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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 132597, 16 pages
doi:10.1155/2012/132597
Research Article

Stabilization of Time-Varying System by Controllers with Internal Loop

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 18 April 2012; Accepted 10 May 2012

Academic Editor: Zidong Wang

Copyright © 2012 Yufeng Lu and Chengkai Shi. 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.

Abstract

We study the concept of stabilization with internal loop for infinite-dimensional discrete time-varying systems in the framework of nest algebra. We originally give a parametrization of all stabilizing controllers with internal loop, and it covers the parametrization of canonical or dual canonical controllers with internal loop obtained before. We show that, in practical application, the controller with internal loop overcomes the awkwardness brought by the extra invertibility condition in the parametrization of the conventional controllers. We also prove that the strong stabilization problem can be completely solved in the closed-loop system with internal loop. Thus the advantage of the controller with internal loop is addressed in the framework of nest algebra.

1. Introduction

The closed-loop system whose stability is achieved by the controller with internal loop has attracted the attention of many authors in recent years (see [15]). This system was originally introduced by Weiss and Curtain in 1997 in [1]. When they extended the theory of dynamic stabilization to regular linear systems (a subclass of the well-posed linear systems), it was shown in Example 6.5 of [1] that even the standard observer-based controller is not a well-posed linear system as needed, correspondingly, its transfer function is not well-posed. To overcome this difficulty, a new type of controller, the so-called stabilizing controller with internal loop, was introduced. This controller is more general and useful than the standard feedback controller. Until now, only a special class of stabilizing controllers with internal loop called canonical controllers is widely investigated. In [1], a procedure was developed to design the canonical controllers for stabilizable and detectable plants. In [6], the parametrization for all canonical controllers is given which is clean and avoids the extra invertibility condition in the parametrization for the controller in the standard feedback system. In [7], the author extended the theory to non-well posed systems, and the robust stabilization problem is considered by using the canonical controller.

In recent years, the study of time-varying systems using modern mathematical methods has come into its own. This was a scientific necessity. After all, many common physical systems are time varying. In [8], A. Feintuch specifically introduced a framework of nest algebra and the control theory for linear time-varying systems was studied in this framework. Meanwhile, many stabilization problems for various nonlinear time-varying systems were widely considered as well (see [916]). Based on these cases, we are motivated to consider the new model of closed-loop feedback system with internal loop for time-varying systems.

In this paper, we study the concept of 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 cases and originally give a parametrization of all stabilizing controllers with internal loop. It is found that the parametrization of the canonical controller obtained in [6] can be viewed as a special case of the parametrization obtained here. As we know, the parametrization of the conventional controller is not clean, and there is always an extra invertibility condition on the parameter. This in turn makes it awkward to use this parametrization to solve the practical problems. While the controller with internal loop overcomes this awkwardness. We take the sensitivity minimization problem as an example to show this advantage of the controller with internal loop. The strong stabilization problem is known as the design of a stable controller which stabilizes the given plant. In the framework of nest algebra, it is still an open problem and only a necessary condition is addressed in [17] for this problem. We prove that any stabilizable plant can be strongly stabilized by the controller with internal loop. This means that the strong stabilization problem can be completely solved in the system with internal loop. We also give a simple example to show how to design the strongly stabilizing controller with internal loop.

This paper is organized as follows. In Section 2, we recall some basic concepts of the linear systems in the framework of nest algebra. In Section 3, we introduce the closed-loop system whose stability is achieved by the controller with internal loop and firstly give a parametrization for all stabilizing controllers with internal loop. In Section 4, we focus on the canonical controller and show the benefit of the controller with internal loop in the practical application. In Section 5, we define the strongly stabilizing controller with internal loop and address an advantage of the controller with internal loop in the framework of nest algebra.

