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This paper presents the method for the construction of tensor-product representation for multivariate switched linear systems, based on a suitable tensor-product representation of vectors and matrices. We obtain a representation theorem for multivariate switched linear systems. The stability properties of the tensor-product representation are investigated in depth, achieving the important result that any stable switched systems can be constructed a stable tensor-product representation of finite dimension. It is shown that the tensor-product representation provides a high level framework for describing the dynamic behavior. The interpretation of expressions within the tensor-product representation framework leads to enhanced conceptual and physical understanding of switched linear systems dynamic behavior.

Switched systems as an important class of hybrid systems [

On the other hand, switched systems cannot be represented in the framework of linear operators due to the logical switching space. For multivariate systems, the relationship between the rows vectors of the matrix space and that between the column vectors might be important for finding a projection to represent the systems operator. Motivated by this idea, switched systems may be represented in the framework of multi-linear algebra. Multi- linear algebra, the algebra of higher-order tensors, may be applied for analyzing the multi-factor structure of switched systems, including input space and logical switching space. Based on the considerations of multi-linear algebra, tensor-product representation is used to investigate the switched systems expression. From the theoretical point of view, this approach extended the subspace method using matrix technique to the multivariate switch- ed systems. The multilinear algebra, the algebra of higher-order tensors (also being a multidimensional or N- way array), which are the higher-order equivalents of vectors (first order) and matrices (second order), i.e., quan- tities of which the elements are addressed by more than two indices, may be applied for analyzing the multifac- tor structure of multivariate switched systems or such class of nonlinear systems.

In this paper, we investigated the more general problem of tensor-product representations for multivariate switched linear systems. We obtain the tensor-product representation theorem for globally table or globally asymptotically stable multivariate switched linear systems which are only defined on subspaces of algebras. This generalization allows us to obtain a Hahn-Banach extension theorem for such maps, which leads to further results on the Haagerup tensor-product norm applied into more general class of nonlinear systems. The stability properties of the tensor-product representation are investigated in depth, which any stable switched system admits a stable tensor-product representation of finite dimension. It is shown that the tensor-product representation provides a high level framework for describing the dynamic behavior of switched systems. The interpretation of expressions within the tensor-product representation framework leads to enhanced conceptual and physical understanding of switched linear systems dynamic behavior.

The main contribution of this paper is to obtain a general expression of tensor-product representation for multivariate switched systems, and the tensor-product representation is conceptually simple and relatively easy to express on the Haagerup tensor-product norm for multivariate switched systems. We focus on the correspondence between completely bounded multilinear maps and completely bounded linear maps on tensor products endowed with the Haagerup norm [^{*}-algebras, after which the techniques developed in [

The rest of this work is organized as follows. Section 2 contains the preliminaries on the tensor-product of vectors and matrices and the basic algebraic tools are provided. Section 3 gives the problem formulation for multivariate switched linear systems expression and its relation with bilinear algebra expression. In Section 4, the tensor-product representation for multivariate switched linear systems are investigated, and the stability pro- perties of the tensor-product representation based on the proposed methods are also investigated, which is followed by concluding conclusion in Section 5.

Notations: The notations throughout this paper are standard. R^{n} denotes the n dimensional Euclidean space. R^{n}^{×}^{m} is the set of all n × m real matrices. G denotes the systems function. I is the identity matrix with appropriate dimensions,

In this section, we give some necessary preliminaries on the tensor-product of vectors and matrices, mainly concerning with basic algebraic tools on definitions and the correspondence between completely bounded bilinear maps and completely bounded linear maps on Haagerup tensor products norm, which will be used in this paper. And, the C^{*}-algebraic technique is used to investigate the tensor-product representation of multivariate switched linear systems.

Let A and B denote unital C^{*} algebras and E and F will denote subspaces. E^{*} and F^{*} will denote the subspaces consisting of the adjoints of elements in E and F, respectively. The Haagerup norm on the algebraic tensor pro- duct

where the infimum is taken over all possible representations

Let ^{*}-algebra of n × n matrices over the field of complex numbers. If V is a vector space then the space

Definition 1 [

It is valuable to note that if L is a matrix of scalars, then

If

where the infimum is taken over all such expressions for

Definition 2 [^{*}-algebras, and let _{n} is the identity operator on M_{n}, for each

and the completely bounded norm

Proposition 1 [

Let

It is easiest to visualize this as formal matrix multiplication of

Lemma 1 [

Proposition 2 [

We consider a collection of discrete-time linear systems described by the equations with parametric uncertainties of process transfer functions and disturbance transfer functions with a class of piecewise constant functions,

where

where

It is assumed that the multivariate switched linear systems is globally table or globally asymptotically stable. This assumption essentially is to guarantee the entire switched systems are completely bounded, which is the very key condition for our study here. For this assumption, an interesting question is: under what conditions it is possible to transform the switched linear system in (6) into descriptive multilinear systems framework and further develop the tensor-product representation theorem. It is also desirable that the tensor-product representation property should promote efficiently numerical computation with respect to the multidimensional time series data analysis for estimating the controller performance of multivariate switched linear systems. The main goal of this paper is to obtain a necessary and sufficient condition for the existence of tensor-product representation for (6) based on such multilinear algebra framework. The problem can be formulated as follows.

