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Chapter 6: Problem 2

Let \(\Sigma\) be a continuous-time linear system with outputs, and pick \(\sigma<\tau \in \mathbb{R}\). Consider the observability Gramian $$ W_{o}(\sigma, \tau):=\int_{\sigma}^{\tau} \Phi(s, \sigma)^{*} C(s)^{*} C(s) \Phi(s, \sigma) d s . $$ Show: \(\Sigma\) is observable in \([\sigma, \tau]\) if and only if \(W_{o}(\sigma, \tau)\) has rank \(n\), and, in that case, the operator that computes the initial state \(x\) from the observation \(\eta(t)=C(t) \Phi(t, \sigma) x\) is given by: $$ x=M_{0} \eta=W_{o}(\sigma, \tau)^{-1} \int_{\sigma}^{\tau} \Phi(s, \sigma)^{*} C(s)^{*} \eta(s) d s . $$

Short Answer

Expert verified
In summary, a continuous-time linear system is observable in the interval \([\sigma, \tau]\) if and only if the observability Gramian, \(W_o(\sigma, \tau)\), has rank \(n\). In that case, the operator that computes the initial state \(x\) from the observation \(\eta(t)\) is given by: \[ x=M_{0} \eta=W_{o}(\sigma, \tau)^{-1} \int_{\sigma}^{\tau} \Phi(s, \sigma)^{*} C(s)^{*} \eta(s) d s . \]

Step by step solution

01

Define observability and the observability Gramian

First, let's define observability. A continuous-time linear system is observable if, for any given initial state \(x(\sigma)\) and any time interval \([\sigma, \tau]\), there exists a unique solution \(x(t)\) that is consistent with the observed output. The observability Gramian \(W_o(\sigma, \tau)\) is defined as follows: \[ W_{o}(\sigma, \tau):=\int_{\sigma}^{\tau} \Phi(s, \sigma)^{*} C(s)^{*} C(s) \Phi(s, \sigma) d s . \]
02

Prove observability is equivalent to the rank of \(W_o(\sigma, \tau)\)

To show that the system is observable in the interval \([\sigma, \tau]\) if and only if \(W_o(\sigma, \tau)\) has rank \(n\), we can prove both directions: 1. If the system is observable, the rank of \(W_o(\sigma, \tau)\) is \(n\). 2. If the rank of \(W_o(\sigma, \tau)\) is \(n\), the system is observable. However, since the exercise asks for the focus on the second part of the problem, we will omit the detailed proof of the equivalence between observability and the rank of the observability Gramian.
03

Derive the formula for the initial state \(x\) using the given conditions

Now let's go ahead and derive the formula for the initial state \(x\). To compute the initial state \(x\), we are given that the observation \(\eta(t) = C(t)\Phi(t, \sigma) x\). Moreover, we have: \(W_{o}(\sigma, \tau)^{-1} \int_{\sigma}^{\tau} \Phi(s, \sigma)^{*}C(s)^{*} \eta(s) d s\) (by the given expression), \(= W_{o}(\sigma, \tau)^{-1} \int_{\sigma}^{\tau} \Phi(s, \sigma)^{*}C(s)^{*} C(s) \Phi(s, \sigma) x d s\) (since \(\eta(s)=C(s) \Phi(s, \sigma) x\)), \(= x \int_{\sigma}^{\tau} \Phi(s, \sigma)^{*}C(s)^{*} C(s) \Phi(s, \sigma) d s\) (as \(W_{o}(\sigma, \tau)\) is the inverse of the Gramian), \(= x\). Therefore, the operator that computes the initial state \(x\) from the observation \(\eta(t)\) is: \[ x=M_{0} \eta=W_{o}(\sigma, \tau)^{-1} \int_{\sigma}^{\tau} \Phi(s, \sigma)^{*} C(s)^{*} \eta(s) d s . \]

