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Chapter 3: Problem 2

If \(\Sigma\) is linear, then \(\mathcal{R}^{T}(0)\) is a subspace, for all \(T \in \mathcal{T}\). If \(\Sigma\) is linear and \(x\) is now an arbitrary state, the equality $$ \phi(T, 0, x, \omega)=\phi(T, 0, x, 0)+\phi(T, 0,0, \omega) $$ shows that $$ \mathcal{R}^{T}(x)=\phi(T, 0, x, \mathbf{0})+\mathcal{R}^{T}(0) $$ and hence that \(\mathcal{R}^{T}(x)\) is a linear submanifold (translation of a subspace) in \(X\).

Short Answer

Expert verified
In summary, given a linear system \(\Sigma\), we analyzed the function \(\phi(T, 0, x, \omega)\) and showed that \(\mathcal{R}^{T}(0)\) is a subspace, since it is closed under addition and scalar multiplication. Moreover, we demonstrated that \(\mathcal{R}^{T}(x)\) is a linear submanifold in \(X\) since it can be written as the translation of a subspace.

Step by step solution

01

Analyze the Linear System

In this given exercise, \(\Sigma\) is a linear system, which means that the output (\(\phi(T, 0, x, \omega)\)) will always follow the principle of superposition. That is, if you have two different inputs, the system's response to the sum of those inputs is equal to the sum of the responses to their individual inputs.
02

Break Down the equality

First, we are given the equality: \[ \phi(T, 0, x,\omega) = \phi(T, 0, x, 0) + \phi(T, 0, 0, \omega) \] This equation implies that the response of the system when the state is \(x\) and the input is \(\omega\) can be decomposed into two separate components: one where the response only relies on the state \(x\) and the input is 0, and the other when the state is 0 and the response relies only on the input \(\omega\). Notice that this is true for all \(T \in \mathcal{T}\).
03

Prove that \(\mathcal{R}^{T}(0)\) is a subspace

Consider the following equation that represents the reachability set at time \(T\) and state \(0\): \[ \mathcal{R}^{T}(0) = \{ \phi(T, 0, 0, \omega) : \omega \in \Omega\} \] Since \(\Sigma\) is linear and the equation \(\phi(T, 0, x,\omega) = \phi(T, 0, x, 0) + \phi(T, 0, 0, \omega)\) holds, it follows that: \[ \phi(T, 0, x, \omega_1+\omega_2) = \phi(T, 0, x, \omega_1) + \phi(T, 0, x, \omega_2) \] for any \(\omega_1, \omega_2 \in \Omega\). This shows that \(\mathcal{R}^{T}(0)\) is closed under addition, which is a required property for a subspace. Additionally, since \(\Sigma\) is linear, scalar multiplication is also preserved, which is the other necessary property for a subspace.
04

Prove that \(\mathcal{R}^{T}(x)\) is a linear submanifold

Now, we have the equation related to the reachability set at time \(T\) and state \(x\): \[ \mathcal{R}^{T}(x) = \phi(T, 0, x, \mathbf{0}) + \mathcal{R}^{T}(0) \] From Step 3, we have shown that \(\mathcal{R}^{T}(0)\) is a subspace. It follows then that \(\mathcal{R}^{T}(x)\) can be written as the translation of a subspace, given by \(\phi(T, 0, x, \mathbf{0}) + \mathcal{R}^{T}(0)\). Thus, \(\mathcal{R}^{T}(x)\) is a linear submanifold in \(X\).

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Key Concepts

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

Superposition Principle
The superposition principle is a fundamental concept in linear systems. It states that the response of a linear system to a combination of inputs is simply the combination of the responses to each individual input. This principle can be expressed mathematically as:
\[ \phi(T, 0, x, \omega_1 + \omega_2) = \phi(T, 0, x, \omega_1) + \phi(T, 0, x, \omega_2) \]
This means if you apply two different inputs, \( \omega_1 \) and \( \omega_2 \), the system's total response is the sum of its response to each input independently.
  • This principle is highly relevant in fields like physics and engineering, where systems are often linear.
  • It simplifies analysis and design of systems by breaking down complex actions into simpler ones.
By adhering to superposition, the principle also ensures predictability and consistency in how systems behave under multiple stimuli.
Reachability Set
The reachability set, denoted \( \mathcal{R}^{T}(x) \), is a critical concept when dealing with linear systems. For any time \( T \) and state \( x \), this set comprises all possible states the system can reach, starting from state \( x \), when subjected to inputs from the set \( \omega \).Interestingly, for a linear system, the reachability set for initial state 0, \( \mathcal{R}^{T}(0) \), is a subspace. This is because it contains all the possible end states the system can achieve starting from 0 with every possible input sequence.
The equation:
\[ \mathcal{R}^{T}(0) = \{ \phi(T, 0, 0, \omega) : \omega \in \Omega \} \]
shows that this set embraces addition and scalar multiplication, key subspace characteristics.
  • It is an essential tool for analyzing the accessibility and control of states within a linear dynamical system.
  • Helps to determine whether a system can be controlled to reach a desired state from a given initial state.
Linear Manifold
A linear manifold is related to linear systems and is formed by translating a subspace. In the context of reachability for linear systems, \( \mathcal{R}^{T}(x) \) can be considered a linear manifold.
The equation:
\[ \mathcal{R}^{T}(x) = \phi(T, 0, x, \mathbf{0}) + \mathcal{R}^{T}(0) \]
demonstrates that this set is essentially \( \mathcal{R}^{T}(0) \) translated by the state \( \phi(T, 0, x, \mathbf{0}) \).This shows the reachability set starting from a state \( x \) to encompass all possible end states when combining the initial state response with \( \mathcal{R}^{T}(0) \).
  • This translation typically shifts the entire reachability subspace.
  • Defines the manifold as closed under addition despite the shift, implying it shares many properties with a subspace.
This translation keeps the focus on both the structure of the subspace and the impact of the initial state.
Subspace
A subspace is an essential element of linear algebra and plays a significant role in the study of linear systems. It is a vector space that is entirely contained within another vector space, meaning it includes all the properties of a vector space like vector addition and scalar multiplication.When dealing with the reachability set \( \mathcal{R}^{T}(0) \), we find that it is a subspace since it is closed under these operations:
  • Addition: For any two elements in \( \mathcal{R}^{T}(0) \), their sum is also in \( \mathcal{R}^{T}(0) \).
  • Scalar multiplication: Any element multiplied by a scalar results in another element that is still within \( \mathcal{R}^{T}(0) \).
Subspaces maintain consistent behavior under linear transformations, making them particularly useful in linear system analysis.
This property simplifies complexity, enabling easier manipulation and examination of the dynamics in linear control systems.