Problem 1 Given the switched linear system described in (6), derive necessary and sufficient conditions for tensor-product representation of systems operators, which show that the algebraic structure of switched linear systems can abstractly represented in tensor space.

In order to let the developed results focus on the tensor-product representation framework of multivariate switched linear systems in this paper, we consider the simplified systems description form with neglecting the effects of disturbances a_{t} in the following,

Then, we can obtain the switched linear systems operator

where U and Y represent the inputs space and outputs space of subsystems, respectively, and the L represents the logical switching space. And the logic switching signals function

where T denotes the logical operators with the same meaning in (7). From the expression (10), we can see that there exists a map from the switched linear systems outputs to the switching signals. Obviously, the expression (9) exactly describes the map of multivariate switched linear systems from bilinear algebra framework. The systems operator description in (8) can be interpreted as dynamical systems behavior in the physical processes, and the equivalent bilinear form in (9) is expressed from the point of view of algebra structure, which is suitable for the technical development of tensor-product representation approach for controller performance measure of multivariate switched linear systems or such more complex class of multivariate nonlinear systems.

In this section, we discuss the tensor-product representation of multivariate switched linear systems from the point of view of systems operator by employing the theory of multilinear algebra. We first give a representation theorem for complete bounded linear operator on

Theorem 3 [

where

Corollary 1 If

Proof. Combine Proposition 2 with Theorem 3, the result is followed.

Now, we focus on studying the tensor-product representation of completely bounded bilinear operators from A × B into

for all

For a linear operator

Let

in

Lemma 2 Let

Proof. First of all, we define completely bounded maps

For completely bounded operators, from Theorem 3 we can obtain

And more details can be found in Theorem 4. If F and G are completely bounded, we have

where

Lemma 3 If

Proof. By Proposition 2,

If

This map need not be bounded unless both

Consider the simplified form of multivariate switched linear systems described in (9), we give the result of the representation theorem as in the following.

Theorem 4 Consider the multivariate switched systems described in (9), the associated linear map

The linear operators

The proof can be found in the Appendix.

Remark 1 There can be no extension theorem for general bilinear maps. If P

is another completely bounded linear operator from

Suppose that

If we let

Proposition 5 Consider multivariate switched linear systems described in (9) and (10) with the associated linear map:

where

Proof. Applying Theorem 3, the result is followed.

From the expressions (11) in Theorem 3 and (23) in Proposition 5, it is shown that the bilinear systems and switched linear systems are the same with the linear systems representation form. However, from (7) of the switched linear systems description in previous section, it can be found that systems outputs determine the logical switching signals described in (10), which shows the contraction T in Proposition 5 is determined in the operator itself (or switched linear systems itself). Furthermore, the logical functions outputs values are the logical values 0 or 1, so the logical functions outputs values are determined for the entire dynamical systems, of course, for the active subsystem, the logical value is 1. More details will be shown in the proof of Theorem 7.

Theorem 6 [

where the supremum is taken over all complete contractions

Lemma 4 Consider multivariate switched linear systems described in (9) and (10). Let

completely bounded. In this case, we have

Proof. Applying Theorem 6, the result is followed.

A necessary and sufficient conditions for tensor-product representation of multivariate switched linear systems can now be presented, under the assumption that the open or closed-loop switched linear systems is globally stable or globally asymptotically stable.

Theorem 7 Consider multivariate switched linear systems described in (9) and (10), the tensor-product representation

the linear function

the logical function

The proof can be found in the Appendix.

It is noted that the F in Theorem 4 is defined to be taken over all representations for the linear operators, but the F in Theorem 7 is defined to be taken over all representations of the subsystems operators for multivariate switched linear systems. This difference naturally lead to the different tensor-product representations for the general bilinear systems and the multivariate switched linear systems, and the n in M_{n} can be interpreted as the integer of the all finite switched subsystems of multivariate switched linear systems. More details can be found in the Examples 1 and 2.

Remark 2 Theorem 7 mainly considers the logic switching space, which exactly describes the tensor-product algebra representation for multivariate switched linear systems, which can extend into the scenario of the time- varying disturbance dynamics. And Theorem 4 just considers the tensor-product algebra representation structure for general bilinear systems operators, which can also extend into more general class of multivariate nonlinear systems in Theorem 10 in this paper. In addition, Theorem 4 shows the tensor-product representation structure for general bilinear operator and Theorem 7 shows the tensor-product representation structure for multi- variate switched linear systems in the framework of bilinear algebra. Furthermore, multivariate switched linear systems essentially can be interpreted as the special class of general bilinear operator with logical linear operator substituting the general linear operators.

