Q1Control System

Question

Q.1. (a) What is control system? Explain difference between open loop & closed loop control system with the help of block diagram. [10]

(b) Find the transfer function relating displacement y & x for the mechanical system shown: a mass-less node A connects to output x through friction f1, connects in parallel to a spring k and friction f2 leading to node B, with f2 and k both grounded at B. [6]

Answer 1

(a) Control System and Open-Loop vs Closed-Loop

A control system is defined as an arrangement of physical components connected or related in such a manner as to command, direct, or regulate itself or another system, so that a desired output (response) is obtained for a given input (reference), despite the presence of disturbances and uncertainties in the system's parameters. Control systems are broadly classified into open-loop and closed-loop configurations based on whether the actual output is used to influence the control action.

Open-Loop Control System

In an open-loop control system, the control action is generated based purely on the reference input, without any measurement or feedback of the actual output — the controller computes the required actuating signal based solely on a predetermined relationship (calibration) between input and expected output, with no mechanism to detect or correct for any deviation between the actual and desired output caused by disturbances, component aging, or modeling inaccuracies.

Closed-Loop (Feedback) Control System

In a closed-loop control system, the actual output is continuously measured (sensed) and fed back to be compared with the reference input, generating an error signal (the difference between desired and actual output) that drives the controller/actuator to correct the output and reduce this error — allowing the system to automatically compensate for disturbances, parameter variations, and non-linearities that an open-loop system cannot correct for.

Feedback: open-loop has none; closed-loop continuously uses output feedback.
Accuracy: open-loop accuracy depends entirely on calibration and is degraded by disturbances/component drift; closed-loop automatically corrects for these, giving higher accuracy.
Sensitivity to disturbances: open-loop is highly sensitive to disturbances since it cannot detect or correct for them; closed-loop actively counteracts disturbances via the error-correcting feedback action.
Stability: open-loop systems are inherently stable if their components are individually stable, requiring no special stability analysis; closed-loop systems can become unstable even with stable individual components, due to the feedback loop's dynamics, requiring careful stability analysis (Routh-Hurwitz, root locus, Bode/Nyquist).
Complexity and cost: open-loop systems are simpler and cheaper (no sensors/comparator needed); closed-loop systems are more complex and costly due to the additional sensing, comparison and feedback circuitry required.
Bandwidth: closed-loop systems generally offer improved bandwidth and reduced effect of parameter variation on overall system gain, compared to open-loop systems.
Examples: open-loop — an electric toaster, a washing machine timer, a traffic light with fixed timing; closed-loop — a room thermostat-controlled heater, a cruise-control system in a car, a servo-motor position control system.

(b) Transfer Function of the Mechanical System

The given mechanical system consists of an input displacement x applied at node A, transmitted through a dashpot (friction/damper) of coefficient f1 to an intermediate massless node, which is connected to ground (node B) through a spring of stiffness k and a second dashpot of coefficient f2, both in parallel, with the output displacement y taken at this intermediate node.

Since the intermediate node is massless, the net force acting on it must be zero at all times (force balance/D'Alembert's principle): the force transmitted through the input dashpot f1 (proportional to the relative velocity between x and y) must exactly equal the sum of the restoring forces developed by the spring k (proportional to displacement y) and the output dashpot f2 (proportional to velocity of y), since both k and f2 connect this node to the fixed ground reference B:

f_{1} (\overset{x}{˙} - \overset{y}{˙}) = k y + f_{2} \overset{y}{˙}

Taking the Laplace transform of both sides (assuming zero initial conditions):

f_{1} s (X (s) - Y (s)) = k Y (s) + f_{2} s Y (s)

Collecting all Y(s) terms on one side:

f_{1} s X (s) = Y (s) [k + f_{2} s + f_{1} s]

Solving for the transfer function Y(s)/X(s):

\frac{Y ( s )}{X ( s )} = \frac{f _{1} s}{( f _{1} + f _{2} ) s + k}

This transfer function shows that the system behaves as a first-order high-pass-like filter relating the two displacements — at very low frequencies (s→0), the ratio Y/X→0, since the spring k dominates and effectively holds the intermediate node close to the fixed ground reference for slowly-varying inputs; at very high frequencies (s→∞), the ratio Y/X → f1/(f1+f2), a constant fraction determined purely by the relative damping coefficients, since the spring's restoring force becomes comparatively negligible relative to the rapidly-changing dashpot forces at high frequency.

