6.3.2. Error Constants

Error Constants:

• Position Error Constant KpK_p: Determines the steady-state error for a step input. Kp=lim s→0G(s)H(s)K_p = \lim_{s \to 0} G(s)H(s)

• Velocity Error Constant KvK_v: Determines the steady-state error for a ramp input. Kv=lim s→0s⋅G(s)H(s)K_v = \lim_{s \to 0} s \cdot G(s)H(s)

• Acceleration Error Constant KaK_a: Determines the steady-state error for a parabolic input. Ka=lim s→0s2⋅G(s)H(s)K_a = \lim_{s \to 0} s^2 \cdot G(s)H(s)

Error constants are crucial for determining the steady-state error of a control system in response to different types of inputs. Each constant corresponds to a specific type of input:

1. Position Error Constant (Kp) helps us understand the error when a step input is applied to the system. Its calculation involves evaluating the transfer function when the complex variable s approaches zero.
2. Velocity Error Constant (Kv) is used for ramp inputs, showing us the steady-state error associated with such inputs. It also involves a limit as s approaches zero but requires multiplying the transfer function by s.
3. Acceleration Error Constant (Ka) relates to parabolic inputs and is calculated similarly to Kv but involves the square of s. Thus, these constants allow engineers to assess how well the system can adapt to various input scenarios.

Think of managing traffic at a signalized intersection. When a sudden rush of cars (step input) occurs, Kp helps determine how many cars are caught at the red light versus how many pass through. For continuous traffic flow (ramp input), Kv provides insight into the steady-state error as the flow increases, while Ka helps understand the impact of sudden surges, like a parade. These constants are like having tools that measure performance under different traffic conditions.

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 depending on the type of input the system receives. The relationships are as follows:

1. Step Input: The steady-state error is directly related to the Position Error Constant (Kp). A higher Kp generally indicates a lower steady-state error, meaning the system performs better.
2. Ramp Input: The steady-state error for this input relates to the Velocity Error Constant (Kv). Again, a higher Kv indicates better system performance for steady-state accuracy.
3. Parabolic Input: The steady-state error for a parabolic input is related to the Acceleration Error Constant (Ka). A higher Ka suggests that the system can better handle changing rates, leading to a lower steady-state error.

Imagine a thermostat in a room. If you set a specific temperature (step input), Kp indicates how close the room temperature can get to that set point. For a heater that adjusts its output in response to continuously running machinery (ramp input), Kv helps measure how accurately the heater can keep up with temperature changes. Lastly, if you have an oven that needs to respond quickly to rapid temperature increases (parabolic input), Ka helps determine how well the oven can adjust without lagging. These constants play a pivotal role in different scenarios, ensuring your appliances work efficiently.

Steady-State Error Formulae:

• Step Input: For a system with a transfer function G(s), the steady-state error for a step input R(s)=1sR(s) = \frac{1}{s} is given by: ess=11+Kpe_{ss} = \frac{1}{1 + K_p}

• Ramp Input: The steady-state error for a ramp input R(s)=1s2R(s) = \frac{1}{s^2} is given by: ess=1Kve_{ss} = \frac{1}{K_v}

• Parabolic Input: The steady-state error for a parabolic input R(s)=1s3R(s) = \frac{1}{s^3} is given by: ess=1Kae_{ss} = \frac{1}{K_a}

The steady-state error can be calculated using specific formulae corresponding to different input types.

1. For a step input, the steady-state error formula is derived as ess = 1 / (1 + Kp). This indicates that as Kp increases, the error decreases.
2. For a ramp input, the steady-state error is expressed as ess = 1 / Kv. Here, a higher Kv means a lower steady-state error as well.
3. For a parabolic input, the formula ess = 1 / Ka shows that a larger Ka will also lead to a smaller error. These formulae are essential tools for engineers to quantify how much the system deviates from the desired level under various conditions.

Consider a car's cruise control system. When you set your speed (step input), the formula ess = 1 / (1 + Kp) helps predict how close your actual speed will be to the target. If you're accelerating smoothly (ramp input), Kv guides how accurately you can keep up with speed changes, measured by ess = 1 / Kv. And if you need to change speed very rapidly (parabolic input), Ka reflects how well the cruise control responds, summarized by ess = 1 / Ka. These formulas ensure that the system can evaluate its performance against what is expected.

Example:

For a system with Kp=10K_p = 10, the steady-state error for a step input is: ess=11+10=0.09e_{ss} = \frac{1}{1 + 10} = 0.09

In this example, we are given a Position Error Constant (Kp) of 10. To find the steady-state error for a step input, we apply the formula for steady-state error: ess = 1 / (1 + Kp) Plugging in the value of Kp: ess = 1 / (1 + 10) ess = 1 / 11 This evaluates to approximately 0.09. This means that when a step input is applied, the system will have a steady-state error of around 9% of the input value, indicating it is relatively effective at reaching its target but not perfectly accurate.

Consider a person trying to hit a target with a dart. If the person's aim is off by 9% (like the steady-state error of 0.09), they can adjust their technique to improve accuracy. Here, Kp of 10 represents the person's skill level; a higher value indicates a better aim, while a lower value would mean greater difficulty in hitting the target. Just like practicing dart throwing can lead to improved aim, adjusting system parameters can help reduce steady-state error.

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