2. Preliminaries

Let be the complex infinite dimensional Hilbert sequence space: 2=𝑥0,𝑥1,𝑥2,𝑥𝑖,𝑖=0||𝑥𝑖||2,<(2.1) where || denotes the standard Euclidean norm on with inner product (𝑥,𝑦)=𝑖=0𝑥𝑖𝑦𝑖. 𝑒 will denote the extended space: 𝑒=𝑥0,𝑥1,𝑥2,𝑥𝑖.(2.2) For each 𝑛0, let 𝑃𝑛 denote the standard truncation projection defined on and 𝑒 by 𝑃𝑛𝑥0,𝑥1,,𝑥𝑛,𝑥𝑛+1=𝑥,0,𝑥1,,𝑥𝑛,0,0,.(2.3) A continuous linear transformation 𝑇 on 𝑒 with the standard seminorm topology ([8, Chapter 5]) is a causal linear system (or a linear system) if for each 𝑛0, 𝑃𝑛𝑇=𝑃𝑛𝑇𝑃𝑛. Let be the set of all linear systems on 𝑒. Then any element of is a lower triangular matrix (with respect to the standard basis, see [8, Chapter 5]).

A linear system 𝑇 is stable if its restriction to is a bounded operator ([8, Chapter 5]). We denote the set of stable systems by 𝒮, then 𝒮 is a weakly closed algebra containing the identity, referred to in the operator algebra literature as a nest algebra ([8, Chapter 5]).

For 𝑃,𝐶, we consider the standard feedback configuration with plant 𝑃 and controller 𝐶 shown in Figure 1.

132597.fig.001
Figure 1: Standard feedback configuration.

𝑢1, 𝑢2 denote the externally applied inputs; 𝑒1, 𝑒2 denote the inputs to the plant and compensator, respectively, and 𝑦1, 𝑦2 denote the outputs of the plant and compensator, respectively. The closed loop system equations are 𝑢1𝑢2=𝑒𝐼𝐶𝑃𝐼1𝑒2.(2.4) The system is well posed if the internal input 𝑒=𝑒1𝑒2 can be expressed as a causal function of the external input 𝑢=𝑢1𝑢2. This is equivalent to requiring that 𝐼𝐶𝑃𝐼 be invertible. This inverse can be easily computed and is given by the transfer matrix 𝐻(𝑃,𝐶)=(𝐼+𝐶𝑃)1𝐶(𝐼+𝑃𝐶)1𝑃(𝐼+𝐶𝑃)1(𝐼+𝑃𝐶)1.(2.5)

Definition 2.1 (see [8]). The closed loop system {𝑃,𝐶} is stable if all the entries of 𝐻(𝑃,𝐶) are stable systems on . The plant 𝑃 is stabilizable if there exists a causal linear system 𝐶 such that {𝑃,𝐶} is stable.
Recall that the graph of a linear transformation 𝑃 with domain 𝒟(𝑃)={𝑥𝑃𝑥} is 𝒢(𝑃)={𝑥𝑃𝑥𝑥𝒟(𝑃)}. Then we can give the definitions of strong right representation and strong left representation.

Definition 2.2 (see [8]). A plant 𝑃 has a strong right representation 𝑀𝑁 with 𝑀 and 𝑁 stable if
(1)𝒢(𝑃)=Ran𝑀𝑁,
(2) there exist 𝑋,𝑌𝒮 such that [𝑌𝑋]𝑀𝑁=𝐼.
A plant 𝑃 has a strong left representation 𝑁[𝑀] with 𝑀 and 𝑁 stable if
(1)𝑁𝒢(𝑃)=Ker[𝑀],
(2) there exist 𝑋,𝑌𝒮 such that 𝑁[𝑀]𝑋𝑌=𝐼.

The following result on strong right representation is proved in [8].

Theorem 2.3 (see [8]). Suppose 𝑀,𝑁𝒮. Then 𝑀𝑁 is a strong right representation of 𝑃 if and only if
(1) there exist 𝑋,𝑌𝒮 such that [𝑌𝑋]𝑀𝑁=𝐼,
(2) M is invertible in .

We say that a plant 𝑃 has a right coprime factorization if there exist 𝑀, 𝑁, 𝑋, 𝑌𝒮 such that 𝑃=𝑁𝑀1 and 𝑌𝑀+𝑋𝑁=𝐼. The proof of Theorem 2.1 in [8] implies that 𝑀𝑁 is a strong right representation of 𝑃 if and only if 𝑁𝑀1 is a right coprime factorization of 𝑃. Similarly, 𝑁[𝑀] is a strong left representation of 𝑃 if and only if 𝑀1𝑁 is a left coprime factorization of 𝑃.

