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Lecture 2: Introduction to Systems

Module by: Thanh Dinh Vu, Truc Pham-Dinh, Anh Tuan Hoang, Tam Huynh-Ngoc

Summary: To describe man-made and natural systems, in forms of mathematical expressions in time and in frequency domains. Many physical systems are described by linear differential equations or by linear difference equations.

Lecture #2:
INTRODUCTION TO SYSTEMS
Motivation: To describe man-made and natural systems, in forms of mathematical expressions in time and in frequency domains. Many physical systems are described by linear differential equations or by linear difference equations.
Outline:
  • Man-made systems — modular, hierarchic design. Natural systems — modular, hierarchic analysis. Dynamic analogies
  • Classification of systems
  • Reducing differential equations to algebraic equations by using complex notations. Role of complex exponential time functions in LTI systems. Sources of linear differential equations. Homogeneous solution — exponential solution, natural frequencies. Particular solution — system function, poles & zeros. Total solution — initial conditions, steady-state.
  • Reducing difference equations to algebraic equations. Role of complex geometric (exponential) time functions in DT LTI systems. Linear difference equations arise as system. Homogeneous solution — geometric (exponential) solution, natural frequencies. Particular solution — system function, poles & zeros. Total solution — initial conditions, steady-state.
Signals and systems
This subject deals with mathematical methods used to describe signals and to analyze and synthesize systems.
  • Signals are variables that carry information
  • Systems process input signals to produce output signals.
Last time — SIGNALS; Today — SYSTEMS.
I. MAN-MADE SYSTEMS — MODULAR, HIERARCHIC DESIGN
Robot car
Figure 1
1/ Robot car block diagram
Hierarchic design — top (1st) level includes: wheel position controller, digital camera, and image processing software.
Figure 2
2/ Wheel position controller block diagram
Hierarchic design — 2nd level is a block diagram of the wheel controller which includes: amplifier, motor, and shaft decoder.
Figure 3
3/ Motor dynamics
Hierarchic design—3rd level includes a more detailed description of the motor. Important quantities are:
  • Current in motor windings i(t),
  • Motor shaft angular displacement θ(t),
  • Motor parameters — viscous damping constant B, moment of inertia J, electromechanical constant k.
The torque balance equation is:
ki ( t ) − B dθ ( t ) dt = J dθ 2 ( t ) dt 2 ki ( t ) − B dθ ( t ) dt = J dθ 2 ( t ) dt 2 size 12{ ital "ki" \( t \) - B { {dθ \( t \) } over { ital "dt"} } =J { {dθ rSup { size 8{2} } \( t \) } over { ital "dt" rSup { size 8{2} } } } } {}
4/ Observation
  • Man-made complex systems are designed in a modular, hierarchical fashion often expressed in nested block diagrams.
  • Block input/output relations provide a communication mechanism for team projects.
  • Optimization of system performance requires excellent tools to characterize signal transformations at each level of the hierarchy.
II. NATURALLY OCCURRING SYSTEMS — MODULAR, HIERARCHIC ANALYSIS
Human speech production system — anatomy
Figure 4
1/ Human speech production system — block diagram
Figure 5
2/ Observation
  • Naturally occurring systems are not designed in a modular fashion — they have evolved.
  • To understand these systems, we impose a hierarchy and parse the system into modules whose function can be characterized.
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.