Next, we give an example (including operator description and unfolding form) of tensor-product representa- tion for multivariate switched linear systems from the perspective of the structured systems models subject to the explicitly bounded disturbance which guarantees the entire systems are completely bounded.

Example 1 Let the k-th subsystem transfer function

For the dynamical systems, the tensor-product representation for the operator systems can be obtained as

where

where

Intuitively, the systems description can be interpreted as considering a linear combination of the linear operators of the original switched linear system that evolves in an asymptotically stable manner in a subspace. The tensor-product representation algebra system evolves in a higher dimensional space to which the original system can be projected under the stability condition of the entire multivariate switched linear systems. From Example 1, we can see that the multivariate switched dynamical systems can be expressed in the form of tensor-product extending the matrix description of multivariate linear systems.

Remark 3 For the entire switched linear systems, the tensor-product representation

Proposition 8 Let

Proof. The proof follows that of Proposition 2.

Theorem 9 [

The construction of efficient representations to multivariate functions and related operators plays a crucial role in the numerical analysis of higher dimensional problems arising in a wide range of modern applications. We generalize our results in Theorem 4 to the k-linear operators.

Theorem 10 Let

The i-th linear operator F_{i} is completely bounded with

The proof can be found in the Appendix.

Remark 4 It is difficult to describe the multivariate nonlinear systems in practical industrial applications. The analytic methods developed in [

Corollary 2 A k-linear operator from A^{k} into

Proof. From the Theorem 9 and Theorem 10, the results is followed.

Next, we extend the multivariate switched linear systems to the case of time varying disturbance scenario. We also assume that disturbance dynamics are piecewise linear time varying. This type of time varying disturbances are reflected as different output trajectories in different subprocess, each subprocess having the same linear time invariant disturbance models. In other words, it is assumed that the disturbance models are the same within each subprocess, and it is required that the switching sequence of disturbance models is also the same within each subprocess. We will give the tensor-product algebraic representation from the point of view of operator theory in the following.

Theorem 11 Consider multivariate switched linear systems described in (9) and (10) with time varying disturbance dynamics, the tensor-product representation

turbance logic switching functionals

From the Lemma 4 and Theorem 9, the results is followed.

Now, we will give an example of tensor-product representation for multivariate switched linear systems with time-varying disturbance dynamics. The representation results is constructive for practical industrial process systems modeling framework.

Example 2 A possible tensor expression of system (6) can be described for all the K MIMO subsystems in the tensor-product representation form. And, the j-th disturbance transfer function

For the dynamical systems, the tensor-product representation for the operator systems can be obtained as

where

For the entire switched systems, the tensor-product representation for the operator systems is given as

where

where

A new tensor-product representation for describing the dynamic behavior of switched linear systems has been presented. The tensor-product representation makes it easy to see the relationship between abstractly algebraic expressions that propagate spatial quantities from subsystem-to-subsystem. Abstract dynamic equations arising from a hybrid systems analysis, can be reinterpreted as equivalent operator-formulated equations. This algebra expression can model the behavior of the switched linear systems by the use of a tensor-product representation, which shows that the algebraic structure can simplify switched linear systems expression. The proposed technique is well suited for real and complex systems, and is capable of providing the tensor-product representation of the class of stable nonlinear system. Furthermore, the interpretation of expressions within the tensor-product representation framework can be enhanced conceptual and physical understanding of switched linear systems, or more general of multivariate nonlinear systems, dynamic behavior expression.

Proof. The necessity and sufficiency proofs are in the following.

Necessity of proof: Let

for

Note that if X is any unitary matrix in M_{n}, then

Such elements are annihilated by

for all unitaries X. This forces the matrix

The element

and

space H and norm one operators_{1} and J_{2} and define completely bounded maps

With these definitions it is easy to check that

where

Sufficiency of proof: It follows, from Lemma 2, that

Since

sentations

for

The element

and

space H and norm one operators_{1} and J_{2} and define com- pletely bounded maps

Since

It follows that f and g are well-defined and we have

This shows that

Proof. To avoid technical complications, we only discuss the case n = 2. In the other words, consider the multivariate switched linear systems with only two subsystems. The calculations are in the same spirit for general

Necessity of proof: Let_{1} and _{2}, a bounded linear operator

for all_{2} such that

are logical functional g_{i} on H_{1} such that

Let _{i} on U such that

for all

for all

Sufficiency of proof: It follows, from Lemma 4, that the tensor-product representation

from

bilinear representation V is completely bounded from

Since_{1} and _{2}, a bounded linear operator

for all

Since very element

for some

for all

And very element

for some

for all

where

lead to

for all

where

Since

Proof. We first consider the case

Since the inclusion

From Lemma 2, we can obtain

combining with inclusion

expression into the formula for