Analytical Expression and Deconstruction of the Volume of the Controllability Ellipsoid ^††thanks: Work supported by the National Natural Science Foundation of China (Grant No. 61273005)

In control theory, linear discrete-time (LDT) systems can be formulated as follows:

where $x_{k}$ and $u_{k}$ are the state variable and input variable, respectively, and matrices $A\in R^{n\times n}$ and $B\in R^{n\times r}$ are the state matrix and input matrix in the system models, respectively, [6] [2]. To investigate the controllability of the LDT systems (1), the controllability Grammian matrix can be defined as follows

That the rank of the Grammian matrix $G_{N}$ is $n$, the dimension of the state space of the LDT systems (1), is the well-known criterion on the state controllability. By the controllability Grammian matrix, the controllability ellipsoid that can describe the maximum controllable region under the total-energy constraint $\left(\sum_{k=0}^{N-1}\|u_{N}\|^{2}_{2}\leq 1\right)$ can be defined as follows [3] [7] [10] [8] [1]

In papers [11], [4], [9], and [5], the determinant value $\det\left(G_{N}\right)$ and the minimum eigenvalue $\lambda_{\textnormal{min}}\left(G_{N}\right)$ of the controllability Grammian matrix $G_{N}$, correspondingly the volume $\textnormal{vol}\left(E_{N}\right)$ and the minimum radius $r_{\textnormal{min}}\left(E_{N}\right)$ of the controllability ellipsoid $E_{N}$, can be used to quantify the control ability of the input variables to the state space, and then be chosen as the objective functions for optimizing and promoting the control ability of the dynamical systems. Due to lack of the analytical computing of the determinant $\det\left(G_{N}\right)$ and eigenvalue $\lambda_{\textnormal{min}}\left(G_{N}\right)$, correspondingly the volume $\textnormal{vol}\left(E_{N}\right)$ and the radius $r_{\textnormal{min}}\left(E_{N}\right)$, these optimizing problems for the control ability are solved very difficulty, and few achievements about that have been made.

In this paper, the analytical volume-computation of the controllability ellipsoid $E_{N}$, when the system (1) is a sigle input system, is studied and an analytical expression on that $N\rightarrow\infty$ will be proven. By deconstructing the analytical volume expression, some factors for the ellipsoid $E_{N}$, such as, the shape factor, the minimum circumscribed rhomboid, etc, can be got. Therefore, the analytical volume and the shape factor of the ellipsoid, the minimum side length of the rhomboid can be used to describe the control ability and can be chosen as the objective functions and the constraint conditions for optimizing and promoting the control ability. Because of the analytical expression of these objective functions and constraint conditions, the optimizing problems will be solved with very effective optimizing computation.

Based on the linear system theory [6], [2], the LDT system (1) can be transformed as the Jordan canonical form, and especially the LDT system (1) with $n$ different eigenvalues can be transformed as the diagonal canonical form. Obviously, if the Jordan canonical form is $\Sigma(P^{-1}AP,P^{-1}B)$ with the Jordan transformation matrix $P$, the corresponding controllability Grammian matrix can be expressed respectively as

And then, the determinant $\det\left(\overline{G}_{N}\right)$ is $\left(\det P\right)^{-2}\det\left(G_{N}\right)$ and the volume of the controllability ellipsoid is $|\det P|^{-1}\textnormal{vol}\left(E_{N}\right)$. In this paper, the determinant and ellipsoid volume for the diagonal canonical form, respectively, are computed analytically and related results can be generalized to the general systems $\Sigma(A,B)$.

When the system $\Sigma(A,B)$ is a sigle-input diagonal canonical form as

the controllability Grammian matrix can be rewritten as

When all eigenvalues $\lambda_{i}\in(-1,1),i=\overline{1,n}$, and $N\rightarrow\infty$, we have

Before discussing the analytical volume computing of the infinite-time controllability ellipsoid $E_{\infty}$ for the diagonal canonical form, a theorem about the determinant of the matrix $G_{\infty}$ is put forward and proven as follows.

Proof. The theorem can be proven by induction method as follows.

(1) When $n=1$ and $2$, we have

And then, for $n=1$ and $2$, Eq. (26) holds.

