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Chapter 5: Problem 1

This problem is a variation of Proposition 5.9.1. Suppose given a smooth control-affine system, and \(V\) as there, so that \((5.43)\) holds, and consider, for each \(x \in X\), the following vector subspace of \(\mathbf{R}^{n}\) $$ \Delta(x):=\operatorname{span}\left\\{f(x), \operatorname{ad}_{f}^{k} g_{i}(x), i=1, \ldots, m, k=0,1,2, \ldots\right\\} $$ (in the differential-geometric terminology of Chapter \(4, \Delta\) defines a "distribution"). Assume that \(\nabla V(x)=0\) implies \(x=0\) and that \(\operatorname{dim} \Delta(x)=n\) for all \(x \neq 0 .\) Show that the feedback law \((5.44)\) stabilizes the system. (Hint: Prove by induction on \(k\) that, if \(\xi\) is a solution of \(\dot{x}=f(x)\) so that \(L_{f} V(\xi(t)) \equiv 0\) and \(L_{g i} V(\xi(t)) \equiv 0\) for all \(i\), then also \(L_{\text {ad } j_{g_{i}}} V(\xi(t)) \equiv 0\), for all \(k\) and all \(i\). You will need to use the facts that \(L_{\text {ad }_{f}^{k+1} g_{i}}\) can be expressed in terms of \(L_{f} L_{\text {ad }_{f}^{k} g_{i}}\) and \(L_{\text {ad }_{f}^{k_{i}}} L_{f}\), and that \(\nabla\left(L_{f} V\right)\) vanishes on the set \(\left\\{x \mid L_{f} V(x)=0\right\\}\) (why?).)

Short Answer

Expert verified
We show that the feedback law stabilizes the system by induction. For the base case, the problem statement provides that \(L_{f} V(\xi(t)) \equiv 0\) and \(L_{g i} V(\xi(t)) \equiv 0\) for all \(i\). Assuming the induction hypothesis for \(k\), we prove the induction step by rewriting \(L_{\operatorname{ad}_{f}^{k+1} g_i} V(\xi(t))\) using Lie derivatives and utilizing the induction hypothesis and given facts. This establishes that \(L_{\operatorname{ad}_{f}^{k+1} g_i} V(\xi(t)) \equiv 0\) for all \(k\) and all \(i\), proving that the feedback law stabilizes the system for all solutions that satisfy the given conditions.

Step by step solution

01

Base Case

Let's start the induction with the base case. Since \(k=0\), we need to show that \(L_{f} V(\xi(t)) \equiv 0\) and \(L_{g i} V(\xi(t)) \equiv 0\) for all \(i\). The problem statement provides these facts, so the base case holds true.
02

Induction Hypothesis

Assume for some \(k\), we have \(L_{f} V(\xi(t)) \equiv 0\) and \(L_{g i} V(\xi(t)) \equiv 0\) for all \(i\), and also \(L_{\text {ad } j_{g_{i}}} V(\xi(t)) \equiv 0\) for all \(k\).
03

Induction Step

We want to show that the induction hypothesis implies \(L_{\operatorname{ad}_{f}^{k+1} g_i} V(\xi(t)) \equiv 0\) for all \(k\) and all \(i\). We know that \(L_{\operatorname{ad}_{f}^{k+1} g_i}\) can be expressed in terms of \(L_{f} L_{\operatorname{ad}_{f}^{k} g_i}\) and \(L_{\operatorname{ad}_{f}^{k} g_i} L_{f}\). Now, we can rewrite \(L_{\operatorname{ad}_{f}^{k+1} g_i} V(\xi(t))\) as: \(L_{\operatorname{ad}_{f}^{k+1} g_i} V(\xi(t)) = L_f L_{\operatorname{ad}_{f}^{k} g_i} V(\xi(t)) - L_{\operatorname{ad}_{f}^{k} g_i} L_f V(\xi(t))\)
04

Using Induction Hypothesis

Let's use the induction hypothesis and the given fact that \(L_{f} V(\xi(t)) \equiv 0\) and \(L_{g i} V(\xi(t)) \equiv 0\) for all \(i\). We have \(L_f L_{\operatorname{ad}_{f}^{k} g_i} V(\xi(t)) = 0\), since \(L_f V(\xi(t)) \equiv 0\), and also \(L_{\operatorname{ad}_{f}^{k} g_i} L_f V(\xi(t)) = 0\) because of our induction hypothesis. Therefore, \(L_{\operatorname{ad}_{f}^{k+1} g_i} V(\xi(t)) = 0 - 0 = 0\).
05

Conclusion

Since we have proved that \(L_{\operatorname{ad}_{f}^{k+1} g_i} V(\xi(t)) \equiv 0\) for all \(k\) and all \(i\), using induction, our result establishes that the feedback law stabilizes the system for all solutions that satisfy the given conditions.

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Key Concepts

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

Stability of Control Systems
When discussing the stability of control systems, we're essentially considering whether a system, when perturbed, will return to its equilibrium state. In the context of control theory, stability often implies that a control system consistently produces predictable output despite any disturbances or initial deviations.

One important notion is that of Lyapunov stability, which involves using a Lyapunov function (like the function \(V\) in the given exercise) to assess stability. A system is stable if small changes in the initial conditions result in small changes in the behavior of the system over time. If this system returns to its starting state, it's said to be asymptotically stable.

For the exercise, the control system is described as control-affine, meaning that it has strong linear behavior with respect to the control input, allowing for more robust stability analysis. The feedback law, mentioned in the solution process, is designed to adjust the control input to maintain stability. By proving that the Lie derivatives of the control vector fields remain zero (demonstrating null growth in certain directions), the system's stability is ensured.
Vector Subspaces and Distributions
In control theory, understanding vector subspaces and distributions is key to analyzing system behaviors. A vector subspace of \(\mathbb{R}^{n}\) is essentially a portion of this real coordinate space that forms a 'smaller' space itself, obeying the same algebraic rules.

