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Chapter 6: Problem 1

The topological system \(\Sigma\) is locally observable about the state \(x^{0} \in X\) if there exists some neighborhood \(V\) of \(x^{0}\) such that every state in \(V\) different from \(x^{0}\) is distinguishable from \(x^{0}\).

Short Answer

Expert verified
To determine if a state \(x^0\) in a topological system \(\Sigma\) is locally observable, find a neighborhood \(V\) of \(x^0\) and test if every state in \(V\) different from \(x^0\) is distinguishable from \(x^0\). If such a neighborhood exists, \(x^0\) is locally observable; otherwise, it is not.

Step by step solution

01

Define the terminology

Start by understanding the terminologies involved in the problem. A topological system, \(\Sigma\), has a special sort of structure, where states can be distinguishable based on topology. In topology, a neighborhood of a point is a subset of the system that includes an open set containing the point. In this context, if there is a neighborhood, \(V\), of \(x^0\) such that every state in \(V\) different from \(x^0\) can be distinguished from \(x^0\), then that state \(x^0\) is said to be locally observable.
02

Identify the state and its neighborhood

The task is to investigate and confirm if a state \(x^0\) is locally observable in \(\Sigma\). This is done by finding a neighborhood \(V\) of \(x^0\) in the system \(\Sigma\).
03

Test for local observability

For a state \(x^0\) to be locally observable, every other state in its neighborhood \(V\) different from it, should be distinguishable from \(x^0\). Here "distinguishable" means that it is possible to find a way to differentiate or separate each of other states in \(V\) from \(x^0\).
04

Conclude based on the test

If one can find at least one neighborhood of \(x^0\), in which every distinct state is distinguishable from \(x^0\), then \(x^0\) is locally observable. If no such neighborhood can be found, \(x^0\) is not locally observable.

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Key Concepts

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

Topological System
When delving into the world of control systems, understanding a topological system is fundamental. In essence, a topological system, often denoted as \( \Sigma \), is a mathematical framework that describes how different points, or 'states,' are arranged and related to one another. It employs the principles of topology, which is a branch of mathematics focusing on the properties of space that are preserved under continuous transformations, such as stretching or bending, without tearing or gluing.

Topology introduces concepts like open and closed sets, which are crucial in defining the system's structure. An open set, for example, is a collection of points where each point has a 'neighborhood' around it that is also within the set. By incorporating these topological notions, it becomes possible to analyze the system's behavior in terms of continuity and connectivity, thereby providing a robust framework for studying various dynamical aspects of control systems.
State Distinguishability
State distinguishability is a term used to describe whether it's possible to tell two states apart within a topological system. To put it simply, if you have two states, \( x \) and \( y \) within the same system, and there is a way to differentiate \( x \) from \( y \)鈥攆or instance, through their responses to certain inputs or by their effect on the system's output鈥攖hen these states are said to be distinguishable.

This concept is fundamental when considering system observability because it allows us to understand if we can uniquely identify the state of the system based on the available information. In practical terms, if each possible state of the system leads to a unique pattern or signature that can be picked up by sensors or inferred through system outputs, we can say that the states are distinguishable.
Neighborhood in Topology
A neighborhood in topology refers to a specific type of subset around a point in a topological space. More formally, if we have a point \( x^{0} \) in a space \( X \) and there exists an open set \( U \) such that \( x^{0} \) is in \( U \) and \( U \) is contained within the subset \( V \)鈥攖hen this subset \( V \) is called a neighborhood of \( x^{0} \).

In the context of control systems, a neighborhood can be thought of as a little bubble around a state that contains all other states 'close enough' to it, according to the topological definition of closeness. The concept is critical when discussing observability, as it helps in identifying the range within which the system's states have to be distinguishable from one another.
System Observability
System observability is a measure of how well the internal states of a system can be inferred from knowledge of its external outputs. In the realm of control systems, observability is a key criterion for determining whether a system can be effectively controlled and monitored. If the state of a system can be completely reconstructed from its outputs, the system is considered to be observable.

Local observability is a more nuanced version of this concept. It entails verifying that within a particular neighborhood of a given state, all other states are distinguishable from the one in question. This means that within this small subset of the system, you can tell every state apart from the others based on the system's outputs. If even a single neighborhood exists that satisfies this condition, then that state is locally observable. It's a property that is essential for pinpointing and correcting potential issues within a small scope of the system's operation.

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Most popular questions from this chapter

If \(m=1\) or \(p=1\) and \(W_{\mathcal{A}}=P / q\) with \(q\) monic of degree equal to the rank of \(\mathcal{A}\), then \(q\) is the (common) characteristic polynomial of the canonical realizations of \(\mathcal{A}\).

