3.2 Steady-state error analysis

Steady-state error analysis is crucial in control systems. It helps engineers assess how well a system tracks desired outputs or rejects disturbances over time. Understanding error constants and system types allows for better design and performance optimization.

The type of input signal and system configuration greatly influence steady-state error. By manipulating system components and control strategies, engineers can reduce or eliminate errors for specific input types, balancing long-term accuracy with transient response.

Steady-state error

• Steady-state error refers to the difference between the desired output and the actual output of a control system when it reaches a steady state after a sufficiently long time
• Analyzing steady-state error helps determine how well a control system tracks a reference input or rejects disturbances in the long term
• The type of input (step, ramp, or parabolic) and the system type (determined by the number of integrators or poles at the origin in the open-loop transfer function) affect the steady-state error

Error constants

Position error constant

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• The position error constant, denoted as $K_p$, is used to determine the steady-state error for a step input in a closed-loop control system
• $K_p$ is defined as the limit of the open-loop transfer function $G(s)H(s)$ as $s$ approaches zero: $K_p = \lim_{s \to 0} G(s)H(s)$
• A higher $K_p$ value indicates a smaller steady-state error for a step input

Velocity error constant

• The velocity error constant, denoted as $K_v$, is used to determine the steady-state error for a ramp input in a closed-loop control system
• $K_v$ is defined as the limit of the product of $s$ and the open-loop transfer function $G(s)H(s)$ as $s$ approaches zero: $K_v = \lim_{s \to 0} sG(s)H(s)$
• A higher $K_v$ value indicates a smaller steady-state error for a ramp input

Acceleration error constant

• The acceleration error constant, denoted as $K_a$, is used to determine the steady-state error for a parabolic input in a closed-loop control system
• $K_a$ is defined as the limit of the product of $s^2$ and the open-loop transfer function $G(s)H(s)$ as $s$ approaches zero: $K_a = \lim_{s \to 0} s^2G(s)H(s)$
• A higher $K_a$ value indicates a smaller steady-state error for a parabolic input

System type

Type 0 systems

• Type 0 systems have no integrators or poles at the origin in their open-loop transfer function
• They exhibit a non-zero steady-state error for step inputs and an infinite steady-state error for ramp and parabolic inputs
• Examples of type 0 systems include first-order systems and second-order systems without an integrator

Type 1 systems

• Type 1 systems have one integrator or pole at the origin in their open-loop transfer function
• They exhibit zero steady-state error for step inputs, a non-zero steady-state error for ramp inputs, and an infinite steady-state error for parabolic inputs
• Examples of type 1 systems include systems with a single integrator, such as a motor with a speed controller

Type 2 systems

• Type 2 systems have two integrators or poles at the origin in their open-loop transfer function
• They exhibit zero steady-state error for step and ramp inputs, and a non-zero steady-state error for parabolic inputs
• Examples of type 2 systems include systems with double integrators, such as a motor with a position controller

Ramp vs step inputs

• Step inputs are sudden changes in the reference input that remain constant over time (unit step function)
• Ramp inputs are reference inputs that increase linearly with time (unit ramp function)
• The steady-state error for a step input depends on the position error constant $K_p$, while the steady-state error for a ramp input depends on the velocity error constant $K_v$
• Parabolic inputs, which increase quadratically with time, are less common but can be analyzed using the acceleration error constant $K_a$

Steady-state error formulas

Formula for step input

• For a step input, the steady-state error $e_{ss}$ is given by: $e_{ss} = \frac{1}{1 + K_p}$
• Where $K_p$ is the position error constant
• A higher $K_p$ value results in a smaller steady-state error for a step input

Formula for ramp input

• For a ramp input, the steady-state error $e_{ss}$ is given by: $e_{ss} = \frac{1}{K_v}$
• Where $K_v$ is the velocity error constant
• A higher $K_v$ value results in a smaller steady-state error for a ramp input

Formula for parabolic input

• For a parabolic input, the steady-state error $e_{ss}$ is given by: $e_{ss} = \frac{1}{K_a}$
• Where $K_a$ is the acceleration error constant
• A higher $K_a$ value results in a smaller steady-state error for a parabolic input

Steady-state error in unity feedback systems

• In a unity feedback system, the feedback gain is equal to 1, simplifying the analysis of steady-state error
• The open-loop transfer function $G(s)H(s)$ reduces to $G(s)$ in unity feedback systems
• The error constants ($K_p$, $K_v$, and $K_a$) can be determined directly from the open-loop transfer function $G(s)$

Steady-state error in non-unity feedback systems

• Non-unity feedback systems have a feedback gain not equal to 1, which affects the steady-state error analysis
• The open-loop transfer function $G(s)H(s)$ must be used to determine the error constants and steady-state error
• The feedback gain $H(s)$ can be designed to improve the steady-state error performance of the system

Improving steady-state error

Increasing system type

• Increasing the system type by adding integrators or poles at the origin can reduce or eliminate steady-state error for certain input types
• Adding an integrator to a type 0 system converts it to a type 1 system, eliminating steady-state error for step inputs
• Adding two integrators to a type 0 system converts it to a type 2 system, eliminating steady-state error for step and ramp inputs

Adding poles at origin

• Adding poles at the origin is another way to increase the system type and improve steady-state error performance
• Poles at the origin contribute to the number of integrators in the system
• Adding a pole at the origin to a type 0 system converts it to a type 1 system, while adding two poles at the origin converts it to a type 2 system

Steady-state error vs transient response

• Steady-state error analysis focuses on the long-term behavior of a control system, while transient response analysis examines the short-term behavior
• Improving steady-state error by increasing the system type or adding poles at the origin can negatively impact the transient response (increased overshoot and settling time)
• A balance must be struck between achieving the desired steady-state error performance and maintaining an acceptable transient response

Steady-state error in PID control

Effect of proportional gain

• Increasing the proportional gain $K_p$ in a PID controller reduces the steady-state error for step inputs
• However, a high proportional gain can lead to increased overshoot and oscillations in the transient response
• The proportional gain does not affect the steady-state error for ramp or parabolic inputs

Effect of integral gain

• The integral gain $K_i$ in a PID controller helps eliminate steady-state error for step and ramp inputs
• Increasing the integral gain reduces the steady-state error but can also lead to slower response times and increased overshoot
• The integral term effectively increases the system type by one, improving steady-state error performance

Effect of derivative gain

• The derivative gain $K_d$ in a PID controller does not directly affect the steady-state error
• However, the derivative term can help improve the transient response by reducing overshoot and settling time
• A well-tuned derivative gain can indirectly improve the overall performance of the system, including its steady-state behavior