I/O Behavior

I/O Behavior

👤 Sébastien Boisgérault

Control Engineering with Python

Symbols

🐍 Code 🔍 Worked Example
📈 Graph 🧩 Exercise
🏷️ Definition 💻 Numerical Method
💎 Theorem 🧮 Analytical Method
📝 Remark 🧠 Theory
ℹ️ Information 🗝️ Hint
⚠️ Warning 🔓 Solution

🐍 Imports

from numpy import *
from numpy.linalg import *
from scipy.linalg import *
from matplotlib.pyplot import *
from mpl_toolkits.mplot3d import *
from scipy.integrate import solve_ivp

🐍 Streamplot Helper

def Q(f, xs, ys):
    X, Y = meshgrid(xs, ys)
    v = vectorize
    fx = v(lambda x, y: f([x, y])[0])
    fy = v(lambda x, y: f([x, y])[1])
    return X, Y, fx(X, Y), fy(X, Y)

🧭 Context

  1. System initially at rest. \(x(0) = 0.\)

  2. Black box. The system state \(x(t)\) is unknown.

  3. Input/Output (I/O). The input determines the output:

    \[ u(t), \, t\geq 0 \; \to \; y(t), \, t\geq 0. \]

The variation of constants method yields

\[ y(t) = \int_0^{t} C e^{A(t-\tau)} B u(\tau) \, d\tau + D u(t). \]

🏷️ Signals & Causality

A signal is a time-dependent function

\[ x(t) \in \mathbb{R}^n, \; t \in \mathbb{R}. \]

It is causal if

\[ t< 0 \; \Rightarrow \; x(t) = 0. \]

📝 Convention

In the sequel, we will assume that time-dependent functions defined only for non-negative times

\[ x(t), \, t \geq 0 \]

are zero for negative times

\[ x(t) = 0, \; t < 0. \]

With this convention, they become causal signals.

🏷️ Heaviside function

The Heaviside function is the causal signal defined by

\[ e(t) = \left| \begin{array}{c} 1 & \mbox{if } \; t\geq 0, \\ 0 & \mbox{if } \; t < 0. \end{array} \right. \]

🏷️ Synonym: (unit) step signal.

🏷️ Impulse Response

The system impulse response is defined by:

\[ H(t) = (C e^{At} B) \times e(t) + D \delta(t) \in \mathbb{R}^{p \times m} \]

📝 Notes

  • the formula is valid for general (MIMO) systems.

    🏷️ MIMO = multiple-input & multiple-output.

  • \(\delta(t)\) is the unit impulse signal, we’ll get back to it (in the meantime, you may assume that \(D=0\)).

📝 SISO Systems

When \(u(t) \in \mathbb{R}\) and \(y(t) \in \mathbb{R}\) the system is SISO.

🏷️ SISO = single-input & single-output.

Then \(H(t)\) is a \(1 \times 1\) matrix.

We identify it with its unique coefficient \(h(t)\):

\[ H(t) \in \mathbb{R}^{1\times 1} = [h(t)], \; h(t) \in \mathbb{R}. \]

💎 I/O Behavior

Let \(u(t)\), \(x(t)\), \(y(t)\) be causal signals such that:

\[ \left| \begin{array}{rcl} \dot{x}(t) &=& A x(t) + B u(t) \\ y(t) &=& C x(t) + Du(t) \end{array} \right., \, t\geq 0 \; \mbox{ and } \; x(0) = 0. \]

Then

\[ y(t) = (H \ast u) (t) := \int_{-\infty}^{+\infty} H(t - \tau) u(\tau) \, d\tau . \]

🏷️ Convolution

The operation \(\ast\) is called a convolution.

🔍 Impulse Response

Consider the SISO system

\[ \left| \begin{array}{ccl} \dot{x} &=& ax + u \\ y &=& x \\ \end{array} \right. \]

where \(a \neq 0\).

We have

\[ \begin{split} H(t) &= (C e^{At} B) \times e(t) + D \delta(t)\\ &= [1]e^{[a]t} [1] e(t) + [0] \delta(t) \\ &= [e(t)e^{at}] \end{split} \]

When \(u(t) = e(t)\) for example,

\[ \begin{split} y(t) &= \int_{-\infty}^{+\infty} e(t - \tau)e^{a(t-\tau)} e(\tau) \, d\tau \\ &= \int_{0}^{t} e^{a(t- \tau)} \, d\tau \\ &= \int_{0}^{t} e^{a \tau} \, d\tau \\ &= \frac{1}{a} \left(e^{a t} - 1 \right) \end{split} \]

🧩 Integrator

Let

\[ \left| \begin{array}{ccc} \dot{x} &=& u \\ y &=& x \\ \end{array} \right. \]

where \(u \in \mathbb{R}\), \(x \in \mathbb{R}\) and \(y \in \mathbb{R}\).

