Stabilization with Internal Loop for Infinite-Dimensional Discrete Time-Varying Systems

Abstract

The concepts of stabilization with internal loop are analyzed for well-posed transfer functions. We obtain some sufficient and necessary conditions such that a stabilizing controller with internal loop stabilizes plant L . We also analyze two special subclasses of stabilizing controllers with internal loop, called canonical and dual canonical controllers, and show that all stabilizing controllers can be parameterized by a doubly coprime factorization of the original transfer function.

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Nai-feng Gan 1, 2 and Yu-feng LU 1 and Ting Gong 1

Academic Editor:Manuel De la Sen

1, School of Mathematical Sciences, Dalian University of Technology, Dalian 116027, China
2, College of Mathematics and Information Science, Anshan Normal University, Anshan 114007, China

Received 22 January 2014; Revised 9 April 2014; Accepted 10 April 2014; 28 April 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Control Theory is a relevant field from the mathematical theoretical point of view as well as in many applications (see [1-6]). What is important, in particular, is the closed-loop stabilization of dynamic system under appropriate feedback control as a minimum requirement to design a well-posed feedback system. In the last twenty years, the closed-loop system whose stability is achieved by the controller with internal loop has attracted the attention of many authors (see [7, 8]). While extending the theory of dynamic stabilization to regular linear systems (a subclass of the well-posed linear systems), it was shown in [7, Example 2.3] that even the standard observer-based controller is not a well-posed linear system and its transfer function is not well-posed. To overcome this, paper [8] proposed another definition of a stabilizing controller which is more general than that has been defined earlier, the so-called stabilizing controller with internal loop. The concept enabled a simple Youla parameterization and has some advantages which turn out to be very important for infinite-dimensional systems. It makes the theory of dynamic stabilization simpler and more natural [8].

Recently, the study of time-varying systems using modern mathematical methods has come into its own. This is a scientific necessity. After all, many common physical systems are time varying (see [9-14]). Paper [15] studied the concept of stabilization with internal loop for infinite-dimensional discrete time-varying systems and gave a parameterization of all stabilizing controllers with internal loop if I - _{K 22} has a well-posed inverse in the framework of nest algebra. But in many cases, the controller C = _{K 11} + _{K 12} ( I - _{K 22} ^{) - 1} _{K 21} will not be well-posed, but C perhaps stabilizes L .

In this paper, we study the stabilization with internal loop for the linear time-varying system under the framework of nest algebra. We extend our study of controllers with internal loop to more general use and give a parameterization of all stabilizing controllers with internal loop even if I - _{K 22} = 0 . It is found that the stabilization with internal loop for the linear time-varying system obtained in [15] can be viewed as a special case of that obtained here. As we know, if the plant is not strictly proper, it is difficult to choose the parameter in such way that the resulting controller will be well-posed. Even if we choose to ignore well-posedness, we still have to ensure that the denominator in the Youla parameterization is invertible. This makes it awkward to use this parameterization to solve the practical problems, while the controller with internal loop overcomes this awkwardness. We obtain canonical and dual canonical controllers and show that all stabilizing controllers can be parameterized by a doubly coprime factorization of the original transfer function.

The rest of this paper is organized as follows. Mathematical background material and notation are introduced in Section 2. In Section 3, we give some sufficient and necessary conditions that a stabilizing controller with internal loop stabilizes plant L . In Section 4, we introduce canonical and dual canonical controllers. We show that a plant L is stabilizable with internal loop by a canonical (dual canonical) controller if and only if L has a right coprime (left coprime) factorization. We give a complete parameterization of all (dual) canonical stabilizing controllers with internal loop. Some conclusions are drawn in Section 5.

2. Preliminaries

We denote by _{... +} the nonnegative integers and by ... the complex numbers. Let H be the complex infinite-dimensional Hilbert sequence space: [figure omitted; refer to PDF] where | · | denotes the standard Euclidean norm on ... . _{H e} will denote the extended space: [figure omitted; refer to PDF]

Definition 1 (see [3]).

A family N of closed subspaces of the Hilbert space H is a complete nest if

(1) { 0 } , H ∈ N .

(2) For _{N 1} , _{N 2} , either _{N 1} ⊆ _{N 2} or _{N 2} ⊆ _{N 1} .

(3) If { _{N α} } is a subfamily in N , then _{∩ α N α} and _{⋁ α N α} are also in N .

