Mathematical Basics of the System Theory PART III

Prac 8. Linearization of Plants and System Models

Examples about finding quilibrium position

Good old controllability and observability

Assignment discussion

Representation of the derivative by the ratio of finite small increments

The recurrent notaion

$x(k+1)=(I+A\triangle t)x(k)+(B\triangle t)u(k),\ y(k)=Cx(k).$

Transfer function “input - output”

$W(s)=\frac{b_1 s+b_0}{(a_4 s+a_3)(a_2 s^2+a_1 s+a_0)}$

Assume that after substitution values of parameters we get:

$W(s,q)=\frac{\frac{1}{4}s+\frac{1}{16}}{s^2+\frac{3}{4}s+\frac{1}{4}}\\ A =\left[\begin{array}{c,c}0&1\\-0.25&-0.75\end{array}\right],B=\left[\begin{array}{c}0\\1\end{array}\right],C=\left[\begin{array}{c,c}0.0625&0.25\end{array}\right]$

Write matrices $\bar{A},\bar{B},\bar{C}$ for the value of the sample time interval $\triangle t = 0.05s $

$\bar{A}\approx I+A\triangle t\\ \bar{B}\approx B\triangle t\\ \bar{C}=C$

Integral model of the continuous-time plant in the “input-output” form

$e^{A\triangle t}=\sum_{i=1}^\infty \frac{1}{i!}(A\triangle t)^i$

Confine to the first k members as you wish.

$\bar{A} =e^{A\triangle t}\\ \bar{B}=(e^{A\triangle t}-I)A^{-1}B\\ \bar{C}=C$

Lec 11. Controllability and Observability

Purpose of control: we need a plant to behave the way to achieve some goal or perform some function

In order to control a plant (a device or a process) we create a system which includes control and measuring devices.

A control system is an interconnection of components forming a system configuration that will provide a desired system response.

Stability

Definition (stability). Stability is a system property, in the presence of which the initial deviation of the dynamic process from its desired course decreases over time.

Definition (stability condition). A system, regardless of its continuous or discrete nature, will be stable provided that the unforced response of the state vector and the output will decrease over time, and in the case of asymptotic stability tend to zero.

Controllability

Consider the problem of existence of a control that can transfer the system from the initial state $x_0(t)=x(t_0)$ to any other desired location $x_f(t)=x(t_f)$ in a finite time.

Observability

Observability refers to the problem of the state vector estimation. It means that we need to determine if it is possible to find the state vector at the time $t_0$ if the control $u(t)$ and output $y(t)$ , $t\in[t_0,t_1],\ t_1>t_0$ are measured.

Prac 9. Autonomous system, Phase portraits

Consider the system

$\dot{x}(t)=Ax(t)+bu(t)$

Find the solution of equation written as

$\dot{x}(t)-Ax(t)=bu(t)$

To do this, multiply both parts by the matrix exponent $e^{-At}$

$e^{-At}\dot{x}(t)-Ae^{-At}x(t)=e^{-At}bu(t)\\ \frac{d}{dt}(e^{-At}x(t))=-Ae^{-At}x(t)+e^{-At}\dot{x}(t)\\=e^{-At}bu(t)$

Integrate

$e^{-At}x(t)=x(0)+\int_0^t e^{-A\tau}bu(\tau)d\tau\\ x(t)=e^{At}x(0)+e^{At}\int_0^t e^{-A\tau}bu(\tau)d\tau$

$e^{At}x(0)$ - unforced (free) component of the movement depends only on the initial conditions

And the rest, it the forced component.

Algo for constructing a phase portrait

Find the eigenvalues of the state matrix A by solving the auxiliary equation
Determine the type of the equilibrium point and the character of stability.
Find the equations of the isoclines:

Vertical: $\frac{dx_1}{dt}=a_{11}x_1+a_{12}x_2$

Horizontal: $\frac{dx_x}{dt}=a_{21}x_1+a_{22}x_2$

If the equilibrium position is a node or a saddle, it is necessary to compute the eigenvectors and draw the asymptotes parallel to the eigenvectors and passing through the origin.
Schematically draw the phase portrait. Show the direction of motion along the phase trajectories (this depends on the stability or instability of the equilibrium point).