/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 Assume that \((A, B)\) is contro... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 5

Assume that \((A, B)\) is controllable and \(\mathcal{U} \subseteq \mathbb{R}^{m}\) is a neighborhood of 0 . Then \(J_{k}^{\mathrm{R}} \subseteq \mathcal{R}_{u}(0)\) for all \(k\).

Short Answer

Expert verified
The pair \((A, B)\) in a Linear Time-Invariant (LTI) system is given to be controllable if it can be transferred from any initial state to any final state in finite time through suitable control inputs. Under the given conditions including a neighborhood \(\mathcal{U}\) of 0, we define reachability as the collective states that can be accessed from a state \(x_0\) in a finite duration using the control inputs from the set \(\mathcal{U}\). Consequently, the mention \(J_{k}^{R} \subseteq \mathcal{R}_{u}(0)\) implies that all states accessible from the zero state after \(k\) steps are contained in the reachability set originating from the zero state. In a controllable system, any state can be accessed from any other state in finite time, confirming the condition \(J_{k}^{R} \subseteq \mathcal{R}_{u}(0)\) for all \(k\). Therefore, the given statement is validated by the definitions and connection between controllability and reachability in an LTI system, denoting that \(J_{k}^{R}\) will indeed be a subset of \(\mathcal{R}_{u}(0)\) for all \(k\) when the system is controllable and the set of possible control inputs approximates 0.

Step by step solution

01

Define Variables and Controllability

Define the variables: - \( (A, B) \): These are matrices that represent a linear time-invariant (LTI) system. In our context, \(A\) is the state matrix and \(B\) is the input matrix. - \(\mathcal{U}\): This is a neighborhood of 0 in \( \mathbb{R}^{m} \), representing possible control inputs to the system. - \( J_{k}^{R} \): This is a set of states that the system can reach after \(k\) steps from the zero state, given feasible control signals in neighborhood \(\mathcal{U}\). Define "controllability": The pair \( (A, B) \) is controllable if and only if it is possible to transfer any initial state to any other target state in finite time via appropriate control inputs.
02

Define Reachability

Define "reachability": The reachability set from a state \(x_0\), denoted \(\mathcal{R}_{u}(x_0)\), contains all the states that can be reached from \(x_0\) in a finite amount of time, using control inputs from the set \(\mathcal{U}\).
03

Understand Given Statement

Given that \(J_{k}^{R} \subseteq \mathcal{R}_{u}(0)\) for all \(k\), it means that all states reachable from the zero state after \(k\) time steps (via controls in \(\mathcal{U}\)) are contained in the reachability set from the zero state.
04

The Connection Between Controllability and Reachability

In a controllable system, the reachability set from the zero state (using controls from neighbor \(\mathcal{U}\)) contains all the states reachable from the zero state after any number of steps (also using controls from neighbor \(\mathcal{U}\)). Therefore, if a system is controllable, then any state is reachable from any other state in a finite time. This fulfills the condition that \(J_{k}^{R} \subseteq \mathcal{R}_{u}(0)\) for all \(k\), because every state reachable in \(k\) steps from the zero state is contained in the general reachability set from the zero state.
05

Conclude About Given Statement

Given the definitions of controllability and reachability, and the connections between them, we can confirm that if a system is controllable, and a set of possible control inputs is a neighbor of 0, then \(J_{k}^{R} \subseteq \mathcal{R}_{u}(0)\) for all \(k\). This confirms the given statement.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Linear Time-Invariant (LTI) Systems
To start with, let's talk about linear time-invariant (LTI) systems, a fundamental concept in control theory. These are mathematical models used to represent dynamic systems that are both linear and time-invariant. But what does that mean?

In essence, linearity implies that the response of the system to an input is directly proportional to the amplitude of the input signal. For instance, if you double the input, the output will also double. Now, let's pair this with the notion of time-invariance. A system is time-invariant if its behavior and structure do not change over time. This means that an input signal will produce the same output regardless of when you apply it.

How Do LTI Systems Function?

Imagine driving a car straight with a constant force applied to the pedals; the car's speed increases at a constant rate, showcasing linearity. Moreover, the car will respond this way whether you do it now or an hour later (time-invariance). In LTI systems, equations characterizing the system's dynamics do not include terms that change with time or non-linear terms like the product or exponent of state variables.
Reachability Set
Next up is the reachability set, which is closely linked with the objective of controlling the system. The reachability set denotes the collection of all states that a system can reach, starting from a specific initial state, by using all possible control inputs within given constraints.