1. 🧮

Compute the impulse response of the system.

🔓 Integrator

1. 🔓

\[ \begin{split} H(t) &= (C e^{At} B) \times e(t) + D \delta(t)\\ &= [1]e^{[0]t} [1] e(t) + [0] \delta(t) \\ &= [e(t)] \end{split} \]

🧩 Double Integrator

Let

\[ \left| \begin{array}{ccc} \dot{x}_1 &=& x_2 \\ \dot{x}_2 &=& u \\ y &=& x_1 \\ \end{array} \right. \]

where \(u \in \mathbb{R}\), \(x=(x_1, x_2) \in \mathbb{R}^2\) and \(y \in \mathbb{R}\).

1. 🧮

Compute the impulse response of the system.

🔓 Double Integrator

1. 🔓

\[ \begin{split} H(t) &= (C \exp(At) B) \times e(t) + D \delta(t)\\ &= \displaystyle \left[\begin{array}{cc} 1 & 0 \end{array}\right] \exp \left( \left[\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right] t \right) \left[\begin{array}{c} 0 \\ 1 \end{array}\right] e(t) + [0] \delta(t) \\ &= \displaystyle \left[\begin{array}{cc} 1 & 0 \end{array}\right] \left[\begin{array}{cc} 1 & t \\ 0 & 1 \end{array}\right] \left[\begin{array}{c} 0 \\ 1 \end{array}\right] e(t) \\ &= [t e(t)] \end{split} \]

🧩 Gain

Let

\[ y = K u \]

where \(u \in \mathbb{R}^m\), \(y \in \mathbb{R}^p\) and \(K \in \mathbb{R}^{p \times m}\).

1. 🧮

Compute the impulse response of the system.

🔓 Gain

1. 🔓

The I/O behavior can be represented by \(\dot{x} = 0x+0u\) and \(y= 0 \times x + K u\) (for example). Thus,

\[ \begin{split} H(t) &= (C \exp(At) B) \times e(t) + D \delta(t)\\ &= 0 + K \delta(t)\\ &= K \delta(t) \end{split} \]

🧩 MIMO System

Let \[ H(t) := \left[ \begin{array}{cc} e^{t} e(t) & e^{-t} e(t) \end{array} \right] \]

1. 🧮

Find a linear system with matrices \(A\), \(B\), \(C\), \(D\) whose impulse response is \(H(t)\).

2. 🧮

Is there another 4-uple of matrices \(A\), \(B\), \(C\), \(D\) with the same impulse response?

3. 🧮

Same question but with a matrix \(A\) of a different size?

🔓 MIMO System

1. 🔓

Since

\[ \exp \left( \left[ \begin{array}{rr} +1 & 0 \\ 0 & -1 \end{array} \right] t \right) = \left[ \begin{array}{rr} e^{+t} & 0 \\ 0 & e^{-t} \end{array} \right], \]

the following matrices work:

\[ A = \left[ \begin{array}{rr} +1 & 0 \\ 0 & -1 \end{array} \right], \; B = \left[ \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right], \; C= \left[ \begin{array}{cc} 1 & 1 \\ \end{array} \right], \; D = \left[ \begin{array}{cc} 0 & 0 \\ \end{array} \right]. \]

2. 🔓

Since \[ \begin{split} H(t) &= (C \exp(At) B) \times e(t) + D \delta(t)\\ &= ((-C) \exp(At) (-B)) \times e(t) + D \delta(t) \end{split} \] changing \(B\) and \(C\) to be \[ B = \left[ \begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array} \right], \; C= \left[ \begin{array}{rr} -1 & -1 \\ \end{array} \right], \; \] doesn’t change the impulse response.

3. 🔓

We can also easily add a scalar dynamics (say \(\dot{x}_3 = 0\)) that doesn’t influence the impulse response.

The following matrices also work

\[ A = \left[ \begin{array}{rrr} +1 & 0 & 0\\ 0 & -1 & 0 \\ 0 & 0 & 0 \end{array} \right], \; B = \left[ \begin{array}{cc} 1 & 0\\ 0 & 1\\ 0 & 0 \end{array} \right], \]

\[ C= \left[ \begin{array}{cc} 1 & 1 & 0 \\ \end{array} \right], \; D = \left[ \begin{array}{cc} 0 & 0\\ \end{array} \right]. \]

🏷️ Laplace Transform

Let \(x(t)\), \(t\in\mathbb{R}\) be a scalar signal.