Every subspace N of H is identifiable with the orthogonal projection _{P n} [figure omitted; refer to PDF] Properties (1) to (3) can be reformulated as follows.

( ^{1 [variant prime]} ) : 0 , I ∈ N .

( ^{2 [variant prime]} ) : For _{P 1} , _{P 2} ∈ N , either _{P 1} ...4; _{P 2} or _{P 2} ...4; _{P 1} .

( ^{3 [variant prime]} ) : If { _{P α} } is a nest in N which converges weakly (equivalently, strongly) to P , then P ∈ N .

Definition 2 (see [3]).

If N is a nest and P is its associated family of orthogonal projections, [figure omitted; refer to PDF] is called a nest algebra, where £ ( H ) is the algebra of all bounded linear operators on H .

A linear transformation T on _{H e} is causal if _{P n} T = _{P n} T _{P n} for n ...5; 0 .

Lemma 3 (see [3]).

The following are equivalent:

(1) T on _{H e} is stable.

(2) T is causal and T |" H is a bound operator.

(3) T is the extension to _{H e} of an operator in Alg R .

This lemma allow us to identify the algebra S of stable operators on _{H e} with the nest algebra Alg R . The restriction of T ∈ S to H is in Alg R and the extension of S ∈ Alg R to _{H e} is in S . Alg R and S are identical.

For L , K ∈ £ , the operator matrix ( I - K - L I ) defined on _{H e} [ecedil]5; _{H e} is called the feedback system with plant L and compensator K .

In Figure 1, L represents a given plant (system) and K = ( _{K 11 K 12 K 21 K 22} ) a compensator or controller; _{e 1} , _{e 2} denote the externally applied inputs; _{u L} , _{u K} denote the inputs to the plant and compensator, respectively; and _{y L} , _{y K} denote the outputs of the compensator and plant, respectively.

Figure 1: The standard feedback system.

[figure omitted; refer to PDF]

The closed-loop system equation are [figure omitted; refer to PDF]

The system is well-posed if the internal input u can be expressed as a causal function of the external input e . This is equivalent to requiring that ( I - K - L I ) be invertible. The inverse is easily computed formally and is given by the matrix as follows: [figure omitted; refer to PDF]

The closed-loop system { L , K } is stable if ( I - K - L I ) has a bound causal inverse defined on H [ecedil]5; H . The stability of the closed-loop system is equivalent to requiring that the four elements of the 2 × 2 matrix H ( L , K ) be in S . L ∈ £ is stabilizable if there exists K ∈ £ such that { L , K } is stable.

3. Stabilization with Internal Loop

In this section, a new type of controller is introduced, the so-called stabilizing controller with internal loop; see [16-18].

The intuitive interpretation of Figure 2 is as follows: L represents the plant and K is the transfer function of the controller from ( _{y k ζ i} ) to ( _{u k ζ 0} ) , when all the connections are open. The connection from _{ξ 0} to _{ξ i} is the so-called internal loop.

Figure 2: The plant L connected to a controller K with internal loop.

[figure omitted; refer to PDF]

Partitioning K into ( _{K 11 K 12 K 21 K 22} ) where _{K i j} ∈ £ , i , j = 1,2 , ... , [figure omitted; refer to PDF] is the transfer function of the closed-loop system from ( _{u L y k ζ i} ) to ( _{e 1 e 2 e 3} ) .

Suppose I - _{K 22} is invertible in £ ; a parameterization of all stabilizing controllers with internal loop is given in [15]. If I - _{K 22} has a well-posed inverse, the internal loop can be closed first and the transfer function from _{y k} to _{u k} is [figure omitted; refer to PDF] But in many cases, the expression (8) is not defined at all (this can happen if I - _{K 22} is nowhere invertible).

Example 4.

Suppose L = I , [figure omitted; refer to PDF] It is easy to see that the transfer function (8) of the controller is undefined since I - _{K 22} = 0 . It is not difficult to check that K stabilizes L with internal loop (this verification can be simplified considerably by using Lemma 10).

In the following, we give some sufficient and necessary conditions such that a stabilizing controller with internal loop stabilizes plant L avoiding the condition that I - _{K 22} is invertible.

Theorem 5.

