Robust Stability Analysis

Abstract

The stability analysis of nonlinear discretized feedback systems is given in this chapter. The concept is based on bounded input and bounded output (BIBO) stability, that is, the small gain theorem. Although the stability analysis of nonlinear feedback systems usually uses Lyapunov methods, the Lyapunov function cannot always be determined for actual systems. There are uncertain parameters (and system orders) and also many exogenous inputs. Therefore, an analysis and design in the frequency domain based on Parseval’s identity is effective. In this book, as a design method, modified Nyquist-Hall and Nichols diagrams are proposed.

Appendices

Appendix A: Fourier-Plancherel Transform

For continuous-time signals, consider the following integral pairs:

$$\begin{aligned} &\hat{x}(j\omega)=\int_{-\infty}^\infty x(t)~\mathrm{e}^{-j\omega t}\mathrm{d}t \end{aligned}$$

(3.97)

$$\begin{aligned} &x(t)=\frac{1}{2\pi}\int_{-\infty}^\infty\hat{x}(j\omega)~\mathrm{e}^{j\omega t} \mathrm{d}\omega, \end{aligned}$$

(3.98)

where ω=2πf.¹⁰ Usually, (3.97) is called the Fourier transform, and (3.98) is called the inverse Fourier transform [5, 6]. When evaluating at s=jω (s: Laplace transform variable, in general, s=σ+jω), the preceding transforms become bilateral as follows:

$$\begin{aligned} &\hat{x}(s)=\int_{-\infty}^\infty x(t)~\mathrm{e}^{-s t}\mathrm{d}t \end{aligned}$$

(3.99)

$$\begin{aligned} &x(t)=\frac{1}{2\pi j}\int_{-j\infty}^{j\infty}\hat{x}(s)~\mathrm{e}^{s t} \mathrm{d}s. \end{aligned}$$

(3.100)

With respect to t∈[0,∞), the following transform is defined:

$$ \hat{x}(s)=\int_0^\infty x(t)~\mathrm{e}^{-s t}\mathrm{d}t. $$

(3.101)

Definition (3.101) is the (unilateral) Laplace transform, which is well known in the field of control engineering.

The value of integrations (3.97) and (3.99) exists when the following inequality holds:

$$|\hat{x}(s)|=\left|\int_{-\infty}^\infty x(t)~\mathrm{e}^{-st}\mathrm{d}t\right| \le \int_{-\infty}^\infty |x(t)|\mathrm{d}t<\infty. $$

In functional analysis, x(t) is said to belong to the L ₁ space, and is written as x(t)∈L ₁. In general, if

$$\int_{-\infty}^\infty |x(t)|^p\mathrm{d}t<\infty, $$

x(t) is said to belong to L _p and is written as x(t)∈L _p or

$$x \in L_p(\mathbb{R}), ~~~~~~\mathbb{R}:=(-\infty, \infty). $$

In the L _p space, the norm is defined as follows:

$$ \|x(t)\|_p:=\left( \int_{-\infty}^\infty |x(t)|^p\mathrm{d}t\right)^{1/p}. $$

(3.102)

Obviously, in regard to the inverse Fourier (Laplace) transform, x(s) must belong to the L ₁ space.

Plancherel’s theorem states that if x(t)∈L ₁∩L ₂, the above transformed function $\hat{x}(s)$ can also be determined similarly.

