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There is a higher order test for observability as well. Here we only present a brief outline of this test. Assume given a continuous-time system affine in controls, $$ \dot{x}=g_{0}(x)+\sum_{i=1}^{m} g_{i}(x) u_{i}, $$ where we assume that all vector fields (that is, vector functions) \(g_{i}\) are of class \(\mathcal{C}^{\infty}\). Consider the vector space spanned by the set of all functions of the type $$ L_{g_{i_{1}}} \ldots L_{g_{i_{k}}} h_{j}(x) $$ over all possible sequences \(i_{1}, \ldots, i_{k}, k \geq 0\), out of \(\\{0, \ldots, m\\}\) and all \(j=\) \(1, \ldots, p\), where \(L_{g} \alpha=\nabla \alpha \cdot g\) for any function \(\alpha\) and any vector field \(g\). This is called the observation space \(\mathcal{O}\) associated to the system. We say that two states \(x_{1}\) and \(x_{2}\) are separated by \(\mathcal{O}\) if there exists some \(\alpha \in \mathcal{O}\) such that \(\alpha\left(x_{1}\right) \neq \alpha\left(x_{2}\right)\). One can prove that if two states are separated by \(\mathcal{O}\) then they are distinguishable. A sketch of the argument is as follows. Assume that \(x_{1}\) is indistinguishable from \(x_{2}\) and consider a piecewise constant control which is equal to \(u^{1}\) on \(\left[0, t_{1}\right)\), equal to \(u^{2}\) on \(\left[t_{1}, t_{1}+t_{2}\right), \ldots\), and equal to \(u^{k}\) on \(\left[t_{1}+\ldots+t_{k-1}, t_{1}+\ldots+t_{k}\right)\). For small enough \(t_{i}\) 's this control is admissible for both \(x_{1}\) and \(x_{2}\), and by indistinguishability we know that the resulting output at time \(t=t_{1}+\ldots+t_{k}\) is equal for both. In general, we denote the \(j\) th coordinate of this output value by $$ h_{j}\left(t_{1}, t_{2}, \ldots, t_{k}, u^{1}, u^{2}, \ldots, u^{k}, x\right) $$ if the initial state is \(x\). It follows that the derivatives with respect to the \(t_{i}\) 's of this output are also equal, for \(x_{1}\) and \(x_{2}\), for every such piecewise constant control. One may prove by induction that $$ \begin{gathered} \left.\frac{\partial^{k}}{\partial t_{1} \ldots \partial t_{k}}\right|_{t_{1}=t_{2}=\ldots=0} \quad h_{j}\left(t_{1}, t_{2}, \ldots, t_{k}, u^{1}, u^{2}, \ldots, u^{k}\right) \\ =L_{X_{1}} L_{X_{2}} \ldots L_{X_{k}} h_{j}(x) \end{gathered} $$ where \(X_{l}(x)=g_{0}(x)+\sum_{i=1}^{m} u_{i}^{l} g_{i}(x)\). This expression is a multilinear function of the \(u_{i}^{l}\), s, and a further derivation with respect to these control value coordinates shows that the generators in (6.16) must coincide at \(x_{1}\) and \(x_{2}\). In the analytic case, separability by \(\mathcal{O}\) is necessary as well as sufficient, because (6.17) can be expressed as a power series in terms of the generators (6.16). The observability rank condition at a state \(x_{0} \in X\) is the condition that the dimension of the span of $$ \left\\{\nabla L_{g_{i_{1}}} \ldots L_{g_{i_{k}}} h_{j}\left(x_{0}\right) \mid i_{1}, \ldots, i_{k} \in\\{0, \ldots, m\\}, j=1, \ldots p\right\\} $$ be \(n\). An application of the Implicit Function Theorem shows that this is sufficient for the distinguishability of states near \(x_{0}\). For more details, see, for instance, \([185],[199]\), and [311].

Step by step solution

01

Explain the continuous-time system affine in controls

Given a continuous-time system affine in controls, characterized by the following equation: \( \dot{x}=g_{0}(x)+\sum_{i=1}^{m} g_{i}(x) u_{i} \), where the vector fields \(g_i\) are of class \( \mathcal{C}^{\infty} \).

02

Define the observation space

The observation space, denoted by \( \mathcal{O} \), is the vector space spanned by the set of all functions of the type: \( L_{g_{i_{1}}} \ldots L_{g_{i_{k}}} h_{j}(x) \), over all possible sequences \( i_{1}, \ldots, i_{k}, k \geq 0 \), out of \( \{0, \ldots, m\} \) and all \( j= 1, \ldots, p \), where \( L_{g} \alpha=\nabla \alpha \cdot g \) for any function \( \alpha \) and any vector field \( g \).

