6.3.4. Steady-State Error Formulae

Once the system has settled and transient effects have subsided, the system reaches a steady-state. The steady-state response describes how the output behaves in response to a constant input after the transient effects die out. Key factors in analyzing steady-state response include:

1. Steady-State Error: The difference between the desired output and the actual output as time approaches infinity.
   • Types of Inputs: Step input, ramp input, or parabolic input.
   • Error Constants: The steady-state error for each type of input can be determined using error constants.

The steady-state error is a measure of how accurately the system responds to a constant input after all transient effects have disappeared. It is defined as the difference between what we want the system to output (the desired output) and what it actually outputs when it has settled. The types of inputs can range from a simple step input to more complex ramp and parabolic inputs. Furthermore, error constants are used to quantify the steady-state error for these common input types, allowing for a deeper analysis of system performance.

Think of a car cruise control system. When you set the speed (a constant input), the actual speed you achieve may differ slightly due to various factors like engine response time or friction losses. Over time, the system stabilizes at a certain speed. The difference between your desired speed and the actual speed when the system is stable represents the steady-state error.

2. Error Constants:
   • Position Error Constant KpK_p: Determines the steady-state error for a step input.
   • Velocity Error Constant KvK_v: Determines the steady-state error for a ramp input.
   • Acceleration Error Constant KaK_a: Determines the steady-state error for a parabolic input.

Error constants are essential tools in control theory that help us quantify the steady-state error for different types of input signals. The Position Error Constant (10) is used for step inputs; it reflects how well the system can reach a specific level and stay there. The Velocity Error Constant (12) applies to ramp inputs, which means the input is constantly increasing. Finally, the Acceleration Error Constant (13) relates to parabolic inputs, where the input changes accelerate over time. Each constant offers insights into the system's ability to maintain accuracy under varied conditions.

Imagine a person trying to catch a ball thrown at different speeds. If the ball is thrown straight (step input), they adjust their hand's position to catch it — this is governed by Kp. If the ball rolls away on a ramp, they must move their hand faster (Kv). For a ball that accelerates toward them (parabolic input), their adjustment becomes increasingly complicated and is represented by Ka. Each scenario requires different adjustments to minimize the catching error.

3. Steady-State Error for Different Inputs:
   • For a step input, the error depends on KpK_p.
   • For a ramp input, the error depends on KvK_v.
   • For a parabolic input, the error depends on KaK_a.

The steady-state error varies based on the type of input applied to the system. For instance, when the system is subjected to a step input, the steady-state error can be directly calculated using the Position Error Constant (10). In the case of a ramp input, the steady-state error can be deduced using the Velocity Error Constant (12). Lastly, for a parabolic input, the Acceleration Error Constant (13) plays a critical role in determining how much error is present as the input escalates over time.

Returning to our car analogy: if you suddenly hit the gas (step input), the cruise control must overcome inertia quickly to match your desired speed, reflecting Kp. If you were to accelerate gradually (ramp input), the control system must keep up with your increasing speed using Kv. Lastly, if you accelerate in a more complicated manner, where the speed continuously increases like a roller coaster (parabolic input), Ka becomes important as the system adjusts to match this dynamic input.

Steady-State Error Formulae:

• Step Input: For a system with a transfer function G(s)G(s), the steady-state error for a step input R(s)=1/sR(s) = 1/s is given by: ess=1/(1+Kp)e_{ss} = 1/(1 + K_p)
• Ramp Input: The steady-state error for a ramp input R(s)=1/s^2R(s) = 1/s^2 is given by: ess=1/Kve_{ss} = 1/K_v
• Parabolic Input: The steady-state error for a parabolic input R(s)=1/s^3R(s) = 1/s^3 is given by: ess=1/KaKae_{ss} = 1/K_a

The steady-state error can be mathematically calculated for different types of inputs. For a step input, the error formula shows that it decreases as the Position Error Constant increases. Similarly, for a ramp input, the steady-state error is inversely related to the Velocity Error Constant, indicating that a higher constant results in a lower error. For parabolic inputs, the Acceleration Error Constant similarly demonstrates that higher values lead to lower steady-state errors. Each formula highlights the relationship between the system's feedback configuration and how it manages to reach or fail to reach desired outputs.

Imagine school grades as inputs to a student's learning system. If a student suddenly gets an A (step input), the difference between where they are and where they need to be (desired grade) is determined by their study habits (Kp). If they are continuously improving (ramp input), consistency in their study techniques (Kv) influences how well they maintain good grades. For their overall academic improvements over time (parabolic input), the more effective their study strategies are (Ka), the better they can manage their overall performance, showing lower steady-state error in their results.

Example: For a system with Kp=10K_p = 10, the steady-state error for a step input is: ess=1/(1+10)=0.09e_{ss} = 1/(1 + 10) = 0.09

In this example, we take a system where the Position Error Constant (Kp) is 10. To find the steady-state error for a step input, we can use the formula ess = 1/(1 + Kp). Plugging in the value, we get ess = 1/(1 + 10) = 1/11, which equals approximately 0.09. This result signifies that when the system is subjected to a step input, the steady-state error is quite low, indicating that the system performs well under such conditions.

Going back to the cruise control example, if the car's control system is very effective (high Kp=10), it means that any difference between the desired speed and its actual speed when settled is only about 9% off from the ideal outcome—the lower the steady-state error, the better the cruise control maintains the desired speed!

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