The following theorem is the classical Youla Parametrization Theorem.

Theorem 2.4 (see [8]). A causal linear system 𝑃 is stabilizable if and only if 𝑃 has a strong right and a strong left representation. If this is the case, the representations can be chosen so that one has the double Bezout identity 𝑁𝑀𝑋𝑁𝑌=𝑋𝑁𝑌𝑁𝑀=𝑌𝑋𝑀𝑀𝑌𝑋𝐼00𝐼.(2.6) A causal linear system 𝐶 stabilizes 𝑃 if and only if it has a strong right representation 𝑌𝑁𝑄𝑋+𝑀𝑄 and a strong left representation [(𝑋+𝑄𝑀)𝑌𝑄𝑁] for some 𝑄𝒮.

3. Controllers with Internal Loop

In this section, we investigate the stabilization of the time-varying system by controllers with internal loop in the framework of nest algebra. This system is illustrated in Figure 2.

132597.fig.002
Figure 2: The plant 𝑃 connected to a controller 𝐾𝐼 with internal loop.

The intuitively interpretation of Figure 2: 𝑃 is the plant and 𝐾𝐼 is a transfer map from 𝑒2𝑒3 to 𝑦2𝑦3 when all the connections are open. Then the connection from 𝑦3 to 𝑒3 is called internal loop. The closed loop system determined by the plant 𝑃 and the controller 𝐾𝐼 with internal loop is denoted by {𝑃,𝐾𝐼}.

Partitioning 𝐾𝐼 into 𝐶11𝐶12𝐶21𝐶22,(3.1) where 𝐶𝑖𝑗,   𝑖,𝑗=1,2, the closed loop system equations are 𝑢1𝑢2𝑢3=𝐼𝐶11𝐶12𝑃𝐼00𝐶21𝐼𝐶22𝑒1𝑒2𝑒3.(3.2)

We say that the system is well posed if 𝐼𝐶11𝐶12𝑃𝐼00𝐶21𝐼𝐶22 is invertible and We denote this inverse by 𝐻(𝑃,𝐾𝐼).

Definition 3.1. The closed loop system {𝑃,𝐾𝐼} determined by the plant 𝑃 and the controller with internal loop 𝐾𝐼 is stable if all the entries of 𝐻(𝑃,𝐾𝐼) are stable. The plant 𝑃 is stabilizable by a controller with internal loop if there exists a 𝐾𝐼 such that 𝐻(𝑃,𝐾𝐼) is stable. In this case, 𝐾𝐼 is called a stabilizing controller with internal loop for 𝑃.
In the previous papers, the study of stabilizing controller with internal loop is mainly focused on the case that 𝑃 and 𝐾𝐼 are both well-posed transfer functions (bounded and analytic on some right half plane). And in all applications, the controller 𝐾𝐼 is assumed to be stable and satisfy two conditions proposed in [1] (refer to Proposition 4.8 in [1]). While, in the framework of nest algebra, we extend the study to the more general case that 𝐶𝑖𝑗, 𝑖,𝑗=1,2 and 𝐾𝐼 need not to satisfy the two conditions proposed in [1].
Suppose 𝐼𝐶22 is invertible in , then we have that 𝐻𝑃,𝐾𝐼=(𝐼+𝐶𝑃)1𝐶(𝐼+𝑃𝐶)1𝑇13𝑃(𝐼+𝐶𝑃)1(𝐼+𝑃𝐶)1𝑇23𝑇31𝑇32𝑇33,(3.3) where 𝐶=𝐶11+𝐶12𝐼𝐶221𝐶21,𝑇13=(𝐼+𝐶𝑃)1𝐶12𝐼𝐶221,𝑇23=𝑃(𝐼+𝐶𝑃)1𝐶12𝐼𝐶221,𝑇31=𝐼𝐶221𝐶21𝑃(𝐼+𝐶𝑃)1,𝑇32=𝐼𝐶221𝐶21(𝐼+𝑃𝐶)1,𝑇33=𝐼𝐼𝐶221𝐶21𝑃(𝐼+𝐶𝑃)1𝐶12𝐼𝐶221.(3.4)

Remark 3.2. Notice that the upper left 2×2 corner of the above transfer matrix 𝐻(𝑃,𝐾𝐼) is just the transfer matrix 𝐻(𝑃,𝐶) of the standard feedback system with the plant 𝑃 and the controller 𝐶=𝐶11+𝐶12(𝐼𝐶22)1𝐶21. This implies that the closed-loop system stabilized by controllers with internal loop is more general than the standard feedback system and its transfer matrix provides more information.
Now we can give a parametrization of all stabilizing controllers with internal loop with 𝐼𝐶22 invertible in .