03

Define separation by the observation space

We say that two states, \(x_{1}\) and \(x_{2}\), are separated by the observation space, \( \mathcal{O} \), if there exists some function \( \alpha \in \mathcal{O} \) such that \( \alpha(x_{1}) \neq \alpha(x_{2}) \). If two states are separated by the observation space, they are distinguishable.

04

Deriving the observability rank condition

The observability rank condition at a state \( x_{0} \in X \) is the condition that the dimension of the span of the following set is equal to \( n \): \( \left\{\nabla L_{g_{i_{1}}} \ldots L_{g_{i_{k}}} h_{j}(x_{0}) \mid i_{1}, \ldots, i_{k} \in\{0, \ldots, m\}, j=1, \ldots p\right\} \). If this condition holds, then the Implicit Function Theorem implies that states near \( x_{0} \) are distinguishable. As a reference for more detailed information about this topic, one can consult the resources \([185]\), \([199]\), and \([311]\).

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Key Concepts

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

Understanding Observability Rank Condition

Observability is one of the key aspects in control theory which pertains to the ability to infer the internal state of a system based on its external outputs. The observability rank condition specifically is an important test that determines if the full state of a system can be observed by measuring its outputs. In essence, this condition checks whether the appropriate derivatives of the system's output functions span the entire state space when evaluated at a particular point. To make this concept clear, imagine you're looking at a complex mechanical clock; the observability rank condition helps you to know if you can understand the position of every gear solely by looking at the clock's hands. This condition is crucial as it guarantees that one can reconstruct the internal state from the outputs under certain conditions. When the condition is met, the system is deemed ‘observable’ at that state, meaning that you have all the information you need to figure out exactly what's going on inside, even if you can't directly see the internal gears or workings.

Continuous-Time System Affine in Controls

When discussing control systems, one encounters various types of models to encapsulate how systems behave over time. A continuous-time system affine in controls is a type of dynamic system model where the rate of change of the state, usually denoted by \( \dot{x} \), is described by a set of continuous functions that have a specific linear dependency on the control inputs. This kind of system is expressed mathematically as \( \dot{x} = g_0(x) + \sum_{i=1}^{m} g_i(x) u_i \), where \( g_0 \) represents the autonomous part of the system's dynamics, and \( g_i \) are functions defining how the control inputs \( u_i \) influence the state's evolution. Think of this like setting the trajectory for a spacecraft. The autonomous part \( g_0 \) could represent the effect of gravity, while \( g_i \) might represent the impact of firing various thrusters (\( u_i \)). This understanding is pivotal for analyzing system behaviors and designing appropriate control strategies.

Exploring the Observation Space

The concept of observation space plays a pivotal role in control theory, as it represents the set of all possible ways we can monitor or 'observe' the state of our system through its outputs. The observation space \( \mathcal{O} \) is defined mathematically by the span of all functions that can be derived from the output functions and the system's vector fields, which are essentially the different possible measurements we can obtain. In the context of the higher-order observability test, this space is spanned by functions in the form of \( L_{g_{i_1}} \ldots L_{g_{i_k}} h_j(x) \), where \( L_{g} \alpha \) denotes the Lie derivative of function \( \alpha \) along the vector field \( g \). The richness of the observation space is directly related to our ability to distinguish between different states of the system. If this space allows you to discern any two distinct states, it's akin to having a security camera system that can give you a clear and complete image of what's happening inside a facility—no blind spots, full coverage.

Applying the Implicit Function Theorem

Finally, the Implicit Function Theorem is a powerful tool in mathematics that can be used in control theory to ensure that under the right conditions, near a particular point, you can solve equations for one set of variables in terms of another set. In terms of observability, this theorem justifies that if the observability rank condition is met at a state \( x_0 \), then locally around that state, you can uniquely determine the system's internal states based on its outputs. It's as if you had a complicated equation describing where every tourist in a theme park is based on their current ride tickets and the theorem assured you that with certain information, you could indeed figure out everyone's location in the park. This theorem becomes a guarantee for distinguishability of states in the mathematical framework of control systems, ensuring that the system can be effectively monitored and controlled. Incorporating this principle helps designers understand the extent to which system states can be determined and manipulated based on available measurements.