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Chapter 3: Problem 6

The system \(\Sigma\) is (completely) controllable on the interval \([\sigma, \tau]\) if for each \(x, z \in X\) it holds that \((x, \sigma) \sim(z, \tau) .\) It is (completely) controllable in time \(T\) if for each \(x, z \in X{X}\) it holds that \(x \underset{T}{\sim} z\). It is just (completely) controllable if \(x \sim z\) for all \(x, z\).

Short Answer

Expert verified
A system is controllable on the interval \([\sigma, \tau]\) if for any initial state \(x\) at time \(\sigma\), it can be steered to any final state \(z\) at time \(\tau\) (\( (x, \sigma) \sim (z, \tau) \) for all \(x, z \in X\)). It is controllable in time \(T\) if any initial state \(x\) can be steered to any final state \(z\) in time \(T\) (\(x \underset{T}{\sim} z\) for all \(x, z \in X\)). A system is completely controllable if it can be steered from any initial state \(x\) to any final state \(z\) in finite time (\(x \sim z\) for all \(x, z\)).

Step by step solution

01

Definition 1: Controllability on an interval

For a system to be controllable on an interval \([\sigma, \tau]\), it must be possible for any initial state \(x\) at time \(\sigma\) to be steered to any final state \(z\) at time \(\tau\). This is expressed mathematically as \((x, \sigma) \sim (z, \tau)\) for all \(x, z \in X\), where \(X\) is the set of all states of the system.
02

Definition 2: Controllability in time

A system is controllable in time \(T\) if it is possible to steer the system from any initial state \(x\) to any final state \(z\) in a fixed time period \(T\). This is written mathematically as \(x \underset{T}{\sim} z\) for all \(x, z \in X\), where \(X\) is the set of all states in the system.
03

Definition 3: Complete Controllability

A system is referred to as completely controllable if for every pair of states \(x, z \in X\) where \(X\) is the set of all states, it is possible to steer the system from \(x\) to \(z\) in finite time. This is often written as \(x \sim z\). The term "completely controllable" is usually used in the context of nonlinear systems, since in the case of linear systems, controllability implies complete controllability.

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Key Concepts

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

Finite Time
Finite time in the context of controllability refers to the ability of a dynamic system to transition between any two states within a designated timeframe. When engineers or mathematicians discuss system controllability, they are fundamentally interested in whether they can guide the system's behavior to a desired state efficiently.

This concept is particularly useful in practical applications where time constraints are critical. We often encounter this in industries that rely on rapid responses, such as aerospace or automotive systems. By ensuring that a system can reach its desired state in finite time, we guarantee operational reliability and efficiency.

Key points regarding finite time controllability include:
  • Fixed Time Intervals: Determining whether a system can move between states within a predefined period.
  • Real-World Applications: Enabling systems to operate effectively under strict time constraints.
  • Control Usefulness: Infinite time systems might theoretically be controllable, but practically, finite time controllability is crucial for applicability.
State Space
The 'State Space' forms the computational framework within which dynamic systems are analyzed and controlled. It represents all possible states of the system as points in a multi-dimensional space. Each dimension of the space corresponds to one of the variables necessary to describe the state of the system completely.

In essence, the state space is a mathematical concept used to model the set of variables that describe the condition of a system at a given time. This technique helps in visualizing how a system evolves over time and assists in predicting future behavior based on current conditions.

Here are a few aspects of state space to consider:
  • System Representation: It provides a complete representation of the system using matrices and vectors.
  • Dimensional Analysis: Each state variable introduces a dimension, expanding the space.
  • Control and Estimation: The state space model is essential for designing control strategies and state estimators.
Dynamic Systems
Dynamic systems are systems that evolve over time and whose state changes in response to inputs, internal mechanisms, or environmental conditions. They exhibit behavior that can generally be described by differential equations or difference equations.

These systems are all around us, from single-component mechanical setups to complex ecosystems, economies, or engineered networks. What makes understanding dynamic systems crucial is their application in designing systems that need stable and predictable behavior over time.

Core elements of dynamic systems comprise:
  • Time Evolution: The state of the system is expected to change over time.
  • Feedback Loops: Many dynamic systems incorporate feedback that affects future states and responses.
  • Stability and Control: Analyzing how the system remains stable or responds to changes is vital for design and safety.

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Most popular questions from this chapter

The following statements are equivalent for \(L, W\) as above: (a) \(L\) is onto. (b) \(L^{*}\) is one-to-one. (c) \(W\) is onto. (d) \(\operatorname{det} W \neq 0\). (e) \(W\) is positive definite. Consider again the situation in Example 3.5.1. Here \(L\) is onto iff the matrix $$ W=\int_{\sigma}^{\tau} k(t)^{*} k(t) d t>0 . $$ Equivalently, \(L\) is onto iff \(L^{*}\) is one-to-one, i.e., there is no \(p \neq 0\) in \(X\) with \(k(t) p=0\) for almost all \(t \in[\sigma, \tau)\), (3.18) or, with a slight rewrite and \(k_{i}:=i\) th column of \(k^{*}\) : \(\left\langle p, k_{i}\right\rangle=0\) for all \(i\) and almost all \(t \Rightarrow p=0 .\) (3.19)

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:

For this problem, we call a time-invariant system output controllable if it holds that for each \(x \in X\) and each \(y \in Y\) there exists a \(T \geq 0\) and a control \(u\) such that \(h(\phi(T, x, u))=y\). Prove that a time-invariant continuous-time linear system \((A, B, C)\) is output controllable iff $$ \operatorname{rank} C R(A, B)=\operatorname{rank}\left[C B, \ldots, C A^{n-1} B\right]=p . $$

If \(x\) is an equilibrium state, then $$ \mathcal{R}^{S}(x) \subseteq \mathcal{R}^{S+T}(x) $$ for each \(S, T \in \mathcal{T}_{+}\).
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