Theorem 3.3. Suppose 𝑃 and there exist 𝑀, 𝑁, 𝑋, 𝑌, 𝑀, 𝑁, 𝑋, 𝑌𝒮 such that 𝑀𝑁 and 𝑁[𝑀] are, respectively, strong right and left representation for 𝑃 that satisfy the double Bezout identity 𝑁𝑀𝑋𝑁𝑌=𝑋𝑁𝑌𝑁𝑀=𝑌𝑋𝑀𝑀𝑌𝑋𝐼00𝐼.(3.5) Then all stabilizing controllers with internal loop 𝐾𝐼=𝐶11𝐶12𝐶21𝐶22 are parameterized by 𝐶11=𝑋+𝑀𝑄𝑌𝑁𝑄1𝑁𝑌𝑄1𝑅1𝑅3+𝑅2𝑌𝑁𝑄1𝑁𝑅11𝑅2𝑌𝑁𝑄1,𝐶12=𝑁𝑌𝑄1𝑅1𝑅3+𝑅2𝑌𝑁𝑄1𝑁𝑅11,𝐶21=𝑅3+𝑅2𝑌𝑁𝑄1𝑁𝑅11𝑅2𝑌𝑁𝑄1,𝐶22𝑅=𝐼3+𝑅2𝑌𝑁𝑄1𝑁𝑅11,(3.6) for some 𝑄,𝑅1,𝑅2,𝑅3𝒮.

In order to prove this theorem clearly, we need the following result which is an improvement of Theorem 2.4. It is interesting that while the two representations for the controller in Theorem 2.4 are independent, the same 𝑄 will in fact work for both.

Theorem 3.4. Suppose 𝑃 satisfies the assumption in Theorem 2.4. Then the stabilizing controller 𝐶 for 𝑃 has the form 𝐶=(𝑌𝑄𝑁)1(𝑋+𝑄𝑀)=(𝑋+𝑀𝑄)(𝑌𝑁𝑄)1 for some 𝑄𝒮.

Proof. Suppose 𝐶 stabilizes 𝑃. By Theorem 2.4, we have that 𝐶 has a right coprime factorization 𝐶=(𝑋+𝑀𝑄)(𝑌𝑁𝑄)1 for some 𝑄𝒮. It is easy to check that 𝑀𝑁𝑀𝑁𝑀=𝑀𝑀𝑁𝑀𝑁=.𝑋+𝑄𝑌𝑄𝑁𝑌𝑁𝑄𝑋+𝑀𝑄𝑁𝑌𝑁Q𝑋+𝑀𝑄𝑋+𝑄𝑌𝑄𝐼00𝐼(3.7) Thus, 𝑀𝑁𝑋+𝑄𝑌𝑄𝑌𝑁𝑄𝑋+𝑀𝑄=0.(3.8) This implies that 𝑀𝑁𝒢(𝐶)=Ran𝑌𝑁𝑄𝑋+𝑀𝑄Ker𝑋+𝑄𝑌𝑄.(3.9) On the other hand, for any 𝑥𝑦Ker[(𝑋+𝑄𝑀)𝑌𝑄𝑁], we have 𝑥𝑦=𝑀𝑀𝑁𝑀𝑁𝑥𝑦=𝑀𝑀𝑁+𝑀𝑁𝑥𝑦=𝑀𝑁𝑥𝑦𝑁𝑌𝑁𝑄𝑋+𝑀𝑄𝑋+𝑄𝑌𝑄𝑁𝑋+𝑄𝑌𝑄𝑌𝑁𝑄𝑋+𝑀𝑄𝑌𝑁𝑄𝑋+𝑀𝑄𝒢(𝐶),(3.10) that is, 𝑀𝑁Ker𝑋+𝑄𝑌𝑄𝒢(𝐶).(3.11) Thus, 𝑀𝑁𝒢(𝐶)=Ker𝑋+𝑄𝑌𝑄.(3.12) Since 𝑀𝑁𝑀𝑋+𝑄𝑌𝑄𝑁=𝐼,(3.13) we obtain that [(𝑋+𝑄𝑀)𝑌𝑄𝑁] is a strong left representation of 𝐶 and 𝐶=(𝑌𝑄𝑁)1(𝑋+𝑄𝑀). This completes the proof.