Suppose that _{K 11} is an admissible feedback transfer function for L . Then F ( K , L ) has a well-posed inverse if and only if I - M is invertible in £ , where M = _{K 22} + _{K 21} L ( I - _{K 11} L ^{) - 1} _{K 12} .

Proof.

Consider the following [figure omitted; refer to PDF] where _{S 11} = ( I - _{K 11} - L I ) , _{S 12} = ( - _{K 12} 0 ) , _{S 21} = ( 0 - _{K 21} ) , _{S 22} = I - _{K 22} , Δ = _{S 22} - _{S 21} ^{S 11 - 1} _{S 12} .

Since _{S 11} = ( I - _{K 11} - L I ) is invertible in _{M 2} ( £ ) , thus ( I - _{K 11} - _{K 12} - L I 0 0 - _{K 21} I - _{K 22} ) is invertible in _{M 3} ( £ ) if and only if [figure omitted; refer to PDF] is invertible in £ .

Further, the condition that F ( K , L ) has a well-posed inverse is equivalent to that K is an admissible feedback transfer function with internal loop for L [7], so we have the following result.

Theorem 6.

Suppose that _{K 11} is a stabilizing controller for L ; then K = ( _{K 11 K 12 K 21 K 22} ) is a stabilizing controller with internal loop for L if and only if

(i) ( I - M ^{) - 1} ∈ S , where M = _{K 22} + _{K 21} L ( I - _{K 11} L ^{) - 1} _{K 12} ,

(ii) there exist _{E 1} , _{E 2} ∈ S such that L _{E 1} ∈ S , _{E 2} L ∈ S , _{E 1} ( I - M ) _{E 2} L ∈ S , _{E 1} ( I - M ) _{E 2} ∈ S , L _{E 1} ( I - M ) _{E 2} ∈ S , L _{E 1} ( I - M ) _{E 2} L ∈ S ,

(iii): _{K 12} = ( I - _{K 11} L ) _{E 1} ( I - M ) ,

(iv) _{K 21} = ( I - M ) _{E 2} ( I - L _{K 11} ) .

Proof.

_{K 11} stabilizes L if and only if ( ( I - _{K 11} L ^{) - 1} _{K 11} ( I - L _{K 11} ^{) - 1} L ( I - _{K 11} L ^{) - 1} ( I - L _{K 11} ^{) - 1} ) ∈ _{M 2} ( S ) .

If there exist _{E 1} , _{E 2} ∈ S that satisfy (i)-(iv), all components in H ( L , K ) = F ( K , L ^{) - 1} are [figure omitted; refer to PDF] Thus, H ( L , K ) ∈ _{M 3} ( S ) , { L , K } is stable.

Conversely, H ( L , K ) = F ( K , L ^{) - 1} , and all components are [figure omitted; refer to PDF] where M = _{K 22} + _{K 21} L ( I - _{K 11} L ^{) - 1} _{K 12} . If H ( L , K ) ∈ _{M 3} ( S ) , then ( I - M ^{) - 1} ∈ S . Let ( I - _{K 11} L ^{) - 1} _{K 12} ( I - M ^{) - 1} = _{E 1} ∈ S , ( I - M ^{) - 1} _{K 21} ( I - L _{K 11} ^{) - 1} = _{E 2} ∈ S ; then _{K 12} = ( I - _{K 11} L ) _{E 1} ( I - M ) , _{K 21} = ( I - M ) _{E 2} ( I - L _{K 11} ) . From ( 3,1 ) ∈ S and ( 2,3 ) ∈ S , we have _{E 2} L ∈ S and L _{E 1} ∈ S . Consider ( 1,1 ) ∈ S , ( 1,2 ) ∈ S , ( 2,1 ) ∈ S , ( 2,2 ) ∈ S , and { L , _{K 11} } are stable; thus all other conditions in (ii) hold.

Remark 7.

{ L , _{K 11} } stable is only sufficient condition for { L , K } stable, but not a necessary condition.

Theorem 8.

