Skip to main content

Continuous system simulation; analog computer example

Short answer

Continuous system simulation studies systems whose state varies smoothly with time, solved by analog or digital computers. An analog computer uses voltage to represent variables and units like integrators and potentiometers. For decay dy/dt=-ky, a single integrator with feedback via pot k produces output y0 e^{-kt} from initial condition -y0.

What is continuous system simulation? Explain analog computer with a practical example.

Continuous system simulation

A continuous system is one whose state variables change continuously with time, such as the position of a vehicle, the temperature of a furnace or the concentration of a chemical. Continuous system simulation is the study of such systems by a model, usually a set of differential equations, solved by a computer. There are two ways:

  • Analog computer: the equation is wired as an electrical circuit.
  • Digital computer: the equation is solved by numerical integration (Euler, Runge-Kutta) in small time steps, using languages such as CSMP or tools such as Simulink.

Examples: aircraft flight, chemical reactors, population growth, electrical circuits.

Analog computer

An analog computer is a computing device that represents each variable by a continuously varying physical quantity (usually voltage) and solves equations by connecting electronic units that behave like the mathematical operations. The analog method solves a differential equation by building a circuit whose voltages obey the same equation (an analogue of the system) and recording the output.

Main units:

UnitFunction
SummerAdds input voltages, output is the negative sum
IntegratorIntegrates the sum of its inputs with respect to time, with sign inversion
Coefficient potentiometerMultiplies a voltage by a constant between 0 and 1
InverterChanges the sign of a voltage
Multiplier / function generatorProduct of two variables, nonlinear functions
Recorder / scopeDisplays the output against time

Practical example: radioactive decay or capacitor discharge

Consider dy/dt = -k y, with y(0) = y0. This may be radioactive decay, or discharge of a capacitor through a resistor with k = 1/RC.

An integrator gives the negative of the integral of its input. If its output is -y, its input must be dy/dt = -k y. This input is obtained by feeding the output -y through potentiometer k.

        +--------[ pot k ]<-------+
        |                          |
        +--------> [ Integrator ]--+--> -y
                   (initial value -y0)

Steps:

  1. Set the integrator's initial condition to -y0.
  2. Connect the integrator output -y to its input through pot k.
  3. Start the run; the output follows -y0 e^(-kt), so the recorder (with an inverter if needed) shows y = y0 e^(-kt).

If k = 0.5 per second and y0 = 10 V, the voltage falls to 3.68 V in 2 s (10 e^(-1)). Changing the pot k changes the decay speed immediately.

For second-order systems such as the car suspension, M x'' + D x' + K x = F, two integrators in series are used.

Advantages and disadvantages

Advantages of the analog computer

  • Operates in parallel, so the solution is fast, even real time.
  • Continuous solution; no step size or numerical round-off error.
  • Interactive: parameters can be changed by a knob and the result seen immediately.
  • Shows the physical meaning of the equation; good for control studies.

Disadvantages

  • Accuracy is limited (about 0.1 to 1 percent) by component tolerance, noise and drift.
  • Amplitude scaling and time scaling are needed, and are tedious.
  • The size of the problem is limited by the number of units available.
  • Hard to implement nonlinearities, logical decisions and data storage.
  • Re-patching for a new problem takes time; poor repeatability and documentation.

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Simulation and Modeling chapter 3 questions →