Now we can give the proof of Theorem 3.3.

Proof of Theorem 3.3. Suppose {𝑃,𝐾𝐼} is stable, then every entry of the matrix (𝐼+𝐶𝑃)1𝐶(𝐼+𝑃𝐶)1𝑃(𝐼+𝐶𝑃)1(𝐼+𝑃𝐶)1(3.14) is in 𝒮 and 𝑇13, 𝑇23, 𝑇31, 𝑇32, 𝑇33𝒮. Note that (3.14) is just the transfer matrix 𝐻(𝑃,𝐶) for the standard feedback system. By Theorem 3.4, we see that 𝐶 has the following representation: 𝑁𝐶=𝑌𝑄1𝑀=𝑋+𝑄𝑋+𝑀𝑄𝑌𝑁𝑄1,(3.15) for some 𝑄𝒮. In this case, 𝑇13=(𝐼+𝐶𝑃)1𝐶12𝐼𝐶221𝑁𝐶=𝑀𝑌𝑄12𝐼C221𝑇𝒮,23=𝑃(𝐼+𝐶𝑃)1𝐶12𝐼𝐶221𝑁𝐶=𝑁𝑌𝑄12𝐼𝐶221𝒮,(3.16) if and only if 𝑁𝐶𝑌𝑄12𝐼𝐶221𝒮.(3.17) It follows that 𝐶12𝐼𝐶221=𝑁𝑌𝑄1𝑅1(3.18) for some 𝑅1𝒮.
In the same way, we obtain that 𝐼𝐶221𝐶21=𝑅2𝑌𝑁𝑄1,(3.19) for some 𝑅2𝒮. So we get 𝐶12=𝑁𝑌𝑄1𝑅1𝐼𝐶22,𝐶21=𝐼𝐶22𝑅2𝑌𝑁𝑄1.(3.20) Since 𝑇33=𝐼𝐼𝐶221𝐶21𝑃(𝐼+𝐶𝑃)1𝐶12𝐼𝐶221=𝐼𝐶221𝑅2𝑌𝑁𝑄1𝑁𝑁𝑁𝑌𝑄𝑌𝑄1𝑅1=𝐼𝐶221𝑅2𝑌𝑁𝑄1𝑁𝑅1𝒮,(3.21) we have 𝐼𝐶221=𝑅3+𝑅2𝑌𝑁𝑄1𝑁𝑅1,(3.22) for some 𝑅3𝒮. Thus, 𝐶22𝑅=𝐼3+𝑅2𝑌𝑁𝑄1𝑁𝑅11.(3.23) Then we can obtain the following representations for 𝐶12 and 𝐶21: 𝐶12=𝑁𝑌𝑄1𝑅1𝑅3+𝑅2𝑌𝑁𝑄1𝑁𝑅11,𝐶21=𝑅3+𝑅2𝑌𝑁𝑄1𝑁𝑅11𝑅2𝑌𝑁𝑄1.(3.24) Substituting the representations of 𝐶, 𝐶12, 𝐶21, and 𝐶22 into 𝐶11=𝐶𝐶12(𝐼𝐶22)1𝐶21, we obtain 𝐶11=𝑋+𝑀𝑄𝑌𝑁𝑄1𝑁𝑌𝑄1×𝑅1𝑅3+𝑅2𝑌𝑁𝑄1𝑁𝑅11𝑅2𝑌𝑁𝑄1.(3.25) This completes the proof.

It was said in [1] that the controller with internal loop was particularly well suited for tracking, and a physical interpretation was given for the system with internal loop. In [6], the author described a seemingly impossible problem, the “intriguing control problem”, which can be easily solved by the system with internal loop. In the next two sections, we will show the other great advantages of the controller with internal loop.