If _{K 11} is an admissible controller for P , then { L , K } is stable if and only if

(i) ^{Δ - 1} = [ I - _{K 22} - _{K 21} L ( I - _{K 11} L ^{) - 1} _{K 12} ^{] - 1} ∈ S ,

(ii) A = ( I - _{K 11} L ^{) - 1} _{K 12} ^{Δ - 1} _{K 21} ( I - L _{K 11} ^{) - 1} + _{K 11} ( I - L _{K 11} ^{) - 1} = ( I - _{K 11} L ^{) - 1} ( _{K 12} ^{Δ - 1} _{K 21} + _{K 11} - _{K 11} L _{K 11} ) ( I - L _{K 11} ^{) - 1} ∈ S ,

(iii): A L ∈ S , L A ∈ S , L A L + L ∈ S , _{K 21} ( I - L _{K 11} ^{) - 1} L ∈ S , L ( I - _{K 11} L ^{) - 1} _{K 12} ∈ S , _{K 21} ( I - L _{K 11} ^{) - 1} ∈ S , ( I - _{K 11} L ^{) - 1} _{K 12} ∈ S .

In fact, the conditions of Theorem 8 are weaker than those of Theorem 6. From the proof of Theorem 6, it is easy to obtain the result of Theorem 8.

We extend the plant G = ( L 0 0 0 _{K 11 K 12} 0 _{K 21 K 22} ) , and L and C are parallel connection. F = ( 0 I 0 I 0 0 0 0 I ) as a feedback operator of G , so we have the following result.

Theorem 9.

K is a stabilizing controller with internal loop for L if and only if I - F G is invertible in _{M 3} ( S ) .

Proof.

F is a stabilizing controller for G if and only if [figure omitted; refer to PDF]

If ( I - F G ^{) - 1} ∈ _{M 3} ( S ) , then F ( I - G F ^{) - 1} = ( I - F G ^{) - 1} F ∈ _{M 3} ( S ) . Since ( I - G F ^{) - 1} = I + G ( I - F G ^{) - 1} F , thus we only need to prove G ( I - F G ^{) - 1} ∈ _{M 3} ( S ) . Consider ^{F 2} = I ; thus G ( I - F G ^{) - 1} = ^{F 2} G ( I - F G ^{) - 1} = F [ F G ( I - F G ^{) - 1} ] = F [ ( I - F G ^{) - 1} - ( I - F G ) ( I - F G ^{) - 1} ] = F ( I - F G ^{) - 1} - F . If ( I - F G ^{) - 1} ∈ _{M 3} ( S ) , then G ( I - F G ^{) - 1} ∈ _{M 3} ( S ) .

Conversely, it is obvious.

4. Canonical and Dual Canonical Controllers

Another motivation for introducing controllers with internal loop is to obtain Youla parameterization. If the plant is not strictly proper, it is difficult to choose the parameter in such way that the resulting controller will be well-posed. Even if we choose to ignore well-posedness, we still have to ensure that the denominator in the Youla parameterization is invertible. By contrast, we can obtain a parameterization for all stabilizing canonical or dual canonical controllers.

The transfer functions of the controllers obtained there were of the form [figure omitted; refer to PDF] We call the controllers of form (15) canonical controllers . Analogously, controllers of the form [figure omitted; refer to PDF] will be called dual canonical controllers .

In following, we analyze the properties of (dual) canonical controllers in some detail. First, we recall Lemma 10 from [15].

Lemma 10 (see [15]).

The canonical controller K = ( 0 I _{K 21 K 22} ) stabilizes L ∈ £ with internal loop if and only if [figure omitted; refer to PDF] is invertible in S and L ^{Δ - 1} ∈ S .

If L ∈ £ has a right-coprime factorization L = N ^{M - 1} , then K stabilizes L with internal loop if and only if [figure omitted; refer to PDF] is invertible in S .

We now turn to the problem of simultaneous stabilization. Given _{L 0} ∈ S and _{L 1} ∈ £ , the following Corollaries 11 and 12 give the conditions that _{L 1} - _{L 0} can be stabilized by some canonical controller.

Corollary 11.

If _{L 0} ∈ S and _{L 1} ∈ £ can be simultaneously stabilized by canonical controller K = ( 0 I _{K 21 K 22} ) , then _{L 1} - _{L 0} can be strongly stabilized by some canonical controller.

Proof.

If ( _{M 1 N 1} ) is a strong right representation of _{L 1} , then ( _{M 1 N 1} - _{L 0 M 1} ) is a strong right representation of _{L 1} - _{L 0} , since [figure omitted; refer to PDF] for _{L 0} ∈ S .

