19. LTI system response#

Usually the response of LTI to 3 different input are studied

19.1. LTI#

\[\begin{split}\begin{cases} \dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} \\ \mathbf{y} = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} \end{cases}\end{split}\]

with initial conditions, \(\mathbb{x}(0) = \mathbf{x}_0\).

19.2. Response of LTI systems#

The response of a LTI can be written as the sum of a free and forced response.

(19.1)#\[\begin{split}\begin{aligned} \mathbf{x}(t) & = e^{\mathbf{A} t} \mathbf{x}_0 + \int_{\tau=0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} \mathbf{u}(\tau) \, d \tau \\ \mathbf{y}(t) & = \mathbf{C} \, e^{\mathbf{A} t} \mathbf{x}_0 + \int_{\tau=0}^{t} \mathbf{C} e^{\mathbf{A}(t-\tau)} \mathbf{B} \mathbf{u}(\tau) \, d \tau + \mathbf{D} \mathbf{u}(t) \\ \end{aligned}\end{split}\]
Proof

todo

Impulse response, \( \mathbf{H}(t)\)

Functions

\[\begin{split}\begin{aligned} \mathbf{H}_x(t) & := e^{\mathbf{A} t} \mathbf{B} \\ \mathbf{H} (t) & := \mathbf{C} e^{\mathbf{A} t} \mathbf{B} + \mathbf{D} \delta(t) \\ \end{aligned}\end{split}\]

are defined impulse response functions for the state \(\mathbf{x}\) and the output \(\mathbf{y}\), as they determine with the evolution of the state and the output of a system with zero initial condition \(\mathbf{x}_0 = \mathbf{0}\) as a consequence of a impulsive force \(\mathbf{u}_{\delta}(t) = \mathbf{u} \delta(t)\). Using Lagrange formula (19.1)

(19.2)#\[\begin{split}\begin{aligned} \mathbf{x}(t) & = e^{\mathbf{A}t} \mathbf{B} \mathbf{u} && = \mathbf{H}_x(t) \mathbf{u} \\ \mathbf{y}(t) & = \left[ \mathbf{C} e^{\mathbf{A}t} \mathbf{B} + \mathbf{D} \delta(t) \right] \mathbf{u} && = \mathbf{H}(t) \mathbf{u} \\ \end{aligned}\end{split}\]

Convolution

Integral in Lagrange formula (19.1) is the convolution of the impulse response function and the input. Changing the integration variable \(z = t - \tau\), \(d\tau = - dz\), \(\tau = 0: \, z = t\), \(\tau = t, \ z = 0\) (and going back from \(z\) to \(\tau\) as the symbol for the dummy integration variable), it’s possible to swap the arguments of the impulsive force and the forcing,

(19.3)#\[\begin{split}\begin{aligned} \mathbf{x}(t) & = e^{\mathbf{A} t} \mathbf{x}_0 + \int_{\tau=0}^{t} \mathbf{H}_x(\tau) \mathbf{u}(t-\tau) \, d \tau \\ \mathbf{y}(t) & = \mathbf{C} \, e^{\mathbf{A} t} \mathbf{x}_0 + \int_{\tau=0}^{t} \mathbf{H}(\tau) \mathbf{u}(t-\tau) \, d \tau \\ \end{aligned}\end{split}\]

Linear but not time-invariant

todo The response of linear system, but not time-invariant can be written with the same expression used by Lagrange fomrula (19.1), with \(\mathbf{H}(t)\) not equal to …, but solution of the ODE…

\[\begin{split}\begin{aligned} \boldsymbol{\Phi} \left( \dot{\mathbf{x}} - \mathbf{A}\mathbf{x} \right) & = \boldsymbol{\Phi} \mathbf{B} \mathbf{u} \\ \dfrac{d}{dt} \left( \boldsymbol{\Phi} \mathbf{x} \right) & = \boldsymbol{\Phi} \mathbf{B} \mathbf{u} \ , \end{aligned}\end{split}\]

with \(\dot{\boldsymbol{\Phi}} = - \boldsymbol{\Phi} \mathbf{A}\), and proper initial conditions \(\boldsymbol{\Phi}(t,0) = \mathbf{I}\).

