2}. \end{aligned}$$

  By adding these inequalities, we have

  $$\begin{aligned} &(|x(1)|+|y(1)|)^2+(|x(2)|+|y(2)|)^2\\ &\le[(|x(1)|^2+|x(2)|^2)^{1/2}+(|y(1)|^2+|y(2)|^2)^{1/2}]\\ &~~~~~~~~\cdot[(|x(1)|+|y(1)|)^2+(|x(2)|+|y(2)|)^2]^{1/2}. \end{aligned}$$

  Thus,

  $$\sqrt{|x(1)+y(1)|^2+|x(2)+y(2)|^2}\le \sqrt{|x(1)|^2+|x(2)|^2}+\sqrt{|y(1)|^2+|y(2)|^2}, $$

  and the equality problem is proved.