For a system described by matrices, it is the set of all potential ending points you can achieve by manipulating your inputs. To illustrate, think of a video game character who can move in a space based on your inputs; every possible point the character can reach by any combination of movements defines the reachability set in the game's universe.

The importance of understanding the reachability set is paramount, as it is directly related to the extent to which you can control the system. If the set includes all possible states, it means you have full control to maneuver the system as desired. Otherwise, there are limits to your influence, and consequently, barriers to achieving certain system states.
State Matrix and Input Matrix
And finally, let's delve into the state matrix and input matrix, the backbone of any LTI system's mathematical model. The state matrix, often denoted as A, represents how the system's current state affects its next state. Meanwhile, the input matrix, represented by B, describes how control inputs influence the state of the system.

Interpreting the Matrices

The state matrix A tells us how the system evolves from one moment to the next without any external input, similar to the natural drift of a boat in the water. In contrast, the input matrix B is like the oars of the boat, dictating how the system responds to control efforts, such as propelling the boat in a specific direction.

To make it intuitive, consider playing with a remote-controlled car. The car moves according to its current speed and direction - that's the state matrix at work. When you use the remote to steer or change speed, you're applying inputs that alter its state, and this interaction is captured by the input matrix. Understanding these matrices is crucial as they enable us to predict and ultimately control the system's future behavior.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

}\( corresponding to \)\sigma=0, \tau=\varepsilon\( sat鈥� # For the system $$ \begin{aligned} &\dot{x}_{1}=x_{2} \\ &\dot{x}_{2}=u \end{aligned} $$ show that the operator \)L^{\\#}\( corresponding to \)\sigma=0, \tau=\varepsilon\( satisfies $$ \left\|L^{\\#}\right\|=O\left(\varepsilon^{-\frac{3}{2}}\right) $$ as \)\varepsilon \rightarrow 0\(. (Hint: Use (3.21) by showing that the smallest eigenvalue of \)W(0, \varepsilon)\( is of the form $$ \frac{\varepsilon^{3}}{12}+o\left(\varepsilon^{3}\right) $$ The power series expansion for \)\sqrt{1+\alpha}$ may be useful here.) We may restate condition (3) in a slightly different form, which will be useful for our later study of observability. Transposing the conclusion (d) of Lemma C.4.1, one has the following:

Show by counterexample that without analyticity the rank condition is not necessary. More precisely, give two examples as follows: (i) A smooth system with \(n=m=1\) that is controllable in some nontrivial interval but for which there is some \(t_{0}\) so that \(M^{(k)}\left(t_{0}\right) \equiv 0\) for all \(k\); and (ii) A smooth system with \(n=m=2\) that is controllable in some nontrivial interval but for which the rank is one for every \(t_{0}\) and \(k\).

Assume that \((A, B)\) is controllable and \(\mathcal{U} \subseteq \mathbb{R}^{m}\) is a convex and bounded neighborhood of 0 . Then there exists a set \(\mathcal{B}\) such that \(\mathcal{R}_{u}(0)=\) \(\mathcal{B}+L\) and \(\mathcal{B}\) is bounded, convex, and open relative to \(M\).

Consider the system with \(\mathbb{K}=\mathbb{R}, n=3, m=1\), and matrices $$ A(t)=\left(\begin{array}{ccc} t & 1 & 0 \\ 0 & t^{3} & 0 \\ 0 & 0 & t^{2} \end{array}\right) \quad B(t)=\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right) $$ This system is smoothly (in fact, analytically) varying on \((-\infty, \infty)\). Since $$ \left[B_{0}(0), B_{1}(0), B_{2}(0), B_{3}(0)\right]=\left(\begin{array}{cccc} 0 & 1 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right), $$ and this matrix has rank 3 , the system is controllable on every nontrivial interval \([\sigma, \tau]\).

Let \(\Sigma\) be a continuous-time linear system, and pick \(\sigma<\tau \in \mathbb{R}\). Consider the controllability Gramian $$ W_{c}(\sigma, \tau):=\int_{\sigma}^{\tau} \Phi(\sigma, s) B(s) B(s)^{*} \Phi(\sigma, s)^{*} d s . $$ Show: \(\Sigma\) is controllable in \([\sigma, \tau]\) if and only if \(W_{c}(\sigma, \tau)\) has rank \(n\), and, in that case, the unique control of minimum square norm that steers \(x\) to 0 is given by the formula \(\omega(t)=-B(t)^{*} \Phi(\sigma, t)^{*} W_{c}(\sigma, \tau)^{-1} x\).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.