…

\[\mathbf{x}(t) = \boldsymbol{\Phi}(t,0) \mathbf{x}_0 + \int_{\tau=0}^{t} \boldsymbol{\Phi}(t, \tau) \mathbf{B}(\tau) \mathbf{u}(\tau) \, d \tau \ .\]

19.3. Equilibria#

Equilibrium solution of a ODE is a stationary solution, independent from time, \(\dot{\overline{\mathbf{x}}} = \mathbf{0}\). With a steady forcing \(\overline{\mathbf{u}}\),

\[\begin{split}\begin{cases} \mathbf{0} = \mathbf{A} \overline{\mathbf{x}} + \mathbf{B} \overline{\mathbf{u}} \\ \mathbf{y} = \mathbf{C} \overline{\mathbf{x}} + \mathbf{D} \overline{\mathbf{u}} \ . \end{cases}\end{split}\]

If \(\mathbf{A}\) is not singular, for every steady forcing \(\overline{\mathbf{u}}\) only one equilibrium exists for the system, \(\overline{\mathbf{x}} = \mathbf{A}^{-1} \mathbf{B} \overline{\mathbf{u}}\).

If \(\mathbf{A}\) is singular…todo rely on Linear algebra

19.4. Stability#

Asymptotic stability to initial perturbations. A system is stable under perturbation of initial conditions if the free response goes to zero as \(t \rightarrow + \infty\),

\[\lim_{t \rightarrow +\infty} \mathbf{x}_{free}(t) = \mathbf{0} \ .\]

todo for all perturbations(?).

If matrix \(\mathbf{A}\) is diagonalizable, a system is asymptotically stable if all the eigenvalues of the matrix \(\mathbf{A}\) have negative real part, \(\text{re}\{\lambda_i(\mathbf{A})\} < 0\).

todo rely on Linear algebra

19.5. Response to integrable input - and Laplace transform#

(19.4)#\[\begin{split}\begin{aligned} \mathbf{x}(s) & = \left(s \mathbf{I} - \mathbf{A} \right)^{-1} \mathbf{x}_0 + \mathbf{H}_x(s) \mathbf{u}(s) \\ \mathbf{y}(s) & = \mathbf{C} \, \left( s \mathbf{I} - \mathbf{A} \right)^{-1} \mathbf{x}_0 + \mathbf{H}(s) \mathbf{u}(s) \end{aligned}\end{split}\]

19.6. Response to periodic input of stable systems - and Fourier transform#

Forced response to periodic input of an stable system can be represented in Fourier domain replacing \(s\) with \(j \omega\) (and dropping \(j\) once it’s clear we’re dealing with Fourier analysis)

(19.5)#\[\begin{split}\begin{aligned} \mathbf{x}(\omega) & = \mathbf{H}_x(\omega) \ \mathbf{u}(\omega) \\ \mathbf{y}(\omega) & = \mathbf{H} (\omega) \ \mathbf{u}(\omega) \end{aligned}\end{split}\]

19.7. Response to stochastic input of stable systems#

19.7.1. Assumptions#

…

19.7.2. Expected value#

\[\boldsymbol{\mu}_{\mathbf{x}}(t) := \mathbb{E}[ \mathbf{x}(t) ]\]

Taking expected value of LTI equations

\[\begin{split}\begin{cases} \mathbb{E}[\dot{\mathbf{x}}] = \mathbf{A} \mathbb{E}[\mathbf{x}] + \mathbf{B} \mathbb{E}[\mathbf{u}] \\ \mathbb{E}[ \mathbf{y} ] = \mathbf{C} \mathbb{E}[\mathbf{x}] + \mathbf{D} \mathbb{E}[\mathbf{u}] \\ \end{cases}\end{split}\]

todo

  • is it possible to swap time derivative and expected value operators?

