Understanding Peak Amplitude

Hello! This is the final lesson in our module on the paper's core argument. Let's get started.

Introduction

In our last lesson, we reached the paper's central conclusion, encapsulated in Theorem 1. We saw that for any fixed time t, the maximum possible amplitude |x_t| is achieved by a very specific difference equation: one whose characteristic roots are all identical and lie on the boundary of the allowed disk, i.e., the characteristic polynomial is f(z) = (z-r)^n.

This lesson builds directly on that result. We will shift our focus from the maximum amplitude at a specific moment t to the maximum peak amplitude, which is the maximum amplitude achieved over all time t ≥ n. This is a subtle but important distinction, moving from a snapshot to the entire history of the system's evolution.

Lesson Goal: By the end of this lesson, you will be able to interpret the paper's result for the peak amplitude sup_{t≥n} |x_t|.

1. From Fixed-Time Maximum to Peak Maximum

First, let's clarify the quantity we are trying to maximize. The paper defines it precisely.

To recap the notation for our simple case where the roots are in the disk D(r) and initial conditions are in the unit disk D_n:

• Maximum amplitude at time t: $$M_t(D(r)|D_n)=\underset{f\in {\mathcal{P}}_{n,D(r)}}{\sup }\underset{x\in {\mathcal{S}}_f,|x_k|\le 1}{\sup }|x_t|$$
• Maximum peak amplitude: $$M_{\text{peak}}(D(r)|D_n)=\underset{f\in {\mathcal{P}}_{n,D(r)}}{\sup }\underset{x\in {\mathcal{S}}_f,|x_k|\le 1}{\sup }\left(\underset{t\ge n}{\sup }|x_t|\right)$$

The order of the sup operations is crucial. At first glance, finding M_peak seems much harder than finding M_t. It's possible that the worst-case equation f could be different for each time t, requiring a much more complex optimization.

However, Theorem 1 provides a powerful simplification. Let's see how.

The key insight is that Theorem 1 shows that the same characteristic polynomial, f(z) = (z-r)^n, maximizes |x_t| for any t ≥ n. The "worst-case" system is the same at all times.

This means we can swap the order of the sup operations:

$$M_{\text{peak}}(D(r)|D_n)=\underset{t\ge n}{\sup }\left(\underset{f\in {\mathcal{P}}_{n,D(r)}}{\sup }\underset{x\in {\mathcal{S}}_f,|x_k|\le 1}{\sup }|x_t|\right)=\underset{t\ge n}{\sup }{M}_t(D(r)|D_n)$$

And since we know M_t(D(r)|D_n) is achieved by f(z) = (z-r)^n, the problem reduces to:

Find the peak amplitude for the single difference equation whose characteristic polynomial is f(z) = (z-r)^n.

This transforms the problem from an optimization over an infinite-dimensional space of functions to analyzing the behavior of a single, well-defined system.

2. Analyzing the Peak Amplitude of the Worst-Case System

Now, let's analyze the behavior of solutions for the characteristic polynomial f(z) = (z-r)^n. As we saw in a previous lesson, when roots have multiplicity n, the general solution is not just c r^t, but involves a polynomial in t:

$$x_t=P_{n-1}(t)\cdot r^t$$

where P_{n-1}(t) is a polynomial of degree n-1 whose coefficients are determined by the initial conditions.

The long-term behavior of |x_t| depends critically on the value of r.

• Case 1: r > 1 (Unstable System)
  The exponential term r^t grows without bound. The amplitude |x_t| will go to infinity. Therefore, M_peak = ∞.

• Case 2: r = 1 (Marginally Stable System)
  The solution is x_t = P_{n-1}(t). Unless n=1 (in which case x_t is constant), the polynomial P_{n-1}(t) will grow without bound as t → ∞. Therefore, M_peak = ∞.

• Case 3: r < 1 (Asymptotically Stable System)
  Here, we have a competition between the polynomial growth of P_{n-1}(t) and the exponential decay of r^t. As you know from your physics background, the exponential term always wins. The amplitude |x_t| will eventually decay to zero. However, it may exhibit a transient growth, an "overshoot," before decaying. This means the peak amplitude M_peak will be finite and will occur at some finite time t_peak ≥ n.

The paper confirms this analysis in the same paragraph we just read, stating: "the peak amplitude is infinite if max_k r_k ≥ 1".

This is the core interpretation: the problem of finding the maximum peak amplitude across all stable systems reduces to analyzing a single system, which has a finite peak amplitude if and only if the system is strictly asymptotically stable (r < 1).

3. The Finite Peak in Asymptotically Stable Systems (r < 1)

Let's look more closely at the interesting case where r < 1. We have reduced the problem to finding sup_{t \ge n} M_t((z-r)^n | D_n).

The paper gives us an explicit formula for M_t for this specific system in Proposition 2.1.

The paper also provides a useful asymptotic estimate for this quantity in Section 2.5, which is easier to analyze.

The estimate given in equation (16) is:

$$M_t((z-r{)}^n|D_n)\approx \left(\genfrac{}{}{0ex}{}{t}{n-1}\right)r^{t-n+1}$$

For large t, the binomial coefficient \binom{t}{n-1} behaves like t^{n-1} / (n-1)!. So, the amplitude behaves like:

$$M_t\propto t^{n-1}{r}^t$$

This functional form is common in the study of damped oscillators and other transient phenomena in physics. To find the time t_peak at which this function reaches its maximum, we can treat t as a continuous variable and set the derivative to zero. Let g(t) = t^{n-1} r^t.

$$\frac{dg}{dt}=(n-1)t^{n-2}{r}^t+t^{n-1}(\ln r)r^t=0$$

Dividing by t^{n-2}r^t (for t>0), we get:

$$(n-1)+t\ln r=0$$ $$t_{\text{peak}}=-\frac{n-1}{\ln r}$$

Since r < 1, ln r is negative, so t_peak is positive. This confirms that the peak occurs at a finite, predictable time that depends on the order of the system n and the radius of the stability disk r. The closer r is to 1, the smaller |\ln r| is, and the later the peak occurs.

Conclusion

This lesson completes our analysis of the paper's main result by extending it from a fixed-time maximum to the overall peak maximum.

Key Takeaways:

• Reduction of Complexity: Theorem 1 is powerful because it proves the same system (f(z)=(z-r)^n) is the "worst-case" for all t. This allows the problem of finding the maximum peak amplitude over all systems to be reduced to analyzing the peak amplitude of just this one system.
• Dependence on Stability (r): The interpretation of the peak amplitude depends entirely on r.
  • If r ≥ 1 (marginally stable or unstable), the maximum peak amplitude is infinite.
  • If r < 1 (asymptotically stable), the maximum peak amplitude is finite.
• Transient Growth: For r < 1, the worst-case system exhibits transient growth, with the amplitude peaking at a finite time t_peak ≈ -(n-1)/ln(r) before decaying exponentially.

You now have a solid grasp of the paper's core problem, its algebraic methodology, and its main conclusion regarding both fixed-time and peak amplitude maximization.

Next Lesson Preview:

We have focused on the "vanilla" problem where the characteristic roots are constrained to a single disk D(r). The paper's results are actually more general and apply to polydiscs, where each root z_k is constrained to its own disk D(r_k). In the next and final module, we will briefly explore this generalization.