Force-Voltage and Force-Current Electrical Analogies

This mechanical network can be verified by drawing its exact electrical-analog circuit, a standard cross-check technique used extensively in control-system courses to relate mechanical translational systems to equivalent electrical networks that obey the same governing differential equation. In the force-voltage (f-v) analogy, force maps to voltage, velocity maps to current, mass maps to inductance, the dashpot (viscous friction) coefficient maps to resistance, and the spring compliance (1/k) maps to capacitance. Under this analogy, the input dashpot f1 (carrying the relative velocity between the input and the intermediate node) becomes a resistor R1=f1 in series with the source, while the output dashpot f2 and the spring k (both connected between the intermediate node and the fixed ground) become a resistor R2=f2 and a capacitor C=1/k in parallel with each other, forming a simple series-parallel voltage-divider-like network whose transfer function has exactly the same algebraic form as Y(s)/X(s)=f1s/[(f1+f2)s+k] derived above, since a parallel R2-C combination has impedance R2/(1+R2Cs), and the potential-divider ratio between the series resistor R1 and this parallel combination reduces to the identical rational function in s once f2 and 1/k are substituted for R2 and C.

In the alternative force-current (f-i) analogy, force maps to current, velocity maps to voltage, mass maps to capacitance, the dashpot coefficient maps to conductance (1/R), and compliance maps to inductance; here, the two dashpots f1 and f2 appear as conductances in a node-equation (nodal-analysis) formulation, which again yields the same transfer function upon direct algebraic reduction. Deriving the same result through either analogy, or through the direct force-balance method used above, confirms the correctness of the transfer function and reinforces the conceptual link between mechanical and electrical network theory that underlies much of classical control system modeling.

Physical Interpretation and Design Implications

The derived transfer function Y(s)/X(s) = f1s/[(f1+f2)s+k] can also be rewritten in standard first-order time-constant form by dividing numerator and denominator by (f1+f2):

\frac{Y ( s )}{X ( s )} = \frac{\frac{f _{1}}{f _{1} + f _{2}} s}{s + \frac{k}{f _{1} + f _{2}}} = \frac{\frac{f _{1}}{f _{1} + f _{2}} τ s}{1 + τ s}, τ = \frac{f _{1} + f _{2}}{k}

which is recognizable as a classical washout-filter (high-pass) transfer function with time constant τ=(f1+f2)/k and high-frequency gain f1/(f1+f2). This form is useful for physical interpretation: for a step change in input displacement x, the output y would jump instantaneously to a fraction f1/(f1+f2) of the step (since a step has infinite initial slope, momentarily dominating the dashpot f1) and then decay exponentially back toward zero with time constant τ, as the spring gradually relaxes the intermediate node back toward its equilibrium position — this washout behavior is precisely why such f1-f2-k mechanical networks (and their electrical analogs) are used in practice as transient-detecting or rate-sensing elements, since they pass rapidly changing (AC-like) components of a signal while blocking the steady-state (DC) component entirely, a property directly evident from the fact that Y(s)/X(s)→0 as s→0 but approaches a nonzero constant as s→∞, exactly as noted above.

It is also worth noting the connection between this two-port mechanical element and general control-system block-diagram reduction: the same result could have been obtained by recognizing the input dashpot f1 as a forward element and the parallel k-f2 combination as a feedback-like impedance loading the node, then applying an impedance-divider argument analogous to the standard feedback formula G/(1+GH) used elsewhere in this subject — reinforcing that the systematic reduction techniques taught for electrical/block-diagram systems apply equally well to mechanical networks once the appropriate force-balance (or equivalently, impedance) equations are written down.