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Chapter 3: Problem 21

Consider the system with \(\mathbb{K}=\mathbb{R}, n=3, m=1\), and matrices $$ A(t)=\left(\begin{array}{ccc} t & 1 & 0 \\ 0 & t^{3} & 0 \\ 0 & 0 & t^{2} \end{array}\right) \quad B(t)=\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right) $$ This system is smoothly (in fact, analytically) varying on \((-\infty, \infty)\). Since $$ \left[B_{0}(0), B_{1}(0), B_{2}(0), B_{3}(0)\right]=\left(\begin{array}{cccc} 0 & 1 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right), $$ and this matrix has rank 3 , the system is controllable on every nontrivial interval \([\sigma, \tau]\).

Short Answer

Expert verified
The given system is smoothly varying and the matrix at \(t=0\) has rank 3, which implies full rank of the attainable subspace. By a lemma, we conclude that the system is controllable on every nontrivial interval \([\sigma, \tau]\).

Step by step solution

01

Lemma

For a time-varying linear system like our system, smooth variation in an interval \([\sigma, \tau]\) implies that its controllability is equivalent to having full rank of the attainable subspace, which is spanned by the set of vectors: $$ \left\{B(t),A(t)B(t), \dots, A^{n-1}(t)B(t) \right\} . $$ To check controllability, we can thus check the rank of the matrix formed by these vectors. In our case, we are given that the rank of this matrix at \(t=0\) is 3.
02

Checking controllability

Now that we have established the lemma, we need to verify that our system is controllable on every nontrivial interval \([\sigma, \tau]\) by checking the rank of the matrix formed by the above vectors. At \(t=0\), the matrix is given as: $$ \left(\begin{array}{cccc} 0 & 1 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right) $$ Since this matrix has rank 3, we know that the system is controllable on the interval \([\sigma, \tau]\) by the lemma we established above.
03

Conclusion

We have shown that the system is smoothly and analytically varying with respect to matrix \(A(t)\) and \(B(t)\). By verifying that the rank of the matrix at \(t=0\) is 3, which implies full rank of the attainable subspace, we have concluded that the given system is controllable on every nontrivial interval \([\sigma, \tau]\).

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Key Concepts

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

Time-varying linear systems
Time-varying linear systems are a fascinating type of dynamic systems where the system's parameters, such as matrices, change with time. Unlike constant systems, where parameters remain fixed, time-varying systems are characterized by variable matrices, often represented as functions of time. This variability allows such systems to adapt to changing conditions, but also makes their analysis more challenging.

One typical representation of these systems is using state-space models, where the system is described by a set of equations involving matrices. For example, in the given exercise, matrices \(A(t)\) and \(B(t)\) vary with time \(t\). These matrices influence the system's behavior and control properties over different time intervals. Unlike time-invariant systems, which can often be analyzed using simpler algebraic methods, time-varying systems usually require more sophisticated mathematical techniques to understand and analyze their controllability and stability.
Matrix rank
The concept of matrix rank is central to understanding the controllability of linear systems. Rank refers to the maximum number of linearly independent column vectors in a matrix. It effectively measures the dimensions of the vector space spanned by its columns. For systems to be controllable, certain matrices associated with them need to fulfill a full rank condition.

In the context of controllability, the rank of the controllability matrix, formed by concatenating the columns \(B(t), A(t)B(t), \ldots, A^{n-1}(t)B(t)\), plays a crucial role. If this matrix achieves full rank (equal to the number of states), it indicates that all states in the system can be controlled, or reached, from any initial state within a finite time period. As demonstrated in the exercise, the matrix at \(t=0\) achieved full rank (rank = 3), indicating the system's controllability over any nontrivial interval \([\sigma, \tau]\).
  • Full rank implies the system is fully controllable.
  • Rank is determined by the number of linearly independent vectors in the matrix.
Attainable subspace
The attainable subspace is a concept reflecting the set of all states that can be reached, or attained, from an initial state by applying suitable inputs over time. In terms of linear control systems, this subspace is spanned by vectors derived from applying the control input through the system's dynamics represented by \(A(t)\) and \(B(t)\).

