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Chapter 4: Problem 3

Let \(\dot{x}=A(t) x+B(t) u\) be a time-varying continuous time linear system, and assume that all the entries of \(A\) and \(B\) are smooth functions of \(t\). Introduce the following system with state-space \(\mathbb{R}^{n+1}\) : $$ \begin{aligned} \dot{x}_{0} &=1 \\ \dot{x} &=A\left(x_{0}\right) x+B\left(x_{0}\right) u . \end{aligned} $$ Explain the relationship between the accessibility rank condition, applied to this system, and the Kalman-like condition for controllability of \(\dot{x}=A(t) x+B(t) u\) studied in Corollary \(3.5 .18 .\) The following easy fact is worth stating, for future reference. It says that the map \(x \mapsto \phi(T, 0, x, \omega)\) is a local homeomorphism, for each fixed control \(\omega .\)

Short Answer

Expert verified
The accessibility rank condition applied to the second system involves evaluating the rank of the Lie algebra generated by the matrices \(A(t + c)\) and \(B(t + c)\). If this condition is satisfied (i.e., the rank is maximum), then the first system is also controllable according to the Kalman-like condition for controllability. Both conditions rely on the rank of the Lie algebra generated by the matrices \(A(t)\) and \(B(t)\), and therefore establish an equivalence between the two in relation to controllability.

Step by step solution

01

Define the accessibility rank condition and Kalman-like condition for controllability

First, let's state the definitions of these two conditions. Accessibility rank condition: A system is said to satisfy the accessibility rank condition if for some \(t_0 \in \mathbb{R}\), the rank of the Lie algebra generated by the matrices \(A(t_0)\) and \(B(t_0)\) is maximum (equal to \(n\)). Kalman-like condition for controllability (Corollary 3.5.18): A continuous-time linear system \(\dot{x} = A(t)x + B(t)u\) is controllable if and only if the matrix \(\int_{0}^{T} \Phi(T, s)B(s) ds\) has full column rank for some \(T > 0\), where \(\Phi(T, s)\) denotes the state-transition matrix of the system.
02

Express the accessibility rank condition for given system

Now, let's modify the accessibility rank condition for the second system. The second system's equations are given by: \[ \begin{aligned} \dot{x}_{0} &=1 \\ \dot{x} &=A\left(x_{0}\right) x+B\left(x_{0}\right) u . \end{aligned} \] From \(\dot{x}_{0}=1\), we have that \(x_{0} = t + c\), where \(c\) is a constant. Applying the accessibility rank condition, the rank of the Lie algebra generated by the matrices \(A(t + c)\) and \(B(t + c)\) must be maximum (equal to \(n\)).
03

Relate the accessibility rank condition with the Kalman-like condition for controllability

By applying the given information that the map \(x \mapsto \phi(T, 0, x, \omega)\) is a local homeomorphism, we know that the mapping \(\phi\) is continuous, invertible, and has a continuous inverse. Now, applying the second system's accessibility rank condition that states that the rank of Lie algebra generated by the matrices \(A(t + c)\) and \(B(t + c)\) must be maximum (equal to \(n\)), we can infer that the first system is also controllable according to the Kalman-like condition for controllability, because the accessibility rank condition of the second system involves the same matrices A and B, making them equivalent criteria for evaluating controllability. To sum up, the accessibility rank condition applied to the second system corresponds to the Kalman-like condition for controllability of the first system, with both of them relying on the rank of the Lie algebra generated by the matrices \(A(t)\) and \(B(t)\). So, if the second system satisfies the accessibility rank condition, then the first system will also satisfy the Kalman-like condition for controllability.

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Key Concepts

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

Accessibility Rank Condition
The accessibility rank condition is a fundamental concept in control theory that addresses the question: Is it possible to reach a desired state from a given initial state using certain control inputs? Specifically, when working with time-varying continuous linear systems like \(\dot{x} = A(t)x + B(t)u\), the accessibility rank condition requires that the collection of all possible system state variations spans the entire state space at some point in time. If we imagine the system as a point moving through space, the accessibility rank condition ensures that at some time \(t_0\), the point has the potential to reach any position in the space it occupies. In mathematical terms, this condition hinges on the rank of the Lie Algebra 鈥� a concept derived from advanced mathematics that deals with structures arising from vector fields and their commutators.

For our system, given by \(\dot{x} = A(t)x + B(t)u\) and the extended system with added \(x0\), this condition is modified to account for the state variable \(x_0\). It is important to note that the accessibility rank condition is strictly local 鈥� it does not assert that we can necessarily reach the entire state space from any starting point, but rather from points in the local vicinity, akin to being able to move freely within your own neighborhood but not necessarily to any location on earth.
Kalman-like Condition
The Kalman-like condition is named after Rudolf E. K谩lm谩n, a mathematician who made significant contributions to control theory. This condition provides us with a reliable test for controllability 鈥� the ability to guide a system to any final state within a finite time period through appropriate control inputs. For our \(\dot{x} = A(t)x + B(t)u\) system, the Kalman-like condition uses a special tool called the state-transition matrix, denoted as \(\Phi(T, s)\), which encapsulates how the system evolves over time.