(2) It is assumed that for a given $k$, Eq. (26) holds for $n=k-1$, that is, we have,

(3) For $n=k$, we have

where

And then, we have

Therefore, by Eq. (31) and Eq. (52), we have, Eq. (26) holds for $n=k$.

In summary, the theorem is proven by the inductive method. \qed

Based on Theorem 1, the determinant of the controllability Grammian matrix and the volume of the controllability ellipsoid for the diagonal canonical form are as follows

where the hypersphere volume-coefficient $H_{n}$ and the Gamma function $\varGamma(s)$ can be defined as

According to the above computation for the diagonal canonical form, a theorem on the the determinant of the controllability Grammian matrix and the volume of the controllability ellipsoid for the general systems $\Sigma(A,B)$ is can be established as follows.

In papers [11], [4], [9], and [5], the determinant value $\det\left(G_{N}\right)$ of the controllability Grammian matrix $G_{N}$ and the volume $\textnormal{vol}\left(E_{N}\right)$ the controllability ellipsoid $E_{N}$ are chosen as the objective functions for optimizing the control ability. Because lack of the analytical computing methods for $\det\left(G_{N}\right)$ and $\textnormal{vol}\left(E_{N}\right)$, these optimizing problems for the control ability are solved very difficulty, and few achievements about that have been made. Based on the above analytical computing of $\det\left(G_{\infty}\right)$ and $\textnormal{vol}\left(E_{\infty}\right)$, these optimizing problems for promoting the control ability can be solved conviently and the controlled plants with the better control abilty and dynamical performance can be designed.

According to the computing equation (60), two factors are deconstructed as follows.

In fact, the above two factors can be used to describe the shape and size of the controllability ellipsoid $E_{N}$ and the eigenvalue evenness of the LDT system.

Theorem 2 for the LDT systems with the real eigenvalues can be generalized to the LDT systems with the complex eigenvalues, and the corresponding the theorem can be stated as follows.

Similar to Section 3, based on these analytical expression, the shape factors of the controllability ellipsoid and the evenness factor of the eigenvalue distribution for LDT systems with complex eigenvalues can be got.

According to the computing equation (67), some factors described the shape and size of the controllability ellipsoid are deconstructed as follows.

Similar to the factors for the matrix $A$ with the real eigenvalues in Eqs. (61) and (62), the above factors can be describe the shape and size of the controllability ellipsoid $E_{N}$ and the eigenvalue evenness factor of the LDT systems.

For the conjugate complex eigenvalue pair $\left(\lambda_{i},\lambda_{i+1}\right)=\left(\lambda_{i},\lambda_{i}^{*}\right)$, the shape factor of the 2-dimensional section of the ellipsoid in the eigen-space is

Fig. 3 shows the 2-dimensional ellipsoids $E_{N}(30)$ generated by the following matrix pairs $(A,b)$ with the complex eigenvalues

where $a=0.1,0.2,0.3$, the corresponding the eirenvalues are $0.8\pm 0.1i,0.8\pm 0.2i,0.8\pm 0.3i$, and the factors $F_{1}=1.032,2.282,4.121$. By Fig. 3, we can see, the bigger the image part of the conjugate complex eigenvalue pair is, the greater the volume of the ellipsoid.

By the diagonal matrix transformation, the diagonal matrix pair is as follows

and then, the analytical computing results about the volume and these factors are as Table 1

Fig. 4 shows the 3-dimensional ellipsoids $E_{N}(60)$ generated by the above matrix pairs $\left(\hat{A},\hat{b}\right)$ in the eigen-space, from three visual angle, and these 3 figures show us the 3 sections of the ellipsoid.

In this article, we present three theorems and develop an effective analytical method to compute the volume of the controllability ellipsoid for the LDT systems with $n$ different eigenvalues. Furthermore, by deconstructing the analytical expression of the volume, some factors on the shape of the ellipsoid, the side length of its circumscribed rhomboid, the evenness of the eigenvalue distribution of the LDT system are constructed. Based on the analytical expression of the volume and these factors, the control ability can be defined, computing, analyzed, and optimized.