In the exercise, the set \(\Delta(x)\) is defined as the span of a collection of vector fields. This 'span' creates a subspace where any vector in the space can be expressed as a combination of these basis vectors 鈥� which includes the vector fields and iterated Lie brackets described in the problem.

A "distribution" in differential geometry, and specifically as in this task, refers to assigning these subspaces of the tangent space at each point \(x\). The dimension of the distribution across the entire space is significant because it informs us about system controllability. If the dimension is equal to \(n\), as required for non-zero \(x\), it indicates that the system is fully controllable at that point.
Differential Geometry in Control Systems
Differential geometry provides the mathematical framework essential for analyzing and designing control systems. This branch of mathematics studies curves, surfaces, and spaces, especially their shapes and properties.

Within control systems, concepts from differential geometry help to articulate the behavior of complex systems through smooth manifolds and vector fields. In the context of the problem, objects like the vector fields \(f(x)\) and Lie derivatives \(\operatorname{ad}_{f}^{k}g_{i}(x)\) exemplify how differential geometry models system dynamics. These constructs are used to describe the influence of control inputs on system states.

The solutions use Lie derivatives to explore how different strategies affect system stability, where the calculation involving \(L_{f}\) and \(L_{g_i}\) expresses how rates of change align or differ across possible control pathways. This provides deeper insights into how control mechanisms can homogeneously drive the system towards desired states while maintaining stability.

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Most popular questions from this chapter

Consider a continuous-time system \(\dot{x}=f(x)\) with no controls and \(x=\mathbb{R}^{n}\). Suppose that \(V: \mathbb{R}^{n} \rightarrow \mathbb{R}\) is proper and positive definite, and satisfies \(\dot{V}(x)=L_{f} V(x)<0\) for all \(x \neq 0\) (this is the Lyapunov condition in Lemma 5.7.4). Show that there exists a continuous function \(\alpha:[0, \infty) \rightarrow[0, \infty)\) which is positive definite (that is, \(\alpha(0)=0\) and \(\alpha(r)>0\) for all \(r>0\) ) such that the following differential inequality holds: \(\nabla V(x) \cdot f(x)=\dot{V}(x) \leq-\alpha(V(x))\) for all \(x \in \mathbb{R}^{n} .\) (Hint: Study the maximum of \(L_{f} V(x)\) on the set where \(V(x)=r_{.}\))

\(\diamond\) A far more restrictive problem is that of asking that \(\Sigma\) be linearizable by means of coordinate changes alone, i.e., that there be some diffeomorphism defined in a neighborhood of \(x^{0}\), and a controllable pair \((A, b)\), so that \(T_{*}(x) f(x)=A T(x)\) and \(T_{*}(x) g(x)=b\). This can be seen as feedback linearization with \(\alpha \equiv 0\) and \(\beta \equiv 1\). Show that such a linearization is possible if and only if \(g\left(x^{0}\right), \operatorname{ad}_{f} g\left(x^{0}\right), \ldots, \operatorname{ad}_{f}^{n-1} g\left(x^{0}\right)\) are linearly independent and \(\left[\operatorname{ad}_{f}^{i} g, \operatorname{ad}_{f}^{j} g\right]=0\) for all \(i, j \geq 0\).

Suppose that \(F: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) is locally Lipschitz when restricted to \(\mathbb{R}^{n} \backslash\\{0\\}\) and satisfies \(F(0)=0\), and assume that \(V\) is a proper and positive definite continuously differentiable function such that \(\nabla V(x) \cdot F(x)<0\) for all nonzero \(x\). (For example, \(F=f+\sum k_{i} g_{i}\), where \(k\), with \(k(0)=0\), is smooth on \(\mathbb{R}^{n} \backslash\\{0\\}\) and (5.54) holds for all nonzero states.) Show that there is a unique solution of \(\dot{x}=F(x)\), for each initial condition, defined for all \(t \geq 0\). Moreover, every solution converges to zero, and for each \(\varepsilon>0\) there is some \(\delta>0\) so that \(\|x(0)\|<\delta\) implies that \(\|x(t)\|<\varepsilon\) for all \(t \geq 0\). That is, the system \(\dot{x}=F(x)\) is globally asymptotically stable. As an illustration, compute the solutions, for each initial condition and each \(t \geq 0\), of the one-dimensional system \(\dot{x}=-x^{\frac{1}{3}}\) \(\left(V(x)=x^{2}\right.\) is as required for this \(\left.F\right)\). (Hint: On the open set \(x:=\mathbb{R}^{n} \backslash\\{0\\}\), there is local existence and uniqueness, and completeness can be proved using properness of \(V\). Properness and positive definiteness also guarantee that 0 is an equilibrium, so behavior such as that of \(\dot{x}=+x^{\frac{1}{3}}\) cannot occur.)

Let \(\Sigma\) be a linear (time-invariant) discrete-time system with no controls over \(\mathbb{R}, x^{+}=A x\), and let \(P>0, P \in \mathbb{R}^{n \times n}\). Prove that the condition $$ A^{\prime} P A-P<0 $$ is sufficient for \(V(x):=x^{\prime} P x\) to be a Lyapunov function for \(\Sigma\).

For any pair \((A, B)\) and any \(F \in \mathbb{K}^{m \times n}\), $$ \mathcal{R}(A+B F, B)=\mathcal{R}(A, B) . $$ In particular, \((A+B F, B)\) is controllable if and only if \((A, B)\) is.
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