It Laplace transform is the function of \(s\) given by:

\[ x(s) = \int_{-\infty}^{+\infty} x(t) e^{-st} \, dt. \]

Domain & Codomain

The Laplace transform of a signal is a complex-valued function; its domain is a subset of the complex plane.

\[ s \in D \, \Rightarrow \, x(s) \in \mathbb{C}. \]

If \(x(t)\) is a causal signal of sub-exponential growth

\[ |x(t)| \leq k e^{\sigma t} e(t), \, t \in \mathbb{R}, \]

(\(k \geq 0\) and \(\sigma \in \mathbb{R}\)), its Laplace transform is defined on an open half-plane:

\[ \Re (s) > \sigma \; \Rightarrow \; x(s) \in \mathbb{C}. \]

⚠️ Notation

We use the same symbol (here “\(x\)”) to denote:

  • a signal \(x(t)\) and

  • its Laplace transform \(x(s)\)

They are two equivalent representations of the same “object”, but different mathematical “functions”.

If you fear some ambiguity, use named variables, e.g.:

\[ x(t=1) \, \mbox{ or } \, x(s=1) \, \mbox{ instead of } \, x(1). \]

Vector/Matrix-Valued Signals

The Laplace transform

  • of a vector-valued signal \(x(t) \in \mathbb{R}^n\) or

  • of a matrix-valued signal \(X(t) \in \mathbb{R}^{m \times n}\)

are computed elementwise.

\[ x_{i}(s) := \int_{-\infty}^{+\infty} x_{i}(t) e^{-st} \, dt. \]

\[ X_{ij}(s) := \int_{-\infty}^{+\infty} X_{ij}(t) e^{-st} \, dt. \]

🏷️ Rational Signals

We will only deal with rational (and causal) signals:

\[ x(t) = \left(\sum_{\lambda \in \Lambda} p_{\lambda}(t) e^{\lambda t} \right) e(t) \]

where:

  • \(\Lambda\) is a finite subset of \(\mathbb{C}\),

  • for every \(\lambda \in \Lambda\), \(p_{\lambda}(t)\) is a polynomial in \(t\).

📝

They are called rational since

\[ x(s) = \frac{n(s)}{d(s)} \]

where \(n(s)\) and \(d(s)\) are polynomials; also

\[ \deg n(s) \leq \deg d(s). \]

🔍 Exponential

Let

\[ x(t) = e^{a t} e(t), \; t\in \mathbb{R} \] for some \(a \in \mathbb{R}\). Then

\[ \begin{split} x(s) &= \int_{-\infty}^{+\infty} e^{at} e(t) e^{-s t} \, dt = \int_0^{+\infty} e^{(a-s) t} \, dt. \\ \end{split} \]

If \(\Re(s) > a\), then \[ \left|e^{(a-s) t}\right| \leq e^{-(\Re (s) -a) t}; \] the function \(t \in \left[0, +\infty\right[ \mapsto e^{(a-s) t}\) is integrable and

\[ x(s) = \left[\frac{e^{(a-s) t}}{a-s} \right]^{+\infty}_0 = \frac{1}{s-a}. \]

💻 Symbolic Computation

import sympy
from sympy.abc import t, s
from sympy.integrals.transforms \
    import laplace_transform    

def L(f):
    return laplace_transform(f, t, s)[0]

>>> from sympy.abc import a
>>> xt = sympy.exp(a*t)
>>> xs = L(xt)
>>> xs
1/(-a + s)

🧩 Ramp

Let

\[ x(t) = t e(t), \; t\in\mathbb{R}. \]

1. 🧮

Compute analytically the Laplace Transform of \(x(t)\).

2. 💻

Compute symbolically the Laplace Transform of \(x(t)\).

🔓 Ramp

1. 🔓

\[ \begin{split} x(s) &= \int_{-\infty}^{+\infty} t e(t) e^{-s t} \, dt \\ &= \int_0^{+\infty} t e^{-s t} \, dt. \\ \end{split} \]

By integration by parts, \[ \begin{split} x(s) &= \left[t\frac{e^{-st}}{-s} \right]^{+\infty}_0 - \int_0^{+\infty} \frac{e^{-s t}}{-s} \, dt \\ &= \frac{1}{s} \int_0^{+\infty} e^{-s t} \, dt \\ &= \frac{1}{s} \left[\frac{e^{-st}}{-s} \right]^{+\infty}_0 \\ &= \frac{1}{s^2} \end{split} \]

2. 🔓

With SymPy, we have accordingly:

>>> xt = t
>>> xs = L(xt)
>>> xs
s**(-2)

🏷️ Transfer Function

Let \(H(t)\) be the impulse response of a system.

Its Laplace transform \(H(s)\) is the system transfer function.

💎

For LTI systems in standard form,

\[ H(s) = C [sI - A]^{-1} B + D. \]

💎 Operational Calculus

\[ y(t) = (H \ast u)(t) \; \Longleftrightarrow \; y(s) = H(s) \times u(s) \]

Graphical Language

Control engineers used block diagrams to describe (combinations of) dynamical systems, with

  • “boxes” to determine the relation between input signals and output signals and

  • “wires” to route output signals to inputs signals.