Plancherel Theorem

When $x\in L_{2}(\mathbb{R})$, the following $\hat{x}(j\omega)$ exists:

$$\begin{aligned} &\hat{x}_A(j\omega)=\int_{-A}^Ax(t)~\mathrm{e}^{-j\omega t}\mathrm{d}t\\ &\int_{-\infty}^{\infty}|\hat{x}(j\omega)-\hat{x}_A(j\omega)|^2 \mathrm{d}\omega\to 0,~~~~~~~~\mathit{for}~~A\to \infty. \end{aligned}$$

With respect to $\hat{x}(j\omega)$, the following relation holds:

$$\begin{aligned} &x_B(t)=\frac{1}{2\pi}\int_{-B}^B\hat{x}(j\omega)~ \mathrm{e}^{j\omega t}\mathrm{d}\omega\\ &\int_{-\infty}^\infty |x(t)-x_B(t)|^2\mathrm{d}t \to 0, ~~~~~~~\mathit{for}~~B\to\infty. \end{aligned}$$

Appendix B: Parseval Identity

Consider the following integral with respect to x ₁(t),x ₂(t)∈L ₁∩L ₂:

$$ I=\int_{-\infty}^\infty \mathrm{d}t x_1(t)x_2(t). $$

(3.103)

By using the inverse Fourier (Laplace) transform, x ₂(t) is given by

$$x_2(t)=\frac{1}{2\pi j}\int_{-j\infty}^{j\infty} \mathrm{d}s \hat{x}_2(s)e^{st}. $$

Substitution of this value of x ₂(t) into (3.103) yields

$$I=\int_{-\infty}^\infty\mathrm{d}t~x_1(t)\frac{1}{2\pi j}\int_{-j\infty}^{j \infty} \mathrm{d}s~\mathrm{e}^{st}\hat{x}_2(s). $$

Interchange the order of the integrations,

$$ I=\frac{1}{2\pi j}\int_{-j\infty}^{j\infty}\mathrm{d}s~ \hat{x}_2(s)\int_{-\infty}^\infty\mathrm{d}t~\mathrm{e}^{st}~x_1(t). $$

(3.104)

By applying the Fourier (two-sided Laplace) transform,

$$\hat{x}_1(s)=\int_{-\infty}^\infty \mathrm{d}t~\mathrm{e}^{-st}~x_1(t). $$

This expression can be written as

$$\hat{x}_1(-s)=\int_{-\infty}^\infty \mathrm{d}t~\mathrm{e}^{st}~x_1(t) $$

Therefore, (3.104) is expressed as

$$ I=\frac{1}{2\pi j}\int_{-j\infty}^{j\infty}\mathrm{d}s~\hat{x}_1(-s)\hat{x}_2(s). $$

(3.105)

For an easier understanding, (3.105) is rewritten for s=jω,

$$ I=\frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{x}_1(-j\omega)\hat{x}_2(j\omega)~\mathrm{d}\omega. $$

(3.106)

When x ₁(t)=x ₂(t)=x(t), (3.106) is given by

$$ I=\frac{1}{2\pi}\int_{-\infty}^\infty \hat{x}(-j\omega)\hat{x}(j\omega)~\mathrm{d}\omega =\frac{1}{2\pi}\int_{-\infty}^\infty |\hat{x}(j\omega|^2~\mathrm{d}\omega. $$

(3.107)

Then, the following equality is obtained from (3.103):

$$ \int_{-\infty}^\infty |x(t)|^2~\mathrm{d}t =\frac{1}{2\pi}\int_{-\infty}^\infty |\hat{x}(j\omega|^2~\mathrm{d}\omega. $$

(3.108)

Here, define the following L ₂ norms:

$$\begin{aligned} &\|x(t)\|_2:= \left(\int_{-\infty}^\infty|x(t)|^2\mathrm{d}t \right)^{1/2}\\ &\|\hat{x}(j\omega)\|_2:= \left(\frac{1}{2\pi}\int_{-\infty}^\infty |\hat{x}(j\omega)|^2\mathrm{d}\omega \right)^{1/2}. \end{aligned}$$

Thus, we obtain

$$\|x(t)\|_2=\|\hat{x}(j\omega)\|_2. $$

This formula is called Parseval’s identity.