4. Canonical Controllers and Dual Canonical Controllers

In this section, we focus on two special classes of controllers with internal loop called canonical controllers and dual canonical controllers, respectively. Here below it is given their definitions in the framework of nest algebra.

Definition 4.1. A controller with internal loop is called the canonical controller for the plant 𝑃 if it is of the form 𝐾𝐼=𝐶0𝐼21𝐶22 with 𝐶21,𝐶22𝒮. Analogously, a controller with internal loop is called a dual canonical controller for the plant 𝑃 if it is of the form 0𝐶12𝐼𝐶22 with 𝐶12, 𝐶22𝒮.
For canonical controllers, we have the following results.

Theorem 4.2. The canonical controller 𝐾𝐼=𝐶0𝐼21𝐶22 stabilizes 𝑃 with internal loop if and only if Δ=𝐼𝐶22+𝐶21𝑃 is invertible in and Δ1, 𝑃Δ1𝒮.
If 𝑃 has a strong right representation 𝑀𝑁, then the canonical controller 𝐾𝐼 stabilizes 𝑃 if and only if 𝐷=𝑀𝐶22𝑀+𝐶21𝑁 is invertible in 𝒮.

Proof. According to the system equations in (3.2), we have that, for the canonical controllers 𝐾𝐼=𝐶0𝐼21𝐶22, the transfer matrix 𝐻(𝑃,𝐾𝐼) can be given by 𝐻𝑃,𝐾𝐼=𝐼Δ1𝐶21𝑃Δ1𝐶21Δ1𝑃𝐼Δ1𝐶21𝑃𝐼𝑃Δ1𝐶21𝑃Δ1Δ1𝐶21𝑃Δ1𝐶21Δ1.(4.1)
Thus, 𝐻(𝑃,𝐾𝐼)𝑀3(𝒮) if and only if Δ1, 𝑃Δ1, Δ1𝐶21𝑃, and 𝑃(𝐼Δ1𝐶21𝑃) are all in 𝒮. Since Δ1𝐶21𝑃=Δ1(Δ+𝐶21𝑃)𝐼=Δ1(𝐼𝐶22)𝐼 and 𝑃(𝐼Δ1𝐶21𝑃)=𝑃Δ1(𝐼𝐶22). We have that all the entries of 𝐻(𝑃,𝐾𝐼) are in 𝒮 if and only if Δ1 and 𝑃Δ1 are in 𝒮. Thus the first statement is proved.
Let us prove the second assertion in the theorem. If 𝑃=𝑁𝑀1 and 𝐷1𝒮, we have that Δ1=𝑀(𝑀𝐶22𝑀+𝐶21𝑁)1=𝑀𝐷1𝒮 and 𝑃Δ1=𝑁𝐷1𝒮. By using the first result, we have that 𝐾𝐼=𝐶0𝐼21𝐶22 stabilizes 𝑃. Conversely, if 𝐾𝐼=𝐶0𝐼21𝐶22 stabilizes 𝑃. By the first result, we have that Δ1, 𝑃Δ1 are both in 𝒮. Suppose that 𝑀, 𝑁, 𝑋, and 𝑌 are as in Definition 2.2, then 𝑌Δ1+𝑋𝑃Δ1=(𝑌𝑀+𝑋𝑁)𝑀𝐶22𝑀+𝐶21𝑁=𝐷1.(4.2) Since 𝑋 and 𝑌 are in 𝒮, we see that 𝐷1𝒮. This completes the proof.

There is a similar result for the dual canonical controller.

Theorem 4.3. The dual canonical controller 𝐾𝐼=0𝐶12𝐼𝐶22 stabilizes 𝑃 with internal loop if and only if Δ=𝐼𝐶22+𝑃𝐶12 is invertible in and Δ1, Δ1𝑃𝒮.
If 𝑃 has a strong left representation 𝑁[𝑀], then the dual canonical controller 𝐾𝐼 stabilizes 𝑃 if and only if 𝐷=𝑀𝑀𝐶22+𝑁𝐶12 is invertible in 𝒮.