Suppose R = ( 0 I _{R 21 R 22} ) stabilizes _{L 1} - _{L 0} ; then by Lemma 10, [figure omitted; refer to PDF] is invertible in S . By Lemma 10, Δ and D are invertible in S : [figure omitted; refer to PDF] Define [figure omitted; refer to PDF] Thus ^{D [variant prime]} is invertible in S , and R = ( 0 I _{R 21 R 22} ) stabilizes _{L 1} - _{L 0} .

Corollary 12.

Suppose _{L 0} ∈ S , _{L 1} ∈ £ , and ( _{M 1 N 1} ) is a strong right representation of _{L 1} . If _{L 1} can be stabilized by canonical controller K = ( 0 I _{K 21 K 22} ) , then _{L 1} - _{L 0} can be stabilized by some canonical controller.

Proof.

Since _{L 0} ∈ S , then ( _{M 1 N 1} - _{L 0 M 1} ) is a strong right representation of _{L 1} - _{L 0} . By Lemma 10, K = ( 0 I _{K 21 K 22} ) stabilizes _{L 1} if and only if D = _{M 1} - _{K 22 M 1} - _{K 21 N 1} is invertible in S . Suppose R = ( 0 I _{R 21 R 22} ) stabilizes _{L 1} - _{L 0} ; then by Lemma 10, [figure omitted; refer to PDF] is invertible in S . Define _{R 21} = _{K 21} ∈ S , _{R 22} = _{K 22} + _{K 21 L 0} ∈ S ; thus ^{D [variant prime]} is invertible in S , and R = ( 0 I _{R 21 R 22} ) stabilizes _{L 1} - _{L 0} .

The conditions of Corollary 12 are weaker than those of Corollary 11. In following, we will discuss the stabilization of { L , K } with coprime factorizations.

Theorem 13.

The canonical controller ( 0 I _{K 21 K 22} ) stabilizes L if and only if Δ = I - _{K 22} - _{K 21} L ∈ £ is invertible in S and L ^{Δ - 1} ∈ S .

Proof.

Let _{K 11} = 0 , _{K 12} = I , _{K 21} , _{K 22} ∈ S ; from Theorem 8, we have that Δ = I - _{K 22} - _{K 21} L ∈ £ is invertible in S and L ^{Δ - 1} ∈ S .

Remark 14.

When _{K 11} = 0 , ( ( I - _{K 11} L ^{) - 1} _{K 11} ( I - L _{K 11} ^{) - 1} L ( I - _{K 11} L ^{) - 1} ( I - L _{K 11} ^{) - 1} ) = ( I 0 L I ) ∈ _{M 2} ( £ ) , thus _{K 11} = 0 is an admissible controller for L ; we do not need to emphasize this in Theorem 13.

Remark 15.

By Remark 14, L ∈ £ , but L ∈ - S , _{K 11} = 0 is not a stabilizing controller for L , but ( 0 I _{K 21 K 22} ) is a stabilizing controller with internal loop for L .

Theorem 16.

If L has right coprime factorization N ^{M - 1} , then L can be stabilized by canonical controller ( 0 I _{K 21 K 22} ) if and only if M - _{K 22} M - _{K 21} N is invertible in S .

Proof.

By Theorem 13, { L , K } is stable if and only if ^{Δ - 1} , L ^{Δ - 1} ∈ S . Consider L = N ^{M - 1} ; then ^{Δ - 1} = M ( M - _{K 22} M - _{K 21} N ^{) - 1} , L ^{Δ - 1} = N ( M - _{K 22} M - _{K 21} N ^{) - 1} . If M - _{K 22} M - _{K 21} N ∈ S , then ^{Δ - 1} , L ^{Δ - 1} ∈ S . Conversely, if ^{Δ - 1} , L ^{Δ - 1} ∈ S and ( Y X ) ( ^{Δ - 1} L ^{Δ - 1} ) = I , then ( Y X ) ( ^{Δ - 1} L ^{Δ - 1} ) = ( M - _{K 22} M - _{K 21} N ^{) - 1} ∈ S .

Theorem 17.

If L has right coprime factorization N ^{M - 1} if and only if L can be stabilized by some canonical controller.

Proof.

If N ^{M - 1} is right coprime factorization of L , there exist Y , X ∈ S such that ( Y X ) ( M N ) = I . Take _{K 21} = - X ∈ S , _{K 22} = I - Y ∈ S ; then M - _{K 22} M - _{K 21} N = Y M + X N = I is invertible in S . By Theorem 13, ( 0 I - X I - Y ) stabilizes L .

