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Chapter 7: Problem 3

If \(\Sigma\) is internally stable, then \(\Lambda_{\Sigma}\) is UBIBO.

Short Answer

Expert verified
Given the internally stable system \(\Sigma\), the interconnected system \(\Lambda_{\Sigma}\) is formed by connecting an input-output map (\(K\)) to it. Since \(\Sigma\) is internally stable, for a given bounded input, the internal states, inputs, and outputs are all bounded. This implies that the exogenous input to the interconnected system \(\Lambda_{\Sigma}\) is bounded, which in turn results in a bounded regulated output. As this property holds uniformly with respect to initial conditions, \(\Lambda_{\Sigma}\) is UBIBO stable.

Step by step solution

01

Define internal stability and UBIBO stability

Before we prove the relationship between internal stability and UBIBO stability, let's define these concepts. - Internal stability: A dynamic system \(\Sigma\) is said to be internally stable if all the trajectories of its internal states, inputs, and outputs are bounded for any initial condition and any bounded input. - UBIBO stability: A dynamic system has bounded-input-bounded-output (BIBO) stability if every bounded input signal results in a bounded output signal. If this property holds uniformly with respect to initial conditions, the system is uniformly bounded-input-bounded-output (UBIBO) stable.
02

Define the interconnected system \(\Lambda_{\Sigma}\)

Now, let's define the interconnected system \(\Lambda_{\Sigma}\) and its relation to the original system \(\Sigma\). The interconnected system \(\Lambda_{\Sigma}\) is formed by connecting an input-output map to the given system \(\Sigma\). The input of this interconnected system \(\Lambda_{\Sigma}\) is the exogenous input and its output is the regulated output. Let the input-output map of the interconnected system \(\Lambda_{\Sigma}\) be represented by the matrix \(K\). The interconnected system \(\Lambda_{\Sigma}\) is given as: \[ \Lambda_{\Sigma} = \Sigma + K \]
03

Show the relation between internal stability and UBIBO stability for \(\Lambda_{\Sigma}\)

Given that the system \(\Sigma\) is internally stable, we need to show that the interconnected system \(\Lambda_{\Sigma}\) is UBIBO stable. The main idea here is that the internal stability of \(\Sigma\) implies that for a given bounded input to the system, the internal states, inputs, and output are all bounded. This means that the interconnected system's exogenous input will be bounded as well, leading to a bounded regulated output. Now, let's consider a bounded input \(u(t)\) to the interconnected system \(\Lambda_{\Sigma}\). Since \(\Sigma\) is internally stable, this bounded input will result in bounded trajectories for its internal states, inputs, and outputs. We know that the interconnected system is represented as \(\Lambda_{\Sigma} = \Sigma + K\). If the bounded input to the interconnected system leads to bounded internal states, inputs, and outputs in \(\Sigma\), then we can infer that it will also lead to bounded regulated output in \(\Lambda_{\Sigma}\), as the matrix \(K\) represents a constant input-output map. Thus, we have shown that given a bounded input to the interconnected system \(\Lambda_{\Sigma}\), the output will be bounded as well. Since this property holds uniformly with respect to initial conditions, we can conclude that \(\Lambda_{\Sigma}\) is UBIBO stable.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

UBIBO Stability
Understanding the concept of UBIBO (Uniformly Bounded-Input-Bounded-Output) stability is crucial for students dealing with dynamic systems. This concept ensures that a system will not react excessively to any given bounded external disturbance.

In essence, UBIBO stability refers to a condition where, regardless of the initial state of the system, if you introduce an input that is limited in magnitude (bounded), the output will also be restricted in magnitude (bounded). It's like saying, no matter how generous your ingredients are, if you're cooking in a pot of a certain size, your soup won't overflow.

Within the context of the exercise, demonstrating that an internally stable system \( \Sigma \) leads to a UBIBO stable interconnected system \( \Lambda_{\Sigma} \) is akin to proving that our well-behaved 'soup' (the system output) remains tame, even as we integrate additional components (the interconnected system \( \Lambda_{\Sigma} \) with input-output map represented by matrix \( K \) ). This assurance is valuable in fields like engineering and control systems where predictability and safe operation are paramount.
Bounded-Input-Bounded-Output Stability
Bounded-Input-Bounded-Output (BIBO) stability is a key property of dynamic systems that ensures predictability and safe operation. At its core, BIBO stability means that if you keep your inputs within a certain range, the outputs will also stay within a set range, no matter what specific inputs you're using.

