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Control Fundamentals

Open-Loop vs Closed-Loop Control

Open-loop control applies a predetermined input without measuring the output. The controller has no knowledge of disturbances or plant variations.

Closed-loop (feedback) control measures the output, computes an error signal, and adjusts the input accordingly.

| Property | Open-Loop | Closed-Loop | |----------|-----------|-------------| | Disturbance rejection | Poor | Good | | Sensitivity to plant variation | High | Low | | Stability risk | Low | Can be unstable | | Complexity | Simple | Higher |

Feedback Principles

Closed-Loop Feedback Control System

The fundamental feedback equation for a single-loop system with forward path G(s) and feedback path H(s):

Closed-loop TF:  T(s) = G(s) / (1 + G(s)H(s))

For unity feedback (H(s) = 1):

T(s) = G(s) / (1 + G(s))

The error transfer function:

E(s)/R(s) = 1 / (1 + G(s)H(s))

Key insight: feedback reduces sensitivity to plant variations by the factor 1 + G(s)H(s), called the return difference.

Block Diagram Algebra

Standard block diagram operations:

Series (cascade):

G_total(s) = G1(s) * G2(s)

Parallel:

G_total(s) = G1(s) + G2(s)

Feedback loop:

G_total(s) = G(s) / (1 ± G(s)H(s))

Negative feedback uses +, positive feedback uses -.

Block diagram reduction rules:

  1. Moving a summing junction ahead of a block: insert the block's inverse in the moved path
  2. Moving a pickoff point ahead of a block: insert the block in the moved path
  3. Moving a summing junction past a block: insert the block in the moved path
  4. Moving a pickoff point past a block: insert the block's inverse in the moved path

Transfer Functions

A transfer function relates the Laplace transform of the output to the input, assuming zero initial conditions:

G(s) = Y(s)/U(s) = (b_m*s^m + ... + b_1*s + b_0) / (a_n*s^n + ... + a_1*s + a_0)

Poles: roots of the denominator -- determine stability and transient behavior. Zeros: roots of the numerator -- affect the shape of the transient response.

A system is proper if m <= n and strictly proper if m < n. Physical systems are always proper.

Factored form:

G(s) = K * (s - z_1)(s - z_2)...(s - z_m) / ((s - p_1)(s - p_2)...(s - p_n))

Partial fraction expansion decomposes G(s) for inverse Laplace transform computation.

Signal-Flow Graphs (Mason's Gain Formula)

An alternative to block diagrams using nodes (signals) and directed branches (gains).

Mason's gain formula:

T = (1/Delta) * sum_k(T_k * Delta_k)

Where:

  • T_k = gain of the k-th forward path
  • Delta = 1 - (sum of all loop gains) + (sum of products of non-touching loop pairs) - ...
  • Delta_k = cofactor of Delta for path k (Delta evaluated with loops touching path k removed)

Example: For a system with two forward paths T1, T2 and three loops L1, L2, L3 where L1 and L2 don't touch:

Delta = 1 - (L1 + L2 + L3) + (L1*L2)

System Classification

Linearity

  • Linear: superposition holds -- f(a*x1 + b*x2) = a*f(x1) + b*f(x2)
  • Nonlinear: saturation, dead zones, backlash, friction, hysteresis

Number of Inputs/Outputs

  • SISO: Single-Input Single-Output -- classical control focus
  • MIMO: Multiple-Input Multiple-Output -- requires matrix transfer functions G(s) where Y(s) = G(s)U(s)

Time Variance

  • Time-invariant (LTI): system parameters do not change with time
  • Time-varying (LTV): parameters are functions of time, e.g., a spacecraft losing mass

Domain

  • Continuous-time: described by differential equations, uses Laplace transform (s-domain)
  • Discrete-time: described by difference equations, uses Z-transform (z-domain)
  • Hybrid: mix of continuous and discrete subsystems (sampled-data systems)

Causality and Memory

  • Causal: output depends only on current and past inputs
  • Memoryless: output depends only on current input
  • Dynamic: output depends on past inputs (has memory/state)

System Type and Error Constants

The system type is the number of pure integrators (poles at s = 0) in the open-loop transfer function.

For unity feedback with open-loop G(s):

| Input | Type 0 | Type 1 | Type 2 | |-------|--------|--------|--------| | Step (1/s) | e_ss = 1/(1+Kp) | 0 | 0 | | Ramp (1/s^2) | infinity | 1/Kv | 0 | | Parabola (1/s^3) | infinity | infinity | 1/Ka |

Error constants:

Kp = lim(s->0) G(s)           (position constant)
Kv = lim(s->0) s*G(s)         (velocity constant)
Ka = lim(s->0) s^2*G(s)       (acceleration constant)

Typical Feedback Control Architecture

R(s) -->(+)--> C(s) --> G_plant(s) --> Y(s)
         ^(-)                           |
         |--- H(s) <-------------------|

Components:

  • R(s): reference input (setpoint)
  • E(s) = R(s) - H(s)Y(s): error signal
  • C(s): controller (compensator)
  • G_plant(s): plant (process to be controlled)
  • H(s): sensor/feedback element
  • D(s): disturbance (enters at various points)

Disturbance rejection transfer function (disturbance at plant input):

Y(s)/D(s) = G_plant(s) / (1 + C(s)*G_plant(s)*H(s))

High controller gain reduces disturbance effect but risks instability.