Conversely, If K = ( 0 I _{K 21 K 22} ) stabilizes L , by Theorem 13, ^{Δ - 1} ∈ S , L ^{Δ - 1} ∈ S . Take M = ^{Δ - 1} , N = L ^{Δ - 1} , Y = I - _{K 22} ∈ S , X = - _{K 21} ∈ S ; then Y M + X N = ( I - _{K 22} ) ^{Δ - 1} - _{K 21} L ^{Δ - 1} = I ; thus, N ^{M - 1} is right coprime factorization of L .

We expect a strong relationship between stabilization with internal loop and the usual concept of stabilization by the parameterization of all stabilizing (dual) canonical controllers.

Theorem 18.

Suppose that L has a doubly coprime factorization; then all canonical controllers that stabilize L with internal loop are parameterized by [figure omitted; refer to PDF] where Q ∈ S , E ∈ S ∩ ^{S - 1} .

Proof.

Take _{K 21} = E ( - X + Q M ^ ) , _{K 22} = I - E ( Y + Q N ^ ) , where E ∈ S ∩ ^{S - 1} , Q ∈ S ; then D = M - _{K 22} M - _{K 21} N = M - ( I - E ( Y + Q N ^ ) ) M - E ( - X + Q M ^ ) N = E ∈ S ∩ ^{S - 1} ; by Theorem 17, K stabilizes L .

Conversely, if K stabilizes L , by Theorem 16, D = M - _{K 22} M - _{K 21} N is invertible in S . Consider I = ^{D - 1} ( I - _{K 22} ) M - ^{D - 1} _{K 21} N ; thus ( ^{D - 1} ( I - _{K 22} ) - ^{D - 1} _{K 21} ) ∈ _{M 1 × 2} ( S ) is a left inverse of ( M N ) . By Theorem 17, there exist Q ∈ S such that ^{D - 1} ( I - _{K 22} ) = Y + Q N ^ , - ^{D - 1} _{K 21} = X - Q M ^ , rewrite these as _{K 21} = D ( - X + Q M ^ ) , _{K 22} = I - D ( Y + Q N ^ ) .

The following Theorem contains the dual statements of Theorems 13, 16, 17 and 18.

Theorem 19.

(a) The dual canonical controller ( 0 _{K ^ 12} I _{K ^ 22} ) stabilizes L if and only if Δ ^ = I - _{K ^ 22} - L _{K ^ 21} ∈ £ is invertible in S and ^{Δ ^ - 1} L ∈ S .

(b) If L has left coprime factorization ^{M ^ - 1} N ^ , then L can be stabilized by canonical controller ( 0 _{K ^ 12} I _{K ^ 22} ) if and only if M ^ - M ^ _{K ^ 22} - N ^ _{K ^ 21} is invertible in S .

(c) If L has left coprime factorization ^{M ^ - 1} N ^ if and only if L can be stabilized by some dual canonical controller.

(d) Suppose that L has a doubly coprime factorization, then all dual canonical controllers that stabilize L with internal loop are parameterized by [figure omitted; refer to PDF] where Q ∈ S , E ∈ S ∩ ^{S - 1} .

The Proof of (c). Suppose L = ^{M ^ - 1} N ^ , there exist X ^ , Y ^ ∈ S such that ( - N ^ M ^ ) ( - X ^ Y ^ ) = I . Let _{K 12} = - X ^ , _{K ^ 22} = I - Y ^ ∈ S , then M ^ - M ^ _{K ^ 22} - N ^ _{K ^ 12} = I ∈ S ∩ ^{S - 1} , L can be stabilized by ( 0 - X ^ I I - Y ^ ) .

Conversely, if L can be stabilized by K ^ , by (1), ^{Δ ^ - 1} , ^{Δ ^ - 1} L ∈ S . Let M ^ = ^{Δ ^ - 1} , N ^ = ^{Δ ^ - 1} L ∈ S , X ^ = - _{K ^ 12} , Y ^ = I - _{K ^ 22} , then L = ^{M ^ - 1} N ^ = ( ^{Δ ^ - 1 ) - 1 Δ ^ - 1} L , N ^ X ^ + M ^ Y ^ = I , thus ( - ^{Δ ^ - 1} L ^{Δ ^ - 1} ) is a left coprime factorization of L .