This property becomes especially important when dealing with real-world systems where you can only allow so much variability before things go awry. For instance, consider an autopilot system. You'd want to be assured that no matter the conditions 鈥� a gentle breeze or strong gusts 鈥� the plane's response remains within operational limits, ensuring safety. BIBO stability guarantees such outcomes.

In dynamic systems, achieving BIBO stability is a bit like making sure that every jump on a trampoline results in a controlled bounce back 鈥� no surprises or unintended somersaults. Students require a solid grasp of this concept as it's foundational to designing systems that respond predictably to inputs, which is often critical to their functional and safe operation.
Dynamic Systems
The study of dynamic systems forms the bedrock of many engineering and science disciplines. A dynamic system is essentially any system that evolves over time based on a set of rules 鈥� for instance, an ecosystem, a climate model, or a spaceship's flight control system.

The fascinating part about dynamic systems is their behavior in response to different types of inputs. It鈥檚 similar to observing how adding different ingredients to a recipe can change the flavor of the dish. With the right understanding, one can predict how the system will evolve over time, manage it, or even control it to achieve desired outcomes.

In the context of our exercise, internal stability is akin to having a trusted recipe that ensures the dish turns out right every time, regardless of slight variations in ingredient amounts or temperatures. When students master the concept of dynamic systems, they are essentially learning how to 'cook' with precision in a complex world of systems that are continuously changing and interacting with each other.

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Most popular questions from this chapter

Let \(\Sigma_{1}\) and \(\Sigma_{2}\) be two systems, and let $$ k_{i}: y_{1} \times y_{2} \rightarrow \mathcal{U}_{i}, \quad i=1,2 $$ be two maps. We say that the interconnection of \(\Sigma_{1}\) and \(\Sigma_{2}\) through \(k_{1}\) and \(k_{2}\) is well-posed if for each \(\left(x_{1}, x_{2}\right) \in X_{1} \times \mathcal{X}_{2}\) there exist unique functions $$ \xi_{i}:[0, \infty) \rightarrow X_{i}, \quad i=1,2 $$ such that, with the notations $$ \begin{aligned} \eta_{i}(t) &:=h_{i}\left(\xi_{i}(t)\right) \\ \omega_{i}(t) &:=k_{i}\left(\eta_{1}(t), \eta_{2}(t)\right) \end{aligned} $$ it holds that \(\xi_{i}=\psi_{i}\left(x_{i}, \omega_{i}\right)\) for \(i=1,2\).

Denote by \(\mathcal{K}\) the class of functions \([0, \infty) \rightarrow[0, \infty)\) which are zero at zero, strictly increasing, and continuous, by \(\mathcal{K}_{\infty}\) the set of unbounded \(\mathcal{K}\) functions, and by \(\mathcal{K} \mathcal{L}\) the class of functions \([0, \infty)^{2} \rightarrow[0, \infty)\) which are of class \(\mathcal{K}\) on the first argument and decrease to zero on the second argument. A continuous-time time-invariant system \(\dot{x}=f(x, u)\) with state space \(X=\mathbb{R}^{n}\) and control-value space \(\mathcal{U}=\mathbb{R}^{m}\) is said to be input-to-state stable (ISS) if there exist a function \(\beta \in \mathcal{K} \mathcal{L}\) and a function \(\gamma \in \mathcal{K}_{\infty}\) so that the following property holds: For each \(T>0\), if \(\omega\) is a control on the interval \([0, T]\) and \(\xi\) is a solution of \(\dot{\xi}=f(\xi, \omega)\) on the interval \([0, T]\), then $$ \|\xi(T)\| \leq \max \left\\{\beta(\|\xi(0)\|, T), \gamma\left(\|\omega\|_{\infty}\right)\right\\} $$

For linear time-invariant continuous-time or discrete-time systems, \((A, C)\) is asymptotically observable if and only if \(\left(A^{\prime}, C^{\prime}\right)\) is asymptotically controllable.

When the system \(\Sigma\) is discrete-time and observable, one may pick a matrix \(L\) so that \(A+L C\) is nilpotent, by the Pole-Shifting Theorem. Thus, in that case a deadbeat observer results: The estimate becomes exactly equal to the state after \(n\) steps.

If \(w\) is stable, then for each \(k \neq 0\) such that \(\frac{1}{k} \notin \Gamma\), the closed-loop system with \(v=k\) is stable if and only if \(\mathrm{cw}\left(\Gamma, \frac{1}{k}\right)=0\).
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