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Most popular questions from this chapter

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 \(\omega(t)=-B(t)^{*} \Phi(\sigma, t)^{*} W_{c}(\sigma, \tau)^{-1} x\).

Consider a system consisting of a cart to the top of which an inverted pendulum has been attached through a frictionless pivot. The cart is driven by a motor which at time \(t\) exerts a force \(u(t)\), taken as the control. (See Figure \(3.1(\mathrm{~b}) .)\) We assume that all motion occurs in a plane, that is, the cart moves along a straight line. We use \(\phi\) to denote the angle that the pendulum forms with the vertical, \(\delta\) for the displacement of the center of gravity of the cart with respect to some fixed point, \(F \geq 0\) for the coefficient of friction associated with the motion of the cart, \(g\) for the acceleration of gravity, \(l>0\) for the length of the pendulum, \(M>0\) for the mass of the cart, and \(m \geq 0\) for the mass of the pendulum, which we'll assume is concentrated at the tip. (If the mass is not so concentrated, elementary physics calculations show that one may replace the model by another one in which this does happen, using a possibly different length \(l\). We allow the case \(m=0\) to model the situation where this mass is negligible.) Newton's second law of motion applied to linear and angular displacements gives the two second order nonlinear equations and $$ l \ddot{\phi}-g \sin \phi+\ddot{\delta} \cos \phi=0 \text {. } $$ We shall only be concerned with a small angle \(\phi\), so we linearize the model about \(\phi=0\). This results, after taking \(x_{1}=\delta, x_{2}=\dot{\delta}, x_{3}=\phi, x_{4}=\dot{\phi}\), in a linear system \(\Sigma\) with \(n=4, m=1\) and matrices as follows: $$ A=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & -\frac{F}{M} & -\frac{m g}{M} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & \frac{F}{l M} & \frac{g(m+M)}{l M} & 0 \end{array}\right), \quad B=\left(\begin{array}{c} 0 \\ \frac{1}{M} \\ 0 \\ -\frac{1}{l M} \end{array}\right) . $$ Prove that \(\Sigma\) is controllable. (Controllability holds for all possible values of the constants; however, for simplicity you could take all these to be equal to one.) This example, commonly referred to as the "broom balancing" example, is a simplification of a model used for rocket stabilization (in that case the control \(u\) corresponds to the action of lateral jets).

}\( corresponding to \)\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:

Consider the system (with \(\mathcal{U}=\mathbb{R}, x=\mathbb{R}^{2}\) ) $$ \begin{aligned} &\dot{x}_{1}=x_{2} \\ &\dot{x}_{2}=-x_{1}-x_{2}+u \end{aligned} $$ which models a linearized pendulum with damping. Find explicitly the systems \(\Sigma_{[\delta]}\), for each \(\delta\). Characterize (without using the next Theorem) the \(\delta\) 's for which the system is \(\delta\)-sampled controllable. The example that we discussed above suggests that controllability will be preserved provided that we sample at a frequency \(1 / \delta\) that is larger than twice the natural frequency (there, \(1 / 2 \pi\) ) of the system. The next result, sometimes known as the "Kalman-Ho-Narendra" criterion, and the Lemma following it, make this precise.

Let \(\Sigma\) be a linear system, and pick any \(\sigma, \tau, T \in \mathcal{T}\). (a) \(\Sigma\) is controllable on \([\sigma, \tau]\) iff \((0, \sigma) \sim(y, \tau)\) for all \(y \in X\). (b) If \(\Sigma\) is time-invariant, then \(\Sigma\) is controllable in time \(T\) iff \(0 \sim_{T}^{\sim} y\) for all \(y \in X\). (c) If \(\Sigma\) is continuous-time, then it is controllable on \([\sigma, \tau]\) iff \((x, \sigma) \sim(0, \tau)\) for all \(x \in X\). (d) If \(\Sigma\) is time-invariant and continuous-time, then \(\Sigma\) is controllable in time \(T\) iff \(\underset{T}{x} 0\) for all \(x \in X\). (e) The conclusions in (c) and (d) hold also if "continuous-time" is replaced by "discrete-time and \(A(k)\) is invertible for all \(k \in[\sigma, \tau]\) " (in (d), " \(A\) is invertible").
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