0.1 Lecture 2: introduction to systems  (Page 2/8)

3/ Conclusion

A system is described structurally by specifying:

• the system topology,
• the rules of interconnection of the elements,
• functional descriptions of the elements — constitutive relations.

III. DYNAMIC ANALOGIES

Physically divergent systems can have similar dynamic properties.

1/ Mechanical free-body diagram

M = mass, B = friction constant,

K = spring constant,

f(t) = external force, and

v(t) = velocity of the mass.

Summing the forces yields

$f(t)=M\frac{\text{dv}(t)}{\text{dt}}+\text{Bv}(t)+K{\int }_{-\infty }^{t}v(\tau )\mathrm{d\tau }$

2/ Electric network

Summing the currents (Kirchhoff’s current law) yields

$i(t)=C\frac{\text{dv}(t)}{\text{dt}}+\frac{v(t)}{R}+\frac{1}{L}{\int }_{-\infty }^{t}v(\tau )\mathrm{d\tau }$

3/ The mechanical and electrical systems are dynamically analogous

$\begin{array}{}f(t)=M\frac{\text{dv}(t)}{\text{dt}}+\text{Bv}(t)+K{\int }_{-\infty }^{t}v(\tau )\mathrm{d\tau }\\ i(t)=C\frac{\text{dv}(t)}{\text{dt}}+\frac{v(t)}{R}+\frac{1}{L}{\int }_{-\infty }^{t}v(\tau )\mathrm{d\tau }\end{array}$

Thus, understanding one of these systems gives insights into the other.

4/ Block diagram

A block diagram using integrators, adders, and gains

$f(t)=M\frac{\text{dv}(t)}{\text{dt}}+\text{Bv}(t)+K{\int }_{-\infty }^{t}v(\tau )\mathrm{d\tau }$

5/ Electronic synthesis of block diagram

The integrator, adder, and gain blocks are other examples of functional descriptions of systems. We can produce a structural model of each of these blocks. For example, the gain block is easily synthesized with an op-amp circuit.

The op-amp itself is a functional model of a device that we can synthesize with an electronic circuit including a number of transistors.

Conclusion: We have seen several types of descriptions of systems

$f(t)=M\frac{\text{dv}(t)}{\text{dt}}+\text{Bv}(t)+K{\int }_{-\infty }^{t}v(\tau )\mathrm{d\tau }$

All four descriptions define a system with the same dynamic properties. We will develop methods that characterize these systems efficiently and that abstract their critical dynamic properties.

IV. CLASSIFICATION OF SYSTEMS

1/ Memoryless systems

The output of a memoryless system at some time to depends only on its input at the same time to. For example, for the resistive divider network,

$v_o(t)=\frac{R_2}{R_1+R_2}{v}_i(t)$

Therefore, $v_o(\text{to})$ depends upon the value of $v_i(\text{to})$ and not on $v_i(t)$ for t ≠ to.

2/ Systems with memory

$i(t)=C\frac{\text{dv}(t)}{\text{dt}}$

$v(t)=\frac{1}{C}{\int }_{-\infty }^{t}i(\tau )\mathrm{d\tau }$

Note that v(t) depends not just on i(t) at one point in time t. Therefore, the system that relates v to i exhibits memory.

3/ Causal and noncausal systems

For a causal system the output at time to depends only on the input for t ≤ to, i.e., the system cannot anticipate the input.

Physical systems with time as the independent variable are causal systems. There are examples of systems that are not causal.

• Physical systems for which time is not the independent variable, e.g., the independent variable is space (x, y, z) as in an optical system.

• Processing of signals where time is the independent variable but the signal has been recorded or generated in a computer. Processing is not in real time.

4/ Stable and unstable systems

Stability can be defined in a variety of ways.

Definition 1: a stable system is one for which an incremental input leads to an incremental output.

An incremental force leads to only an

incremental displacement in the stable

system but not in the unstable system.

Definition 2: A system is BIBO stable if every bounded input leads to a bounded output. We will use this definition.