Feedback Block-Diagram

  • Triangles denote gains (scalar or matrix multipliers),

  • Adders sum (or substract) signals.

  • LTI systems can be specified by:

    • (differential) equations,

    • the impulse response,

    • the transfer function.

Equivalent Systems

🧩 Feedback Block-Diagram

Consider the system depicted in the Feedback Block-Diagram picture.

1. 🧮

Compute its transfer function.

🔓 Feedback Block-Diagram

1. 🔓

The diagram logic translates into:

\[ y(s) = \frac{1}{s} \left(u(s) - k y(s)\right), \]

and thus

\[ \left(1 - \frac{k}{s}\right) y(s) =\frac{1}{s} u(s) \]

or equivalently

\[ y(s) = \frac{1}{s- k} u(s). \]

Thus, the transfer function of this SISO system is

\[ h(s) = \frac{1}{s- k}. \]

🤔 Impulse Response

Why refer to \(h(t)\) as the system “impulse response”?

By the way, what’s an impulse?

Impulse Approximations

Pick a time constant \(\varepsilon > 0\) and define

\[ \delta_{\varepsilon}(t) := \frac{1}{\varepsilon} e^{-t/\varepsilon} e(t). \]

🐍

def delta(t, eps):
    return exp(-t / eps) / eps * (t >= 0)

📈

figure()
t = linspace(-1, 4, 1000)
for eps in [1.0, 0.5, 0.25]:
    plot(t, delta(t, eps), 
         label=rf"$\varepsilon={eps}$")
xlabel("$t$"); title(r"$\delta_{\varepsilon}(t)$") 
legend()

In the Laplace Domain

\[ \begin{split} \delta_{\varepsilon}(s) &= \int_{-\infty}^{+\infty} \delta_{\varepsilon}(t) e^{-st} \, dt \\ &= \frac{1}{\varepsilon} \int_{0}^{+\infty} e^{-(s + 1/\varepsilon)t} \, dt \\ &= \frac{1}{\varepsilon} \left[ \frac{e^{-(s+1/\varepsilon)t}}{-(s+1/\varepsilon)} \right]^{+\infty}_0 = \frac{1}{1 + \varepsilon s}\\ \end{split} \]

(assuming that \(\Re(s) > -1/\varepsilon\))

  • The “limit” of the signal \(\delta_{\varepsilon}(t)\) when \(\varepsilon \to 0\) is not defined as a function (issue for \(t=0\)) but as a generalized function \(\delta(t)\), the unit impulse.

  • This technicality can be avoided in the Laplace domain where \[ \delta(s) = \lim_{\varepsilon \to 0} \delta_{\varepsilon}(s) = \lim_{\varepsilon \to 0} \frac{1}{1 + \varepsilon s} = 1. \]

Thus, if \(y(t) = (h \ast u)(t)\) and

  1. \(u(t) = \delta(t)\) then

  2. \(y(s) = h(s) \times \delta(s) = h(s) \times 1 = h(s)\)

  3. and thus \(y(t) = h(t)\).

Conclusion: the impulse response \(h(t)\) is the output of the system when the input is the unit impulse \(\delta(t)\).

🏷️ I/O Stability

A system is I/O-stable if there is a \(K \geq 0\) such that

\[ \|u(t)\| \leq M, \, t\geq 0 \]

\[ \Rightarrow \]

\[ \|y(t)\| \leq K M, \, t\geq 0. \]

🏷️ More precisely, BIBO-stability (“bounded input, bounded output”).

🏷️ Transfer Function Poles

A pole of the transfer function \(H(s)\) is a \(s \in \mathbb{C}\) such that for at least one element \(H_{ij}(s)\),

\[ |H_{ij}(s)| = +\infty. \]

💎 I/O-Stability Criteria

A system is I/O-stable if and only if all its poles are in the open left-plane, i.e. such that

\[ \Re (s)< 0. \]

💎 Internal Stability \(\Rightarrow\) I/O-Stability

If the system \(\dot{x} = A x\) is asymptotically stable, then for any matrices \(B\), \(C\), \(D\) of compatible shapes,

\[ \begin{split} \dot{x} &= A x + B u \\ y &= C x + Du \end{split} \]

is I/O-stable.

🔍 Fully Actuated & Measured System

If \(B=I\), \(C=I\) and \(D=0\), that is

\[ \dot{x} = A x +u, \; y = x \]

then \(H(s) = [sI-A]^{-1}\).

Therefore, \(s\) is a pole of \(H\) iff it’s an eigenvalue of \(A\).

Thus, in this case, asymptotic stability and I/O-stability are equivalent.

(This equivalence actually holds under much weaker conditions.)