Figure 6
Summing the forces yields
f ( t ) = M dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ f ( t ) = M dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ size 12{f \( t \) =M { { ital "dv" \( t \) } over { ital "dt"} } + ital "Bv" \( t \) +K Int rSub { size 8{ - infinity } } rSup { size 8{t} } {v \( τ \) dτ} } {}
2/ Electric network
Figure 7
Summing the currents (Kirchhoff’s current law) yields
i ( t ) = C dv ( t ) dt + v ( t ) R + 1 L ∫ − ∞ t v ( τ ) dτ i ( t ) = C dv ( t ) dt + v ( t ) R + 1 L ∫ − ∞ t v ( τ ) dτ size 12{i \( t \) =C { { ital "dv" \( t \) } over { ital "dt"} } + { {v \( t \) } over {R} } + { {1} over {L} } Int rSub { size 8{ - infinity } } rSup { size 8{t} } {v \( τ \) dτ} } {}
3/ The mechanical and electrical systems are dynamically analogous
f ( t ) = M dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ i ( t ) = C dv ( t ) dt + v ( t ) R + 1 L ∫ − ∞ t v ( τ ) dτ f ( t ) = M dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ i ( t ) = C dv ( t ) dt + v ( t ) R + 1 L ∫ − ∞ t v ( τ ) dτ alignl { stack { size 12{f \( t \) =M { { ital "dv" \( t \) } over { ital "dt"} } + ital "Bv" \( t \) +K Int rSub { size 8{ - infinity } } rSup { size 8{t} } {v \( τ \) dτ} } {} # i \( t \) =C { { ital "dv" \( t \) } over { ital "dt"} } + { {v \( t \) } over {R} } + { {1} over {L} } Int rSub { size 8{ - infinity } } rSup { size 8{t} } {v \( τ \) dτ} {} } } {}
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 dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ f ( t ) = M dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ size 12{f \( t \) =M { { ital "dv" \( t \) } over { ital "dt"} } + ital "Bv" \( t \) +K Int rSub { size 8{ - infinity } } rSup { size 8{t} } {v \( τ \) dτ} } {}
Figure 8
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.
Figure 9
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
Figure 10
f ( t ) = M dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ f ( t ) = M dv ( t ) dt + Bv ( t ) + K ∫ − ∞ t v ( τ ) dτ size 12{f \( t \) =M { { ital "dv" \( t \) } over { ital "dt"} } + ital "Bv" \( t \) +K Int rSub { size 8{ - infinity } } rSup { size 8{t} } {v \( τ \) dτ} } {}
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 ) = R 2 R 1 + R 2 v i ( t ) v o ( t ) = R 2 R 1 + R 2 v i ( t ) size 12{v rSub { size 8{o} } \( t \) = { {R rSub { size 8{2} } } over {R rSub { size 8{1} } +R rSub { size 8{2} } } } v rSub { size 8{i} } \( t \) } {}
Therefore, vo(to)vo(to) size 12{v rSub { size 8{o} } \( "to" \) } {}depends upon the value of vi(to)vi(to) size 12{v rSub { size 8{i} } \( "to" \) } {}and not on vi(t)vi(t) size 12{v rSub { size 8{i} } \( t \) } {}for t ≠ to.
2/ Systems with memory
Figure 11
i ( t ) = C dv ( t ) dt i ( t ) = C dv ( t ) dt size 12{i \( t \) =C { { ital "dv" \( t \) } over { ital "dt"} } } {}
v ( t ) = 1 C ∫ − ∞ t i ( τ ) dτ v ( t ) = 1 C ∫ − ∞ t i ( τ ) dτ size 12{v \( t \) = { {1} over {C} } Int rSub { size 8{ - infinity } } rSup { size 8{t} } {i \( τ \) dτ} } {}
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.
Figure 12
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.
Figure 13
For the resistor, if i(t) is bounded then so is v(t), but for the capacitance this is not true. Consider i(t) = u(t) then v(t) = tu(t) which is unbounded.
5/ Linear systems
Figure 14
for all x1(t)x1(t) size 12{x rSub { size 8{1} } \( t \) } {}, x2(t)x2(t) size 12{x rSub { size 8{2} } \( t \) } {}, a, and b.
6/ Time-invariant systems
Figure 15
for all x(t) and τ.
7/ Linear and time-invariant (LTI) systems
  • Many man-made and naturally occurring systems can be modeled as LTI systems.
  • Powerful techniques have been developed to analyze and to characterize LTI systems.
  • The analysis of LTI systems is an essential precursor to the analysis of more complex systems.
Problem — Multiplication by a time function
A system is defined by the functional description
Figure 16
  • Is this system linear?
  • Is this system time-invariant?