  • does time derivative of a stochastic process always exist? What’s the right way to interpret time derivative in stochastic equations? As an increment?

    \[d \mathbf{x} = \mathbf{A} \mathbf{x} \, d t + \mathbf{B} \mathbf{u} \, d t\]

    so that if \(\mathbf{u}(t) \sim \boldsymbol{\xi}(t)\), then \(\mathbf{u} \, dt \sim d \mathbf{W}\)

19.8. Analysis of response of LTI in Fourier domain#

Energy or power. For

  • time integrable signals: don’t average in time; energy

  • periodic or stochastic non-integrable signals: average in time; power

Stationary stochastic process. Strong-sense (joint probability independent from time) or wide-sense (moments up to order \(m\) are independent on time)

Ergodic process. Average over random process can be replaced with average over time.

Example of stationary non-ergodic process

As an example of stationary, but not ergodic process is a random process whose realizations are

\[X_t = \theta \ , \qquad \forall t \in [0,T]\]

with \(\theta\) drawn from a random variable \(\Theta\), with average \(\mu\). Average in time over a realization gives \(\theta\), while average over realizations gives \(\mu\) for all \(t\),

\[\begin{split}\begin{aligned} \text{Avg. in time over a realization: } & \quad \dfrac{1}{T} \int_{t=0}^{T} X_t \, dt = \dfrac{1}{T} \int_{t=0}^{T} \theta \, dt = \theta \\ \text{Avg. in probability over realizations: } & \quad \mathbb{E}[X_t] = \mathbb{E}[\Theta] = \mu \ . \end{aligned}\end{split}\]

19.8.1. Stationary Ergodic processes#

Auto-correlation

\[\mathbf{R}_{XX}(t_1, t_2) = \mathbb{E}\left[ \mathbf{X}_{t_1} \, \mathbf{X}_{t_2}^* \right]\]

Auto-covariance

\[\mathbf{K}_{XX}(t_1, t_2) = \mathbb{E}\left[ (\mathbf{X}_{t_1} - \boldsymbol{\mu}_{t_1} ) (\mathbf{X}_{t_2} - \boldsymbol{\mu}_{t_2} )^* \right]\]

For stationary processes, both auto-correlation and auto-covariance don’t depend on two time instant \(t_1\), \(t_2\) but only on the time-shift between them, \(t_2 = t_1 + \tau\),

\[\mathbf{R}_{XX}(\tau) = \mathbb{E}\left[ \mathbf{X}_{t} \, \mathbf{X}_{t + \tau}^* \right] \ , \qquad \forall t \ .\]
Autocorrelation and ESD of time integrable processes

With causal, \(\mathbf{x}(t \le 0) = 0\), and integrable processes, autocorrelation is defined as

\[\tilde{\mathbf{R}}_{xx}(\tau) = \int_{t = 0}^{+\infty} \mathbf{x}(t) \mathbf{x}^*(t+\tau) \, d t \ ,\]

Energy Spectral Density (ESD) is defined as its Fourier transform

\[\tilde{\boldsymbol{\Phi}}_{xx}(f) = \mathscr{F} \left\{ \mathbf{R}_{xx}(\tau) \right\} (f) \ ,\]
Autocorrelation and PSD of infinite-time processes

Using average over time as an approximation of the probability average

\[\mathbb{E}[ f(t) ] \sim \lim_{T \rightarrow +\infty} \dfrac{1}{T} \int_{t=0}^{T} f(\tau) \, d \tau \ ,\]

Power Spectral Density (PSD), defined as the Fourier transform of the auto-correlation, reads

\[\begin{split}\begin{aligned} \boldsymbol{\Phi}_{YY}(f) & = \mathscr{F}[ \mathbf{R}_{YY}(\tau) ](f) = \\ & = \int_{\tau = -\infty}^{+\infty} \mathbb{E} \left[ \mathbf{y}(t) \mathbf{y}^*(t+\tau) \right] \, e^{-i 2 \pi f \tau} \, d \tau \ . \end{aligned}\end{split}\]

