15.1. Final Value Theorem

In the analysis of systems, particularly in engineering and control theory, it is often useful to know the long-term or steady-state behavior of a system's response. The Final Value Theorem (FVT) is a mathematical tool in the Laplace transform domain that allows us to find this steady-state value without performing the full inverse Laplace transform. This can be especially helpful when analyzing electrical circuits, mechanical systems, and control systems.

The Final Value Theorem (FVT) serves a critical purpose in understanding the behavior of systems over time. When engineers or scientists analyze systems—like electrical circuits or mechanical devices—they often want to know how these systems behave after a long period (the steady state). The FVT provides a shortcut: it allows us to determine this long-term behavior using Laplace transforms without having to compute the entire inverse transform, saving time and effort.

Imagine you're waiting for a pot of water to boil. Instead of standing there the entire time, you set a timer for five minutes and return to check if it's boiling. The FVT is like that timer; it gives us a way to predict the steady state of a system without having to monitor it continuously.

If a function f(t) has a Laplace transform F(s), and the limit lim t→∞ f(t) exists, then the Final Value Theorem states: lim f(t)=lim sF(s) t→∞ s→0

This definition describes the mathematical basis of the Final Value Theorem. To apply the theorem, two conditions must be met: the function must have a Laplace transform (denoted F(s)), and it must stabilize or have a limit as time (t) approaches infinity. The theorem expresses that the final value of f(t) as time approaches infinity can be found instead by evaluating the limit of sF(s) as s approaches 0.

Think of a company's profits over time. If you know the profits have been calculated into a function over some period and you want to know the ultimate profit when everything stabilizes, instead of analyzing all the years, you can just look at a formula (like F(s)) that models these profits. That way, you can predict the final profit more easily.

The Final Value Theorem can only be applied if:

1. All poles of sF(s) lie in the left half of the complex plane, except possibly at the origin.
2. The function f(t) converges to a finite value as t→∞.

If f(t) has oscillatory or diverging behavior, the FVT does not apply.

For the Final Value Theorem to work correctly, certain conditions must be satisfied. Firstly, the poles of the transformed function sF(s) need to be located in the left half of the complex plane, meaning they should not lead to instability in the system. Secondly, the original function f(t) must approach a finite limit. If f(t) oscillates or diverges, we cannot use the FVT, as it would give inaccurate results.

Consider a car's speed as you drive. If the car is speeding (going beyond limits), trying to predict your final position using a model that's unstable (like FVT with a right pole) won't give useful information. You need to ensure that your speed model is steady (left poles) and converges at certain distances for predictions to be valid.

To apply the Final Value Theorem:

1. Find F(s) – the Laplace transform of f(t).
2. Multiply by s to get sF(s).
3. Take the limit as s→0.

Applying the Final Value Theorem involves a straightforward three-step process. First, you need to compute the Laplace transform of your function, yielding F(s). Next, you multiply this transform by s, creating the expression sF(s). Finally, you evaluate the limit of this new expression as s approaches zero, which will provide the steady-state value of the original time function f(t).

Imagine baking a cake. First, you gather your ingredients (finding F(s)). Then, you mix them together (multiply by s). Finally, after letting it bake, you check to see if it's done (taking the limit as s→0). Each of these steps is crucial for achieving your desired result at the end: a delicious cake (the steady-state value).

Example 1: Simple Exponential Decay Let f(t)=1−e−2t Step 1: Find F(s) F(s)=L{1−e−2t }= − 1 s s+2 Step 2: Multiply by s sF(s)=s − =1− s s+2 s+2 Step 3: Take the limit as s→0 lim s →0 1− =1 s+2 ✅ Therefore, lim t→∞f(t)=1 Example 2: Non-Converging Function (Invalid Case) Let f(t)=sin(t) F(s)= 1 s2 +1 s sF(s)= 1 s2 +1 lim s →0 sF(s)=0 But in reality, lim t→∞sin(t) does not exist. Example 3: Rational Function Let f(t)=(5s+3)/(s(s+2)) Then, sF(s)=5s+3/(s+2) lim s→0 5s+3/(s+2)=2.

The examples illustrate how to utilize the Final Value Theorem in practice. In the first example with exponential decay, the process confirms that the function approaches a steady value of 1. The second example showcases a classic invalid application of FVT with a sinusoidal function, demonstrating that its limit does not exist as it oscillates indefinitely. The third example with a rational function shows how FVT can successfully predict values for such functions as well.

Think of the first example as tracking the temperature of a hot cup of coffee cooling down — it steadily approaches room temperature (1 degree). In contrast, the second example is like trying to predict the unpredictable — the endless ups and downs of a roller coaster. Finally, the third example is like measuring how fast a vehicle approaches a steady speed over time from a jittery start.

• Control Systems: Determine steady-state error. • Electrical Circuits: Find final voltage or current without full inverse transform. • Signal Processing: Evaluate the limit of a time-domain signal. • Mechanical Systems: Predict final displacement, velocity, etc.

The Final Value Theorem finds practical use across several fields. In control systems, it helps in assessing how much error may persist in steady-state conditions. In electrical engineering, it aids in determining the final state of electrical parameters like voltage and current without complex calculations. Similarly, in signal processing, it aids in finding the limit of signals over time. In mechanical engineering, it predicts how far an object will move or its speed as it stabilizes.

Consider the FVT in control systems as a GPS recalculating your route to a destination as traffic changes. In electrical circuits, it's akin to reading your smartphone's battery life prediction without having to analyze every little fluctuation in usage.

• FVT is useful only for systems that stabilize over time. • The Initial Value Theorem is a counterpart for t→0+: lim t→0+ f(t)=lims F(s) s→∞. • If poles of sF(s) are in the right half plane or on the imaginary axis (except at 0), then FVT is not valid.

It's crucial to remember that the applicability of the Final Value Theorem is limited to stable systems. If a system does not settle at a definite value, the theorem cannot provide meaningful results. Additionally, there exists an Initial Value Theorem for analyzing behavior as time approaches zero, highlighting the importance of understanding different limits. Lastly, if poles exist in the problematic areas (right half-plane or on the imaginary axis), FVT becomes irrelevant.

Think of a ship navigating a storm. The FVT won't help if the ship keeps drifting unpredictably (unstable system). But if the ship eventually enters calm waters, then FVT gives useful answers about where it will settle. The Initial Value Theorem, on the other hand, is like checking the ship's position right before the storm hit.

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