Linear Dynamical Systems

Discrete-time Control Systems:

• Model of a (discrete-time) dynamical system: $x(t+1)=f(x(t),u(t),t),y(t)=g(x(t),u(t),t)$ where:
  • $\nabla x(t)\in {\mathbb{R}}^n$ is the “state” at time $t$;
  • $u(t)\in {\mathbb{R}}^p$ is the “control input”;
  • $y(t)\in {\mathbb{R}}^q$ is the “measured output”;
  • $f,g$ are given maps.
• If $f,g$ do not depend explicitely on time, the system is said to be time-invariant.
• ”Continuous-time” version involves differential equations; can be approximated by the above via “discretization”

Linear models

• It is often convenient to approximate the model via linearization: $x(t+1)=A(t)x(t)+B(t)u(t),y(t)=C(t)x(t)+D(t)u(t)$ where $A(t),B(t),C(t),D(t)$ are matrices obtained by first-order Taylor expansion of the maps $f,g$.
• Linear, time-invariant (LTI) system:
  • correspond to the case when $A,B,C,D$ are constant: $x(t+1)=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t)$
  • Given an initial state $x(0)$, and control input $u(t),t=0,\dots ,T-1$ we can express the state at time $T\ge 1$ as a linear map:
    • $x(T)=A^{T}x(0)+{\sum }_{t=0}^{T-1}{A}^{T-t-1}Bu(t)$
    • where $A^k$ is the matrix $A$ multiplied by itself $k$ times.
  • Proof: we check that $x(T+1)=Ax(T)+Bu(T)$.

Applications:

Population dynamics

• Yearly population dynamics modeled via $x(t+1)\approx Ax(t)$ where
  • $A=\left(\begin{array}{cccc}{b}_1& b_2& \cdots & b_{99}\\ 1-d_1& 0& \cdots & 0\\ 0& 1-d_2& \cdots & 0\\ ⋮& \ddots & & \\ 0& 0& \cdots & 1-d_{99}\end{array}\right)$
• and:
  • state vector $x(t)\in {\mathbb{R}}^{100}$, with $x_i(t)$ the population with age $i-1$ at time $t$;
  • $b_i$ is is the average number of births per person with age $i-1$;
  • $d_i$ is the portion of those aged $i-1$ who will die this year;
  • the model assumes a maximum life span of 100 years.
  • A control input can be used to model interventions (say, public health decisions)
• Time-invariance means that we assume the birth and death rates to be constant over time; the model also assumes that there is no immigration or other “inputs”.
• In the previous model, we now assume that, at any point in time, there is an influx $u(t)$ of immigrants in the age group $25-26$. Can the situation be modeled via $x(t+1)\approx Ax(t)+bu(t)$ where $b$ is a vector?
  1. Yes, with $b$ all ones, except zero at index 25 .
  2. Yes, with $b$ all zeroes, except one at index 25 .
  3. Yes, with $b$ all zeroes, except one at index 26 .

Supply chain models:

• Consider a simple supply network for a single commodity, stored at n warehouses.
  • Each location has a target (desired) level or amount of the commodity.
  • Commodity is transported over m transportation links, and also enters and exits the nodes through purchases (from suppliers) and sales (to end-users).
• Supply network dynamics
  • x(t) is n-vector of deviations of commodity levels from target.
  • Commodity flow m-vector is $f(t)$.
  • Purchases and sales are recorded in $n$-vectors $p(t)$, $s(t)$
  • Then: $x(t+1)=x(t)+Af(t)+p(t)-s(t)$ where $A$ is the $n\times m$ incidence matrix of the network.

Continuous-time control system:

• Model of a (continuous-time) dynamical system: $\frac{d}{dt}x(t)=f(x(t),u(t),t),y(t)=g(x(t),u(t),t)$ where
  • $\nabla x(t)\in {\mathbb{R}}^n$ is the “state” at time $t$;
  • $u(t)\in {\mathbb{R}}^p$ is the “control input”;
  • $y(t)\in {\mathbb{R}}^q$ is the “measured output”;
  • $f,g$ are given maps.
• The continuous-time aspect often originates from physical laws, e.g. equations of motion $(f=ma)$.
• example: Cart on rail:
  • Consider a cart of mass $m$, moving along a horizontal rail, where it is subject to viscous damping (a damping which is proportional to the velocity) of coefficient $\beta$.
  • We let $p(t)$ denote the position of the center of mass of the cart, and $u(t)$ the force applied to the center of mass, where $t$ is the (continuous) time.
  • The Newton dynamic equilibrium law then prescribes that $u(t)-\beta \dot{p}(t)=m\ddot{p}(t)$
    • which is the second-order differential equation governing the dynamics of this system.
  • If we introduce variables (states) $x_1(t)=p(t),x_2(t)=\dot{p}(t)$, we may rewrite the Newton equation in the form of a system of two coupled differential equations of first order: $\begin{array}{rl}& {\dot{x}}_1(t)=x_2(t),\\ & {\dot{x}}_2(t)=\alpha x_2(t)+bu(t),\end{array}$
    • where we defined $\alpha \doteq -\frac{\beta }{m},b\doteq \frac{1}{m}$
  • The system can then be recast in compact matrix form as $\dot{x}(t)=A_{c}x(t)+B_{c}u(t)$ where $A_c=\left[\begin{array}{ll}0& 1\\ 0& \alpha \end{array}\right],B_c=\left[\begin{array}{l}0\\ b\end{array}\right]$
  • With this system we can also associate an output equation, representing a signal $y$ that is of some particular interest, e.g., the position of the cart itself: $y(t)=Cx(t),C=\left[\begin{array}{ll}1& 0\end{array}\right]\text{. }$
  • The model is a so-called continuous-time linear time-invariant (LTI) system, of the form $\dot{x}(t)=A_{c}x(t)+B_{c}u(t),y(t)=Cx(t)$
• Lagrange formula
  • Given the value of the state at some instant $t_0$, and given the input $u(t)$ for $t\ge t_0$, there exists an explicit formula (usually known as Lagrange’s formula) for expressing the evolution in time of the state of the system: $x(t)={\mathrm{e}}^{A\left(t-t_0\right)}x\left(t_0\right)+{\int }_{t_0}^t{\mathrm{e}}^{A(t-\tau )}Bu(\tau )\mathrm{d}\tau ,t\ge t_0,$
  • where, for a square matrix $M$, we define the matrix exponential as $e^M\doteq \sum _{k=1}^{\infty }\frac{1}{k!}{M}^k\approx I+M\text{. }$
• Discretization

  • Often, a continuous-time systems must be analyzed and controlled by means of digital devices, such DSP and computers. It is therefore common to “convert” a continuous-time system into its discrete-time version, by “taking snapshots” of the system at time instants $t=k\Delta$, where $\Delta$ is referred to as the sampling interval, assuming the the input signal $u(t)$ remains constant between two successive sampling instants, that is $u(t)=u(k\Delta ),\forall t\in [k\Delta ,(k+1)\Delta ).$

  • This discrete-time conversion can be done as follows: given the state of the continuous system (1) at time $t=k\Delta$ (which we denote by $x(k)=x(k\Delta )$ ), we can use equation (2) to compute the value of the state at instant $(k+1)\Delta$ : $x(k+1)={\mathrm{e}}^{A\Delta }x(k)+{\int }_0^{\Delta }{\mathrm{e}}^{A\tau }B\mathrm{d}\tau u(k)$

  • The sampled version of the continuous-time system (1) evolves according to a discrete-time recursion of the form $\begin{array}{rl}x(k+1)& =A_{\Delta }x(k)+B_{\Delta }u(k),\\ y(k)& =Cx(k),\\ A_{\Delta }={\mathrm{e}}^{A\Delta },& B_{\Delta }={\int }_0^{\Delta }{\mathrm{e}}^{A\tau }B\mathrm{d}\tau .\end{array}$

  • For the cart example, with $m=1\mathrm{Kg},\beta =0.1\mathrm{Ns}/\mathrm{m}$, and $\Delta =0.1\mathrm{s}$, we obtain a discrete-time system

$$\begin{array}{c}x(k+1)=Ax(k)+Bu(k),\\ y(k)=Cx(k)\\ A={\mathrm{e}}^{A_c\Delta }=\left[\begin{array}{cc}1& \frac{1}{\alpha }\left({\mathrm{e}}^{\alpha \Delta }-1\right)\\ 0& {\mathrm{e}}^{\alpha \Delta }\end{array}\right]=\left[\begin{array}{ll}1& 0.0995017\\ 0& 0.9900498\end{array}\right]\\ B=\frac{b}{\alpha }\left[\begin{array}{c}\frac{1}{\alpha }\left({\mathrm{e}}^{\alpha \Delta }-1\right)-\Delta \\ {\mathrm{e}}^{\alpha \Delta }-1\end{array}\right]=\left[\begin{array}{l}0.0049834\\ 0.0995016\end{array}\right].\end{array}$$