For asymptotically stable systems subject to periodic forcing, response to non-homogenoeus initial conditions vanishes while the forced response can be written either in Fourier or in time domain, exploiting the property of the transform of convolution Property 15.1, as

\[\begin{aligned} \mathbf{y}(f) = \mathbf{H}(f) \, \mathbf{u}(f) \qquad , \qquad \mathbf{y}(t) = \mathbf{H} \ast \mathbf{u}(t) = \int_{\tau=-\infty}^{+\infty} \mathbf{H}(\tau) \mathbf{u}(t - \tau) \, d \tau \ . \end{aligned}\]

Inserting harmonic response into the expression of PSD, it becomes

\[\begin{split}\begin{aligned} \boldsymbol{\Phi}_{YY}(f) & = \int_{\tau = -\infty}^{+\infty} \mathbb{E} \left[ \left( \int_{\xi=-\infty}^{+\infty} \mathbf{H}(\xi) \, \mathbf{u}(t - \xi) \, d\xi \right) \left( \int_{\chi=-\infty}^{+\infty} \mathbf{H}(\chi) \, \mathbf{u}(t+\tau-\chi) \, d \chi \right)^* \right] \, e^{-i 2 \pi f \tau} \, d \tau = && (1) \\ & = \int_{\tau = -\infty}^{+\infty} \, \int_{\xi=-\infty}^{+\infty} \, \int_{\chi=-\infty}^{+\infty} \mathbf{H}(\xi) \mathbb{E} \left[ \mathbf{u}(t - \xi) \mathbf{u}^*(t+\tau-\chi) \right] \mathbf{H}^*(\chi) \, e^{-i 2 \pi f \tau} \, d \tau \, d \chi \, d \xi = && (2) \\ & = \int_{\tau = -\infty}^{+\infty} \, \int_{\xi=-\infty}^{+\infty} \, \int_{\chi=-\infty}^{+\infty} \mathbf{H}(\xi) \mathbb{E} \left[ \mathbf{u}(t ) \mathbf{u}^*(t+\tau-\chi+\xi) \right] \mathbf{H}^*(\chi) \, e^{-i 2 \pi f \tau} \, d \tau \, d \chi \, d \xi = && (3) \\ & = \int_{z = -\infty}^{+\infty} \, \int_{\xi=-\infty}^{+\infty} \, \int_{\chi=-\infty}^{+\infty} \mathbf{H}(\xi) \mathbb{E} \left[ \mathbf{u}(t ) \mathbf{u}^*(t+z) \right] \mathbf{H}^*(\chi) \, e^{-i 2 \pi f \left( z - \xi + \chi \right)} \, d z \, d \chi \, d \xi = \\ & = \int_{\xi = -\infty}^{+\infty} \mathbf{H}(\xi) \, e^{i 2 \pi f \xi} \, d \xi \, \int_{z=-\infty}^{+\infty} \mathbf{R}_{uu}(z) \, e^{-i 2 \pi f z } \, dz \, \int_{\chi=-\infty}^{+\infty} \mathbf{H}^*(\chi) \, e^{-i 2 \pi f \chi } \, d \chi = (4) \\ & = \overline{\mathbf{H}}(f) \, \boldsymbol{\Phi}_{uu}(f) \, \mathbf{H}^T(f) \ . \end{aligned}\end{split}\]

having used \((1)\) the fact that the impulse response function is deterministic, \((2)\) time shift in the autocorrelation for a stationary random process \(\mathbf{u}(t)\) as an input, and \((3)\) the change of coordinates \((z, \xi, \chi) = (\tau-\chi+\xi, \xi, \chi)\), and \((4)\) a slight notation abuse ndicating Fourier transform with the same symbols of the functions in time domain.