Theorem 20.

If the canonical controller ( 0 I _{K 21 K 22} ) stabilizes L , then L can be stabilized by the dual canonical controller ( 0 ^{Δ - 1} _{K 21} ^{M ^ - 1} I I - ( I + L ^{Δ - 1} _{K 21} ) ^{M ^ - 1} ) .

Proof.

If L can be stabilized by the canonical controller ( 0 I _{K 21 K 22} ) , by Theorems 13 and 17, ^{Δ - 1} = ( I - _{K 22} - _{K 21} L ^{) - 1} ∈ S , L ^{Δ - 1} ∈ S , and L has a right coprime factorization. From [17], we known that L has a left coprime factorization ^{M ^ - 1} N ^ and there exist X ^ , Y ^ ∈ S such that N ^ X ^ + M ^ Y ^ = I . Let _{K ^ 12} = ^{Δ - 1} _{K 21} ^{M ^ - 1} , _{K ^ 22} = I - ( I + L ^{Δ - 1} _{K 21} ) ^{M ^ - 1} . In the following, we need to prove (1) _{K ^ 12} , _{K ^ 22} ∈ S and (2) K ^ stabilizes L .

_{K ^ 12} = ^{Δ - 1} _{K 21} ^{M ^ - 1} I = ^{Δ - 1} _{K 21} ^{M ^ - 1} ( N ^ X ^ + M ^ Y ^ ) = ^{Δ - 1} _{K 21} ^{M ^ - 1} N ^ X ^ + ^{Δ - 1} _{K 21} Y ^ = ^{Δ - 1} _{K 21} L X ^ + ^{Δ - 1} _{K 21} Y ^ = [ ^{Δ - 1} ( I - _{K 22} ) - I ] X ^ + ^{Δ - 1} _{K 21} Y ^ ∈ S .

Since ( I + L ^{Δ - 1} _{K 21} ) ^{M ^ - 1} = ( I + L ^{Δ - 1} _{K 21} ) ^{M ^ - 1} ( N ^ X ^ + M ^ Y ^ ) = ( L + L ^{Δ - 1} _{K 21} L ) X ^ + ( I + L ^{Δ - 1} _{K 21} ) Y ^ = L ^{Δ - 1} ( I - _{K 22} ) X ^ + ( I + L ^{Δ - 1} _{K 21} ) Y ^ ∈ S , so _{K ^ 22} = I - ( I + L ^{Δ - 1} _{K 21} ) ^{M ^ - 1} ∈ S .

I - _{K ^ 22} - L _{K ^ 12} = I - I + ( I + L ^{Δ - 1} _{K 21} ) ^{M ^ - 1} - L ^{Δ - 1} _{K 21} ^{M ^ - 1} = ^{M ^ - 1} is invertible in S , ^{Δ ^ - 1} L = M ^ · ^{M ^ - 1} N ^ = N ^ ∈ S . By Theorem 19(a), K ^ stabilizes L .

Notice that if a canonical K stabilizes L with internal loop, then _{K 21} and I - _{K 22} are left coprime, since ( I - _{K 22} ^{Δ - 1} ) - _{K 21} L ^{Δ - 1} = I . Theorem 20 has a dual statement for right-coprime factorizations K .

There is a similar result for the dual canonical controller.

Theorem 21.

If the dual canonical controller ( 0 _{K ^ 21} I _{K ^ 22} ) stabilizes L , then L can be stabilized by the dual canonical controller ( 0 I ^{M - 1} _{K ^ 12} ^{Δ ^ - 1} I - ^{M - 1} ( I + _{K ^ 12} ^{Δ ^ - 1} L ) ) .

The proof of Theorem 21 is similar to that of Theorem 20, and we omit it.

5. Conclusion

In this paper, we investigate the dynamic stabilization of a large class of transfer functions in the framework of nest algebra. To obtain a natural generalization of dynamic stabilization, we introduce a new concept of stabilization by a controller with internal loop. The concept enables a simple Youla parameterization and has some advantages which turn out to be very important for infinite-dimensional systems. It makes the theory of dynamic stabilization simpler and more natural.

We also analyze canonical and dual canonical controllers, which are controllers with internal loop of a special (simple) structure. We have found that these are closely related to (doubly) coprime factorization, and we have given a complete parameterization of all stabilizing controllers with internal loop which are (dual) canonical.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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