On the other hand, as for discrete-time signals, consider the following summation of discrete signals:

$$ J=\sum_{k=1}^\infty x_1(k)x_2(k). $$

(3.109)

By using the inverse z-transform,

$$ x_2(k)=\frac{1}{2\pi j}\int_{|z|=1}\hat{x}_2(z)z^{k-1}\mathrm{d}z, $$

(3.110)

where z=e^h(σ+jω). Substitution of this x ₂(k) into (3.109) yields

$$J=\sum_{k=0}^\infty x_1(k)~ \frac{1}{2\pi j}\int_{|z|=1}\hat{x}_2(z)z^{k-1}\mathrm{d}z. $$

Here, for σ=0,

$$\mathrm{d}z=jh\mathrm{e}^{j\omega h}~\mathrm{d}\omega. $$

Based on the z-transform,

$$\hat{x}_1(z)=\sum_{k=0}^\infty x_1(k)~z^{-k}, $$

the following can be defined:

$$\hat{x}_1(\overline{z})=\sum_{k=1}^\infty x_1(k)~z^k. $$

Thus

$$ J=\frac{1}{2\pi j}\int_{|z|=1}\hat{x}_1(\overline{z})\hat{x}_2(z)z^{-1}\mathrm{d}z =\frac{h}{2\pi}\int_{-\pi}^\pi \hat{x}_1(\mathrm{e}^{-j\omega h})~ \hat{x}_2(\mathrm{e}^{j\omega h})~\mathrm{d}\omega. $$

(3.111)

When x ₁(k)=x ₂(k)=x(k), (3.111) is given by

$$J=\frac{h}{2\pi}\int_{-\pi}^\pi \hat{x}(\mathrm{e}^{-j\omega h})~\hat{x}(\mathrm{e}^{j\omega h})~\mathrm{d}\omega. =\frac{h}{2\pi}\int_{-\pi}^\pi |\hat{x}(\mathrm{e}^{-j\omega h})|^2~\mathrm{d}\omega. $$

Then, the following equality is obtained from (3.109):

$$ \sum_{k=0}^\infty |x(k)|^2 =\frac{h}{2\pi} \int_{-\pi}^\pi |\hat{x}(\mathrm{e}^{j\omega h})|^2 \mathrm{d}\omega. $$

(3.112)

If the norm expressions in the ℓ ₂ space,

$$\begin{aligned} &\|x(k)\|_2:=\left(\sum_{k=0}^\infty|x(k)|^2\right)^{1/2}\\ &\|\hat{x}(\mathrm{e}^{j\omega h})\|_2:= \left(\frac{h}{2\pi}\int_{-\pi}^\pi |\hat{x}(\mathrm{e}^{j\omega h})|^2\mathrm{d}\omega\right)^{1/2} \end{aligned}$$

are used, the relation

$$\|x(k)\|_2=\|\hat{x}(\mathrm{e}^{j\omega h})\|_2 $$

can be obtained. The result is called Parseval’s identity [7, 12].

Appendix C: Bilinear Transformation and Mapping

The relationship between z and δ with respect to

$$\delta=\frac{2}{h}\cdot\frac{z-1}{z+1},~~~{\mathrm{that~~is},~~~} z=\frac{1+\frac{h}{2}\delta}{1-\frac{h}{2}\delta}, $$

is as shown in Figs. 3.24(a) and (b). Furthermore, from the following equality:

$$\delta(\mathrm{e}^{j\omega h})=j\Omega(\omega) =j\frac{2}{h}\tan\left(\frac{\omega h}{2}\right), $$

the relationship between ω and Ω is illustrated in Fig. 3.25.

