Fundamentals of Steady State Error
Control systems are engineered to ensure a system’s output accurately follows a desired input signal. The performance of these systems is often evaluated by how closely the actual output matches the target. After any initial adjustments or transient behaviors subside, the system reaches a stable condition known as the steady state.
Steady state error represents the difference that persists between the desired output and the actual output once the system has settled. This deviation signifies how well the control system can track its input under stable conditions. Understanding and minimizing steady state error is a primary consideration in the design and tuning of effective control systems.
In a typical feedback control system, the error signal is generated by comparing the desired input with the system’s measured output. This error signal then drives the controller to adjust the system’s behavior. The “steady state” refers to the system’s behavior as time approaches infinity, meaning all transient responses have decayed.
To analyze steady state error systematically, control engineers use three standard test inputs: the unit step, unit ramp, and unit parabolic functions. A unit step input represents a sudden, constant desired output, akin to setting a thermostat to a new temperature. A unit ramp input signifies a desired output that increases linearly over time, like tracking a constant velocity. A unit parabolic input represents a desired output that accelerates over time, which can model scenarios requiring constant acceleration.
The Final Value Theorem
The Final Value Theorem (FVT) provides a mathematical approach to determine the steady state value of a time-domain function directly from its Laplace transform. This theorem eliminates the need to perform an inverse Laplace transform, which can often be a complex process. It is a valuable tool for analyzing the long-term behavior of systems.
The general formula for the Final Value Theorem is expressed as: `lim(t→∞) f(t) = lim(s→0) sF(s)`. Here, `f(t)` represents the function in the time domain, and `F(s)` is its corresponding Laplace transform. This theorem is particularly useful for finding the steady state error, `e_ss`, by applying it to the Laplace transform of the error signal, `E(s)`.
For the Final Value Theorem to be applicable, certain conditions must be met. Specifically, all poles of `sF(s)` must lie in the left-half of the complex s-plane, with the possible exception of a simple pole at the origin. This condition ensures that the function `f(t)` settles to a finite, constant value as time approaches infinity.
System Type and Error Constants
The ability of a control system to eliminate or reduce steady state error significantly depends on its “system type.” The system type is determined by the number of pure integrators, or poles at the origin, present in the open-loop transfer function `G(s)H(s)`. These integrators represent the system’s capacity to accumulate or sum past errors.
A system with no integrators in its open-loop transfer function is classified as a Type 0 system. If there is one integrator, it is a Type 1 system, and with two integrators, it becomes a Type 2 system. Each system type exhibits different steady state error characteristics when subjected to the standard test inputs. This classification helps predict how well a system will track various types of command signals.
To quantify the steady state error for different system types and inputs, control theory utilizes static error constants. The positional error constant, `Kp`, is used for Type 0 systems and is defined as `lim(s→0) G(s)H(s)`. The velocity error constant, `Kv`, applies to Type 1 systems and is calculated as `lim(s→0) sG(s)H(s)`. Finally, the acceleration error constant, `Ka`, is relevant for Type 2 systems and is given by `lim(s→0) s²G(s)H(s)`.
These error constants directly relate to the magnitude of the steady state error. A higher value for `Kp` for a Type 0 system, for instance, indicates a smaller steady state error when the system is subjected to a step input. Similarly, larger `Kv` and `Ka` values correspond to improved tracking performance for ramp and parabolic inputs, respectively. The system type and the associated error constants provide a quick assessment of a system’s steady state accuracy.
Calculating Error for Standard Inputs
Calculating steady state error involves combining the Final Value Theorem with the appropriate error constants for a given system type and input. This process allows engineers to predict a system’s long-term accuracy when subjected to various command signals.
For a unit step input, which has a Laplace transform of `1/s`, the steady state error `e_ss` for a Type 0 system is `1 / (1 + Kp)`. For example, if a Type 0 system has an open-loop transfer function `G(s)H(s) = 10 / (s + 2)`, then `Kp = lim(s→0) 10 / (s + 2) = 5`. The steady state error for a unit step input would be `1 / (1 + 5) = 1/6`. For a Type 1 system, the steady state error `e_ss` is zero. This occurs because the integrator in the system can effectively eliminate the constant error from a step input.
For a unit ramp input, which has a Laplace transform of `1/s²`, a Type 1 system will exhibit a finite steady state error of `1 / Kv`. If a Type 1 system has `G(s)H(s) = 5 / (s(s + 1))`, then `Kv = lim(s→0) s [5 / (s(s + 1))] = 5`. The steady state error for a unit ramp input would be `1/5`. For a Type 2 system, the steady state error `e_ss` is zero.
For a unit parabolic input, which has a Laplace transform of `1/s³`, a Type 2 system will have a finite steady state error of `1 / Ka`. Consider a Type 2 system with `G(s)H(s) = 8 / (s²(s + 3))`. Here, `Ka = lim(s→0) s² [8 / (s²(s + 3))] = 8/3`. The steady state error for a unit parabolic input would be `1 / (8/3) = 3/8`.
This relationship between system type, input type, and steady state error can be summarized: a Type 0 system has a finite error for a step input and infinite error for ramp and parabolic inputs. A Type 1 system achieves zero error for a step input, a finite error for a ramp input, and infinite error for a parabolic input. A Type 2 system demonstrates zero error for both step and ramp inputs, but a finite error for a parabolic input. Higher system types generally result in better tracking performance for higher-order inputs, but they can also introduce challenges in system stability.