To satisfy the Kalman-like condition, it's required that the integral \(\int_{0}^{T} \Phi(T, s)B(s) ds\) has full column rank for some time duration \(T > 0\). Think of this matrix integral as checking if there is a sequence of control inputs over time (from \(0\) to \(T\)) that can influence all the states. If so, we can say the system is controllable. This is like having a universal remote that can adjust every imaginable setting on your television 鈥� if such a remote exists, you have full control over the TV, just as the system's control inputs can guide it to any state.
State-transition Matrix
The state-transition matrix, or \(\Phi(T, s)\), plays a pivotal role in analyzing time-varying linear systems. Imagine a theater where the stage set changes from one scene to another, the state-transition matrix reflects the choreography that transitions the system from one state to another over time. It encapsulates how the initial condition \(x(s)\) at time \(s\) will evolve to become \(x(T)\) at time \(T\), given the system's dynamics. Mathematically, it is the solution to the matrix differential equation \(\frac{d}{dt}\Phi(t, s) = A(t)\Phi(t, s)\) with the initial condition \(\Phi(s, s) = I\), where \(I\) is the identity matrix. This matrix is a bridge between various times in a system's life, outlining each step the state would take under the system's rules. The significance of the state-transition matrix within the Kalman-like condition lies in its encapsulation of the system's potential to be controlled鈥攅very possible maneuver is documented within \(\Phi\).
Lie Algebra
Lie algebra, named after the Norwegian mathematician Sophus Lie, is an algebraic structure that studies the symmetries and transformations of mathematical systems. In the realm of control theory, Lie algebra comes into play when dealing with the accessibility rank condition. The matrices \(A(t)\) and \(B(t)\) from our system can be thought of as operators that inform us how the system states can be altered at a particular instant.

But in dynamical systems, we're not just interested in instantaneous changes; we must consider how sequences of changes can compound or negate each other, leading to new possible states. This is where the concept of commutators within the Lie algebra comes in. They represent the additive effect of applying one transformation after another, and consequently, the reverse sequence. For a control system to meet the accessibility rank condition, the Lie algebra generated by the matrices must be sufficiently 'rich'鈥攈aving enough 'operators' to span the entire state space. To draw an analogy, if you think of the system transformations as moves in a dance, Lie algebra is interested in all the new moves you can create by combining and reordering your existing choreography.

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Most popular questions from this chapter

Suppose that \(\Delta=\Delta_{f_{1}, \ldots, f_{r}}\) is involutive of constant rank \(r\). Then, for each \(x^{0} \in \mathcal{O}\) there exists some open subset \(\mathcal{O}_{0}\) containing \(x^{0}\), an \(\varepsilon>0\), and a diffeomorphism \(\Pi: \mathcal{O}_{0} \rightarrow(-\varepsilon, \varepsilon)^{n}\), so that the following property holds. If we partition $$ \Pi=\left(\begin{array}{l} \Pi_{1} \\ \Pi_{2} \end{array}\right), \quad \Pi_{2}: \mathcal{O}_{0} \rightarrow(-\varepsilon, \varepsilon)^{n-r}, $$ then \(\Delta(x)=\operatorname{ker}\left(\Pi_{2}\right)_{*}(x)\) for all \(x \in \mathcal{O}_{0}\).

A distribution on the open subset \(\mathcal{O} \subseteq \mathbb{R}^{n}\) is a map \(\Delta\) which assigns, to each \(x \in \mathcal{O}\), a subspace \(\Delta(x)\) of \(\mathbb{R}^{n}\). A vector field \(f \in \mathbb{V}(\mathcal{O})\) is pointwise in \(\Delta\), denoted \(f \in_{p} \Delta\), if \(f(x) \in \Delta(x)\) for all \(x \in \mathcal{O}\). A distribution is invariant under a vector field \(f \in \mathbb{V}(\mathcal{O})\) if $$ g \in_{p} \Delta \Rightarrow[f, g] \in_{p} \Delta, $$ and it is involutive if it is invariant under all \(f \in_{p} \Delta\), that is, it is pointwise closed under Lie brackets: $$ f \in_{p} \Delta \text { and } g \in_{p} \Delta \quad \Rightarrow \quad[f, g] \in_{p} \Delta . $$ The distribution generated by a set of vector fields \(f_{1}, \ldots, f_{r} \in \mathrm{V}(\mathcal{O})\) is defined \(b y\) $$ \Delta_{f_{1}, \ldots, f_{r}}(x):=\operatorname{span}\left\\{f_{1}(x), \ldots, f_{r}(x)\right\\} $$ for each \(x \in \mathcal{O}\). A distribution has constant rank \(r\) if \(\operatorname{dim} \Delta(x)=r\) for all \(x \in \mathcal{O}\).