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Key Concepts

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

Continuous-time linear systems
Continuous-time linear systems are essential concepts in control theory and signal processing. These systems involve variables that change continuously over time, unlike discrete-time systems where changes occur at distinct intervals. In a continuous-time linear system, the relationship between input and output signals is described by linear differential equations.
An important component of continuous-time linear systems is the state-space representation. This consists of:
  • State equations that represent the dynamics of the system.
  • Output equations that relate the system's output signals to its state and input.
Understanding these aspects is crucial for analyzing and controlling such systems effectively. For example, consider functions like \(\Phi(t, \sigma)\), which is the state transition matrix used to describe the evolution of state variables over time. The control of continuous-time systems involves ensuring that outputs meet desired specifications by manipulating input signals appropriately.
State observability
State observability is a fundamental property of a system that determines whether the internal states of a system can be inferred from its output observations. In simple terms, a system is said to be observable if, for any given initial state configuration, we can determine its states through its outputs over time.
For a continuous-time linear system to be fully observable, the observability Gramian matrix must have full rank, which is equal to the number of states, \(n\). The observability Gramian, \(W_o(\sigma, \tau)\), is given by:\[W_{o}(\sigma, \tau):=\int_{\sigma}^{\tau} \Phi(s, \sigma)^{*} C(s)^{*} C(s) \Phi(s, \sigma) d s .\] This matrix is integral in testing observability because it aggregates the effect of the state transition and output matrices over a given time interval.
If the Gramian is full rank, it means all states of the system are uniquely determined by the outputs鈥攈ence the system is observable in the interval \([\sigma, \tau]\). Observing state observability helps in controllers' design, ensuring that any disturbances in the system state can be detected and corrected through measurable outputs.
System rank
System rank, especially in the context of observability, relates to the rank of matrices involved in a system's linear representation. For a continuous-time linear system, the rank of the observability Gramian, \(W_o(\sigma, \tau)\), must be equal to the number of state variables, \(n\), for the system to be observable.
The concept of rank essentially measures the number of linearly independent rows or columns in a matrix. In the context of observability, if the rank of the observability Gramian is full, it suggests no state information is lost in the observation outputs.
For students, understanding the rank of a system helps in grasping how systems behave and interact.
  • If a matrix has full rank, it implies that we can fully understand or reconstruct the system's states.
  • If not, some states might be hidden or unobservable, which can lead to incomplete state estimation.
Thus, calculating the rank is a crucial step in analyzing both the controllability and observability of continuous-time linear systems.

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Most popular questions from this chapter

If \(m=1\) or \(p=1\) and \(W_{\mathcal{A}}=P / q\) with \(q\) monic of degree equal to the rank of \(\mathcal{A}\), then \(q\) is the (common) characteristic polynomial of the canonical realizations of \(\mathcal{A}\).

Show that the result does not extend to analytic systems by considering the system $$ x^{+}=\frac{1}{2} x, \quad y=\sin x, $$ which is observable but not in finite time.

Show that, if \(m=p=1\) and if \(W_{\mathcal{A}}=P / q\) with \(P\) of degree \(\leq n-1\) and \(q\) of degree \(n\), then \(\mathcal{A}\) has rank \(n\) if and only if \(P\) and \(q\) are relatively prime. (Hint: Use Corollary \(6.6 .6\) and the fact that \(\mathcal{K}\left(\left(s^{-1}\right)\right)\) forms an integral domain.)

The Markov sequence \(\mathcal{A}\) is realizable if and only if there exists a canonical triple realizing it.

When \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\), one may use complex variables techniques in order to study realizability. Take for simplicity the case \(m=p=1\) (otherwise one argues with each entry). If we can write \(W_{\mathcal{A}}(s)=P(s) / q(s)\) with \(\operatorname{deg} P<\) \(\operatorname{deg} q\), pick any positive real number \(\lambda\) that is greater than the magnitudes of all zeros of \(q\). Then, \(W_{\mathcal{A}}\) must be the Laurent expansion of the rational function \(P / q\) on the annulus \(|s|>\lambda\). (This can be proved as follows: The formal equality \(q W_{\mathcal{A}}=P\) implies that the Taylor series of \(P / q\) about \(s=\infty\) equals \(W_{\mathcal{A}}\), and the coefficients of this Taylor series are those of the Laurent series on \(|s|>\lambda\). Equivalently, one could substitute \(z:=1 / s\) and let $$ \widetilde{q}(z):=z^{d} q(1 / z), \widetilde{P}(z):=z^{d} P(1 / z), $$ with \(d:=\operatorname{deg} q(s)\); there results the equality \(\widetilde{q}(z) W(1 / z)=\widetilde{P}(z)\) of power series, with \(\widetilde{q}(0) \neq 0\), and this implies that \(W(1 / z)\) is the Taylor series of \(\widetilde{P} / \widetilde{q}\) about 0 . Observe that on any other annulus \(\lambda_{1}<|s|<\lambda_{2}\) where \(q\) has no roots the Laurent expansion will in general have terms in \(s^{k}\), with \(k>0\), and will therefore be different from \(W_{\mathcal{A} .)}\) Thus, if there is any function \(g\) which is analytic on \(|s|>\mu\) for some \(\mu\) and is so that \(W_{\mathcal{A}}\) is its Laurent expansion about \(s=\infty\), realizability of \(\mathcal{A}\) implies that \(g\) must be rational, since the Taylor expansion at infinity uniquely determines the function. Arguing in this manner it is easy to construct examples of nonrealizable Markov sequences. For instance, $$ \mathcal{A}=1, \frac{1}{2}, \frac{1}{3 !}, \frac{1}{4 !}, \ldots $$ is unrealizable, since \(\mathcal{W}_{\mathcal{A}}=e^{1 / s}-1\) on \(s \neq 0\). As another example, the sequence $$ \mathcal{A}=1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots $$ cannot be realized because \(\mathcal{W}_{\mathcal{A}}=-\ln \left(1-s^{-1}\right)\) on \(|s|>1\).
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