The attainable subspace links closely with controllability. If the attainable subspace covers the entire state space, the system is controllable. Essentially, this means that any state can be achieved through appropriate control inputs. In our exercise, the computation shows that the rank, or dimension, of this subspace reaches its maximum possible value, underscoring the full controllability of the system.
  • An attainable subspace spanning the state space indicates system controllability.
  • It represents the set of states reachable by the system.
Analytic functions
Analytic functions are mathematical entities that express smooth and infinitely differentiable behaviors across an interval. Such functions have a power series that converges to the function within some radius around every point in its domain.

In the setting of time-varying linear systems, analyticity of the matrices, such as \(A(t)\) and \(B(t)\), implies that they vary in a smooth and predictable manner. This ensures that the system transitions smoothly over time, which is crucial for defining solutions and evaluating their stability and controllability.
  • An analytic function can be expanded into a power series.
  • This property guarantees smooth behavior over time intervals.

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Most popular questions from this chapter

Let \(\Sigma\) be a continuous-time system as in Definition \(2.6 .7\) and let \(\sigma<\tau\). With the present terminology, Lemma \(2.6 .8\) says that \((x, \sigma) \sim\) \((z, \tau)\) for the system \(\Sigma\) iff \((z, \sigma) \sim(x, \tau)\) for the system \(\Sigma_{\sigma+\tau}^{-}\). This remark is sometimes useful in reducing many questions of control to a given state to analogous questions (for the time-reversed system) of control from that same state, and vice versa. Recall that a linear system is one that is either as in Definition 2.4.1 or as in Definition 2.7.2.

Let \(\Sigma\) be a continuous-time linear system, analytically varying on \(\mathcal{I}\). Prove that if \(\Sigma\) is controllable on any nontrivial subinterval \([\sigma, \tau]\) then $$ \operatorname{rank}\left[B_{0}(t), B_{1}(t), \ldots, B_{n-1}(t)\right]=n $$ for almost all \(t \in \mathcal{I}\). (Hint: First prove that if rank \(M^{(k)}(t)=\operatorname{rank} M^{(k+1)}(t)\) for \(t\) in an open interval \(J \subseteq \mathcal{I}\), then there must exist another subinterval \(J^{\prime} \subseteq J\) and analytic matrix functions $$ V_{0}(t), \ldots, V_{k}(t) $$ on \(J^{\prime}\) such that $$ M_{k+1}(t)=\sum_{i=0}^{k} M_{i}(t) V_{i}(t) $$ on \(J^{\prime} .\) Conclude that then rank \(M^{(k)}(t)=\operatorname{rank} M^{(l)}(t)\) for all \(l>k\) on \(J^{\prime}\). Argue now in terms of the sequence \(\left.n_{k}:=\max \left\\{\operatorname{rank} M^{(k)}(t), t \in \mathcal{I}\right\\} .\right)\)

Let \(\Sigma\) be a continuous-time linear system, and pick \(\sigma<\tau \in \mathbb{R}\). Consider the controllability Gramian $$ W_{c}(\sigma, \tau):=\int_{\sigma}^{\tau} \Phi(\sigma, s) B(s) B(s)^{*} \Phi(\sigma, s)^{*} d s . $$ Show: \(\Sigma\) is controllable in \([\sigma, \tau]\) if and only if \(W_{c}(\sigma, \tau)\) has rank \(n\), and, in that case, the unique control of minimum square norm that steers \(x\) to 0 is given by the formula \(\omega(t)=-B(t)^{*} \Phi(\sigma, t)^{*} W_{c}(\sigma, \tau)^{-1} x\).

Let \(\mathcal{U} \subseteq \mathbb{R}^{m}\) and pick any two \(S, T \geq 0\). Then $$ \mathcal{R}_{u}^{T}(0)+e^{T A} \mathcal{R}_{u}^{S}(0)=\mathcal{R}_{u}^{S+T}(0) . $$

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)
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