Suppose that \(\Delta=\Delta_{X_{1}, \ldots, X_{r}}\) has constant rank \(r\), is involutive, and is invariant under the vector field \(f\). Pick any \(x^{0} \in \mathcal{O}\), and let \(\mathcal{O}_{0}\) and II be as in Frobenius' Lemma 4.4.16. Define, for \(z \in(-\varepsilon, \varepsilon)^{n}, g(z):=\) \(\Pi_{*}\left(\Pi^{-1}(z)\right) f\left(\Pi^{-1}(z)\right)\), and partition \(g=\left(g_{1}, g_{2}\right)^{\prime}\) and \(z=\left(z_{1}, z_{2}\right)^{\prime}\) as in the proof of Theorem 11 . Show that \(g_{2}\) does not depend on \(z_{1}\), that is to say, the differential equation \(\dot{x}=f(x)\) transforms in the new coordinates \(z=\Pi(x)\) into: $$ \begin{aligned} &\dot{z}_{1}=g_{1}\left(z_{1}, z_{2}\right) \\ &\dot{z}_{2}=g_{2}\left(z_{2}\right) \end{aligned} $$ Explain how, for linear systems \(\dot{x}=A x\), this relates to the following fact from linear algebra: if \(A\) has an invariant subspace, then there is a change of coordinates so that \(A\) is brought into upper triangular form consistent with that subspace. (Hint: (For the proof that \(\partial g_{2} / \partial z_{1}=0\).) We have that \(g_{2}(\Pi(x))=\Pi_{2 *}(x) f(x)\). On the other hand, each row of \(\Pi_{2 *}(x) f(x)\) is of the form \(L_{f} \psi_{i}\), where \(\psi_{i}\) 's are the rows of \(\Pi_{2}\). We know that \(L_{X_{j}} \psi_{i}=0\) for all \(i_{1} j\) (this is what Lemma \(4.4 .16\) gives), and also \(L_{\left[f, X_{j}\right]} \psi_{i}=0\) (because \(\Delta\) is invariant under \(f)\), so conclude that \(L_{X_{j}}\left(L_{f} \psi_{i}\right)=0\). This gives that the directional derivatives of the rows of \(g_{2}(\Pi(x))\) along the directions \(e_{j}(x):=\Pi_{*}(x) X_{j}(x)\) are all zero. Now observe that the vectors \(e_{i}(x)\) are all of the form \(\left(e_{i 1}, 0\right)^{\prime}\), and they are linearly independent.)

(a) Show that for systems of dimension one: \(\dot{x}=f(x, u), x=\) \(\mathbb{R}\), the accessibility rank condition holds at a state \(x^{0}\) if and only if \(\mathcal{R}\left(x^{0}\right)\) has a nonempty interior. (b) Consider the following system with \(\mathcal{X}=\mathbb{R}^{2}\) and \(\mathcal{U}=\mathbb{R}^{2}\) : $$ \begin{aligned} &\dot{x}_{1}=u_{1} \\ &\dot{x}_{2}=\varphi\left(x_{1}\right) u_{2} \end{aligned} $$ where \(\varphi: \mathbb{R} \rightarrow \mathbb{R}\) is a \(C^{\infty}\) function with the property that \(\varphi(z) \neq 0\) for each \(z \neq 0\) but \(\frac{d^{j} \varphi}{d z^{j}}(0)=0\) for all \(j=0,1,2, \ldots\) (for instance, \(\varphi(z):=e^{-1 / z^{2}}\) for \(z \neq 0, \varphi(0):=0)\). Show that, for all \(x^{0}, x^{0} \in\) int \(\mathcal{R}_{\mathcal{V}}^{\leq T}\left(x^{0}\right)\), but, on the other hand, the accessibility rank condition does not hold at the points of the form \(\left(0, x_{2}\right)\).

Suppose that \(f_{1}, \ldots, f_{r}\) are smooth vector fields on an open set \(\mathcal{O} \subseteq \mathbb{R}^{n}\), and that \(f_{1}(x), \ldots, f_{r}(x)\) are linearly independent for each \(x \in \mathcal{O}\). Show that the following two properties are equivalent: \- \(\left[f_{i}, f_{j}\right]=0\) for each \(i, j \in\\{1, \ldots, r\\} .\) \- For each \(x^{0} \in \mathcal{O}\) there is an open subset \(\mathcal{O}_{0} \subseteq \mathcal{O}\) which contains \(x^{0}\) and a diffeomorphism \(\Pi: \mathcal{O}_{0} \rightarrow \mathcal{V}\) into some open subset \(\mathcal{V} \subseteq \mathbb{R}^{n}\) such that \(\left(\Pi_{*} f_{i}\right)(z)=e_{i}\) for each \(z \in \mathcal{V}\), where \(e_{i}\) is the \(i\) th canonical basis vector. (That is, the vector fields commute if and only if there is a local change of variables where they all become \(\left.f_{i} \equiv e_{i} \cdot\right)\)
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