Fig. 3.24

Relation between z-plane and δ-plane, and contours

Fig. 3.25

Distorted frequency characteristic

Appendix D: The Hall Diagram

Consider a (unity feedback) closed-loop characteristic,

$$ W(z)=\frac{G(z)}{1+G(z)}, $$

(3.113)

as shown in Fig. 3.26. When we define the following frequency characteristic for ω<ω _c as shown in (3.32):

$$G(\mathrm{e}^{j\omega h})=U(\omega)+jV(\omega), $$

and

$$W(\mathrm{e}^{j\omega h})=M\mathrm{e}^{j\varphi}, $$

obviously,

$$M=|W(\mathrm{e}^{j\omega h})|~~~\mathrm{and}~~~ \varphi=\angle W(\mathrm{e}^{j\omega h}). $$

Therefore,

$$ M=\frac{|U+jV|}{|1+U+jV|} $$

(3.114)

and

$$ \varphi=\tan^{-1}\frac{V}{U}- \tan^{-1}\frac{V}{1+U}. $$

(3.115)

By rearranging (3.114), the following equation is obtained:

$$ \left(U+\frac{M^2}{M^2-1}\right)^2+V^2=\left(\frac{M}{M^2-1}\right)^2, ~~~\mathrm{for}~~~M\neq 1. $$

(3.116)

Here, when M=1, the equation is given as

$$ U=-\frac{1}{~2~}. $$

(3.117)

Equation (3.117) is a (purple) line in Fig. 3.27, and (3.116) (in blue) circles the right side to (3.117) for M<1 and the left side to the line for M>1.

Fig. 3.26

Discrete-time unity feedback system

Fig. 3.27

Hall diagram

On the other hand, with respect to (3.115) the following circles equation is obtained by setting N=tanφ:

$$ \left(U+\frac{1}{2}\right)^2+\left(V-\frac{1}{2N}\right)^2 =\frac{1}{4}\left(\frac{N^2+1}{N^2}\right). $$

(3.118)

Equation (3.118) also becomes circles, shown in light blue in Fig. 3.27. Such type of diagram is often called the Hall diagram [2].

The derivations of (3.116) and (3.118) from (3.114) and (3.115) are left for the reader.

Appendix E: The Nichols Diagram

Next, consider the open-loop characteristic G(e^jωh) in polar coordinates as follows:

$$ G(\mathrm{e}^{j\omega h})=\rho\cdot\mathrm{e}^{j\theta}. $$

(3.119)

Obviously, the closed-loop characteristic is given by

$$ W(\mathrm{e}^{j\omega h}) =\frac{\rho\cdot\mathrm{e}^{j\theta}}{1+\rho\cdot\mathrm{e}^{j\theta}}. $$

(3.120)

Therefore,

$$ M=|W(\mathrm{e}^{j\omega h})| =\frac{\rho}{\sqrt{(1+\rho\cos\theta)^2+\rho^2\sin^2\theta}}. $$

(3.121)

By rearranging (3.121), the following quadratic equation is obtained for M≠1:

$$ \rho^2+2\frac{M^2\cos\theta}{M^2-1}\cdot\rho+\frac{M^2}{M^2-1}=0. $$

(3.122)

Then,

$$ \rho=\frac{1}{M^2-1}\left(-M^2\cos\theta\pm M\sqrt{\cos^2\theta-(M^2-1)}\right). $$

(3.123)

Curves (3.123) are drawn in blue in Fig. 3.28. When M=1, note that ρ is simply written as follows:

$$ \rho=-\frac{1}{2\cos\theta}. $$

(3.124)

This curve is drawn in purple in the figure.

Fig. 3.28

Nichols diagram

On the other hand, for phase φ,

$$ \varphi=\angle W(\mathrm{e}^{j\omega h}) =\theta-\tan^{-1}\left(\frac{\rho\sin\theta}{1+\rho\cos\theta}\right). $$

(3.125)

From (3.125),

$$N=\tan\varphi=\frac{\sin\theta}{\cos\theta+\rho}. $$

Then,

$$ \rho=\frac{\sin\theta}{\tan\varphi}-\cos\theta. $$

(3.126)

Curves (3.125) are drawn in sky blue in the figure. Here, the derivations of (3.123), (3.124), and (3.126) from (3.121) and (3.125) are left for the reader.

The diagram shown in Fig. 3.28 is the well-known Nichols diagram [4].