Solution — Multiplication by a time function
Let
{ y 1 ( t ) = g ( t ) x 1 ( t ) y 2 ( t ) = g ( t ) x 2 ( t ) { y 1 ( t ) = g ( t ) x 1 ( t ) y 2 ( t ) = g ( t ) x 2 ( t ) size 12{ left lbrace matrix { y rSub { size 8{1} } \( t \) =g \( t \) x rSub { size 8{1} } \( t \) {} ## y rSub { size 8{2} } \( t \) =g \( t \) x rSub { size 8{2} } \( t \) } right none } {}
By definition the response to
x ( t ) = ax 1 ( t ) + bx 2 ( t ) x ( t ) = ax 1 ( t ) + bx 2 ( t ) size 12{x \( t \) = ital "ax" rSub { size 8{1} } \( t \) + ital "bx" rSub { size 8{2} } \( t \) } {}
Is
y ( t ) = g ( t ) ( ax 1 ( t ) + bx 2 ( t ) ) y ( t ) = g ( t ) ( ax 1 ( t ) + bx 2 ( t ) ) size 12{y \( t \) =g \( t \) \( ital "ax" rSub { size 8{1} } \( t \) + ital "bx" rSub { size 8{2} } \( t \) \) } {}
This can be rewritten as
y ( t ) = ag ( t ) x 1 ( t ) + bg ( t ) x 2 ( t ) y ( t ) = ag ( t ) x 1 ( t ) + bg ( t ) x 2 ( t ) size 12{y \( t \) = ital "ag" \( t \) x rSub { size 8{1} } \( t \) + ital "bg" \( t \) x rSub { size 8{2} } \( t \) } {}
y ( t ) = ay 1 + by 2 ( t ) y ( t ) = ay 1 + by 2 ( t ) size 12{y \( t \) = ital "ay" rSub { size 8{1} } + ital "by" rSub { size 8{2} } \( t \) } {}
Therefore, the system is linear.
Now suppose that x1(t)= x(t)x1(t)= x(t) size 12{x rSub { size 8{1} } \( t \) =" x" \( t \) } {}and x2(t)= x(t - τ)x2(t)= x(t - τ) size 12{x rSub { size 8{2} } \( t \) =" x" \( "t - "τ \) } {}, and the response to these two inputs are y1(t)y1(t) size 12{y rSub { size 8{1} } \( t \) } {}and y2(t)y2(t) size 12{y rSub { size 8{2} } \( t \) } {}, respectively. Note that
y 1 ( t ) = y ( t ) = g ( t ) x ( t ) y 1 ( t ) = y ( t ) = g ( t ) x ( t ) size 12{y rSub { size 8{1} } \( t \) =y \( t \) =g \( t \) x \( t \) } {}
And
y 2 ( t ) = g ( t ) x ( t − τ ) ≠ y ( t − τ ) y 2 ( t ) = g ( t ) x ( t − τ ) ≠ y ( t − τ ) size 12{y rSub { size 8{2} } \( t \) =g \( t \) x \( t - τ \) <> y \( t - τ \) } {}
Therefore, the system is time-varying.
Problem — Addition of a constant
Suppose the relation between the output y(t) and input x(t) is y(t) = x(t)+K, where K is some constant. Is this system linear?
Solution — Addition of a constant
Note, that if the input is x1(t)+x2(t)x1(t)+x2(t) size 12{x rSub { size 8{1} } \( t \) +x rSub { size 8{2} } \( t \) } {}then the output will be y(t)= x1(t)+x2(t)+K ≠ y1(t)+y2(t)=(x1(t)+K)+(x2(t)+K).y(t)= x1(t)+x2(t)+K ≠ y1(t)+y2(t)=(x1(t)+K)+(x2(t)+K). size 12{y \( t \) =" x" rSub { size 8{1} } \( t \) +x rSub { size 8{2} } \( t \) +"K " <> " y" rSub { size 8{1} } \( t \) +y rSub { size 8{2} } \( t \) = \( x rSub { size 8{1} } \( t \) +K \) + \( x rSub { size 8{2} } \( t \) +K \) "." } {}
Therefore, this system is not linear.
In general, it can be shown that for a linear system if x(t) = 0 then y(t) = 0. Using the definition of linearity, choose a = b = 1 and