Let \(\omega_{i}, i=1, \ldots, k\) be \(k\) different positive real numbers. Show that there is some continuous-time time-invariant linear system with outputs and no inputs \(\Sigma=(A, 0, C)\) such that: \- \(\Sigma\) is observable, and \- for each set of \(2 k\) real numbers \(a_{i}, \varphi_{i}, i=1, \ldots, k\), there is some initial state \(x\) so that $$ \lambda_{x}^{0, t}=\eta(t)=\sum_{i=1}^{k} a_{i} \sin \left(2 \pi \omega_{i} t+\varphi_{i}\right) $$ for all \(t \geq 0\). Conclude from the above discussion that, if $$ \frac{1}{\delta}>2 \max _{i=1, \ldots, m}\left|\omega_{i}\right| $$ then the complete function \(\eta(t)\) can be recovered from the values $$ \eta(0), \eta(\delta), \eta(2 \delta), \ldots $$ for every set of \(a_{i}\) 's and \(\varphi_{i}\) 's.

The Markov sequence \(\mathcal{A}\) is realizable if and only if there exists a canonical triple realizing it.

Consider the discrete-time time-invariant finite system with $$ x=\\{a, b, c, d\\}, \quad \mathcal{U}=\\{u, v\\}, \quad y=\\{\alpha, \beta\\}, $$ local-in-time transition function \(\mathcal{P}\) given by the following table: \begin{tabular}{|c|c|c|} \hline \(\mathcal{P}\) & \(u\) & \(v\) \\ \hline\(a\) & \(c\) & \(c\) \\ \hline\(b\) & \(c\) & \(b\) \\ \hline\(c\) & \(a\) & \(a\) \\ \hline\(d\) & \(d\) & \(a\) \\ \hline \end{tabular} and output function \(h(a)=h(b)=\alpha\) and \(h(c)=h(d)=\beta\). The pairs $$ \\{a, c\\},\\{a, d\\},\\{b, c\\},\\{b, d\\} $$ are instantaneously distinguishable. To distinguish $$ \\{a, b\\} $$ one may use any sequence starting with \(v\) and to distinguish $$ \\{c, d\\} $$ any sequence starting with \(u\). In summary, every pair of distinct states can be distinguished in time 1 , and the system is observable in time \(1 .\) There is however no single "universal" sequence \(\omega \in \mathcal{U}^{[\sigma, \tau)}\) so that for every pair \(x \neq z\) it would hold that $$ \lambda_{x}^{\sigma, t}\left(\left.\omega\right|_{[\sigma, t)}\right) \neq \lambda_{z}^{\sigma, t}\left(\left.\omega\right|_{[\sigma, t)}\right) $$ for some \(t\) (that is allowed to depend on \(x\) and \(z\) ). This is because if \(\omega\) starts with \(u\), then all outputs when starting at \(a\) or \(b\) coincide, since the state trajectories coincide after the first instant, but if \(\omega\) starts with \(v\), then \(\omega\) cannot separate \(c\) from \(d\).

We consider again the parity check example discussed in Example 2.3.3. In particular, we shall see how to prove, using the above results, the last two claims in Exercise 2.3.4. The behavior to be realized is \(\lambda(\tau, 0, \omega)=\) $$ \begin{cases}1 & \text { if } \omega(\tau-3)+\omega(\tau-2)+\omega(\tau-1) \text { is odd and } 3 \text { divides } \tau>0 \\ 0 & \text { otherwise }\end{cases} $$ and we take the system with $$ x:=\\{0,1,2\\} \times\\{0,1\\} $$ and transitions $$ \mathcal{P}((i, j), l):=(i+1 \bmod 3, j+l \bmod 2) $$ for \(i=1,2\) and $$ \mathcal{P}((0, j), l):=(1, l) \text {. } $$ The initial state is taken to be \((0,0)\), and the output map has \(h(i, j)=1\) if \(i=0\) and \(j=1\) and zero otherwise. (The interpretation is that \((k, 0)\) stands for the state " \(t\) is of the form \(3 s+k\) and the sum until now is even," while states of the type \((k, 1)\) correspond to odd sums.) This is clearly a realization, with 6 states. To prove that there is no possible (time-invariant, complete) realization with less states, it is sufficient to show that it is reachable and observable. Reachability follows from the fact that any state of the form \((0, j)\) can be obtained with an input sequence \(j 00\), while states of the type \((1, j)\) are reached from \(x^{0}\) using input \(j\) (of length one) and states \((2, j)\) using input \(j 0\). Observability can be shown through consideration of the following controls \(\omega_{i j}\), for each \((i, j):\) $$ \omega_{01}:=0, \omega_{00}:=100, \omega_{10}:=10, \omega_{11}:=00, \omega_{21}:=0, \omega_{20}:=0 . $$ Then, \(\omega_{01}\) separates \((0,1)\) from every other state, while for all other pairs \((i, j) \neq\) \((0,1)\), $$ \lambda_{(i, j)}\left(\omega_{\alpha \beta}\right)=1 $$ if and only if \((i, j)=(\alpha, \beta)\).
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