In [6], the parametrization of all canonical controllers and dual canonical controllers is given and it can be easily extended to our framework.

Theorem 4.4. Suppose 𝑃 satisfies the assumption of Theorem 3.3. Then all canonical controllers that stabilize 𝑃 are parameterized by 𝐸𝑀𝑁0𝐼𝑋+𝑄𝐼𝐸𝑌𝑄,(4.3) where 𝑄𝒮 and 𝐸 is invertible in 𝒮.
Analogously, all dual canonical controllers that stabilize 𝑃 are parameterized by 0𝑅𝑅𝑋+𝑀𝑄𝐼𝐼𝑌𝑁𝑄,(4.4) where 𝑄𝒮 and 𝑅 is invertible in 𝒮.

Remark 4.5. Indeed, if we choose the parameters in Theorem 3.3 such that 𝑅1=𝐸1, 𝑅2=𝑋+𝑀𝑄, and 𝑅3=𝑀𝐸1, we can obtain the same result of the above theorem. This implies that the result derived in [6] can be regarded as a special case of Theorem 3.3.
The following theorem gives a strong relation between the stabilization with canonical controller and the usual concept of stabilization.

Theorem 4.6. Suppose 𝐼𝐶22 is invertible in , then 𝑃 can be stabilized by a canonical controller with internal loop if and only if 𝑃 is stabilizable in the framework of standard feedback system.

Proof . Suppose 𝑃 is stabilized by a canonical controller 𝐾𝐼=0𝐶12𝐼𝐶22 with 𝐼𝐶22 invertible in . Then all entries of 𝐻(𝑃,𝐾𝐼) in (4.1) are in 𝒮. By computation, we can easily obtain that the upper left 2×2 corner of the transfer matrix 𝐻(𝑃,𝐾𝐼) is just the transfer matrix 𝐻(𝑃,C) in the standard feedback system with the plant 𝑃 and the controller 𝐶=(𝐼𝐶22)1𝐶21. It follows that 𝑃 is stabilizable in the standard feedback system.
On the other hand, suppose 𝑃 is stabilizable in the standard feedback system. Then, from Theorem 2.4, all the stabilizing controllers can be given by 𝐶=(𝑌𝑄𝑁)1(𝑋+𝑄𝑀) with some 𝑄𝒮. Let 𝐶21𝑀=𝑋+𝑄, 𝐶22𝑁=𝑌𝑄, then we obtain a canonical controller 𝐾𝐼=0𝐼𝑋+𝑄𝑀𝑌𝑄𝑁 and it is easy to verify that 𝐾𝐼 stabilizes 𝑃.

Naturally, there exists a dual result for the dual canonical controller.

Now we can explain the advantage of the controller with internal loop in the practical application.

Recall the parametrization of the conventional controllers in Theorem 2.4, it is not clean and an extra invertibility condition is imposed on the Youla parameter. This in turn makes it awkward to use this parametrization to solve the practical problems. For example, in [8, Section 7], the sensitivity minimization problem for the system described in Figure 3 is studied. The weighted sensitivity operator for this system is defined by 𝑆𝑊=(𝐼+𝑃𝐶)1𝑊 and the weighted sensitivity minimization problem is to find 𝑆inf𝑊𝐶stabilizes𝑃.(4.5)

132597.fig.003
Figure 3: Standard feedback system with outside disturbance 𝑑 and 𝑊 invertible.

Suppose 𝑃 satisfy the condition in Theorem 2.4, then all stabilizing controllers can be given by 𝐶=(𝑋+𝑀𝑄)(𝑌𝑁𝑄)1 with 𝑄𝒮 and 𝑌𝑁𝑄 is invertible in . By simple computation, we can obtain that the weighted sensitivity minimization problem is to find 𝑌inf𝑀𝑊𝑁𝑄𝑀𝑊𝑄𝒮,𝑌𝑁𝑄isinvertiblein.(4.6) Obviously, the extra condition that 𝑌𝑁𝑄 is invertible in makes the practical control engineers difficult to continue their computations. So they have to choose to ignore the fact that the Youla parameter can not be taken for all the elements in 𝒮.

Fortunately, the controller with internal loop overcomes this awkwardness. Let us consider the sensitivity minimization problem for the system with internal loop as described in Figure 4. We consider this problem for the dual canonical controller 𝐾𝐼=0𝐶12𝐼𝐶22 with 𝐶12, 𝐶22𝒮. When 𝑢1=𝑢2=𝑢3=0, it is easy to check that the weighted sensitivity operator for this system is 𝑆𝑊=𝐼𝑃𝐶12𝐼𝐶22+𝑃𝐶121𝑊,(4.7) and the weighted sensitivity minimization problem is to find 𝑆inf𝑊𝐾𝐼=0𝐶12𝐼𝐶22stabilizes𝑃.(4.8) By using the parametrization of the dual canonical controller given in Theorem 4.4, the weighted sensitivity minimization problem is to find 𝑌inf𝑀𝑊𝑁𝑄𝑀𝑊𝑄𝒮.(4.9) Obviously, it avoids the extra invertibility condition for the parameter 𝑄 as it appears in the standard feedback system. This overcomes the difficulty arisen in the standard feedback system.

132597.fig.004
Figure 4: System with internal loop with outside disturbance 𝑑 and 𝑊 invertible.

5. Strong Stabilization with Internal Loop

Practicing control engineers is reluctant to use unstable compensators for the purpose of stabilization. This motivated considering the strong stabilization problem whether among the stabilizing controllers for a given stabilizable plant 𝑃, there exist stable ones. If there exists such a controller, 𝑃 is said to be strongly stabilizable and the stable controller is called the strongly stabilizing controller. In this section, we consider the strong stabilization problem for the system with internal loop and address another advantage of the controller with internal loop.

Definition 5.1. 𝑃 is said to be strongly stabilizable with internal loop if it can be stabilized by the controller 𝐾𝐼=𝐶11𝐶12𝐶21𝐶22 with 𝐶𝑖𝑗𝒮, 𝑖,𝑗=1,2. This controller 𝐾𝐼 is called the strongly stabilizing controller with internal loop.
Obviously, the canonical controller and dual canonical controller are both the strongly stabilizing controller with internal loop. From the parametrization of controllers with internal loop given in Theorem 3.3, we see that the strongly stabilizing controller with internal loop can be characterize by choosing the parameters 𝑄, 𝑅1, 𝑅2, and 𝑅3 in Theorem 3.3 such that 𝑌𝑁𝑄 and 𝑅3+𝑅2(𝑌𝑁𝑄)1𝑁𝑅1 are invertible in 𝒮. The following theorem shows the existence of the strongly stabilizing controller with internal loop.

Theorem 5.2. Suppose 𝑃 is stabilizable, then 𝑃 can be strongly stabilized by the controller with internal loop.

Proof. Suppose 𝑃 is stabilizable. From Theorem 2.4, the controller stabilizes 𝑃 has the parametrization that 𝐶=(𝑌𝑄𝑁)1(𝑋+𝑄𝑀) for some stable 𝑄. Set 𝐶11=0, 𝐶12=𝐼, 𝐶21=(𝑋+𝑄𝑀) and 𝐶22=𝐼(𝑌𝑄𝑁), then 𝐾𝐼=0𝐼𝑋+𝑄𝑀𝐼(𝑌𝑄𝑁) is a stable controller with internal loop and it strongly stabilizes the plant 𝑃.

Remark 5.3. In [17], it is proved that a given plant 𝑃 with left coprime factorization 𝑀𝑃=1𝑁 can be strongly stabilized in the standard feedback system if 𝑁 is compact. While in the case where 𝑁 is not compact, it is still an open problem whether or not there exists a stable controller that stabilizes the plant. Theorem 5.2 shows that any stabilizable plant can be stabilized by a stable controller with internal loop. It implies that the strong stabilization problem can be completely solved by the controller with internal loop. And this addresses an advantage of the controller with internal loop in the framework of nest algebra.
From the proof of Theorem 5.2, we see that it also provides a method to design the strongly stabilizing controller with internal loop. We end our paper with a simple example to show this design method.

Example 5.4. Suppose 𝑃=𝐼, it is obviously stabilizable. Take 101𝑁=𝑁=𝑀=𝑀=𝐼,𝑌=𝑌=2100310004001,𝑋=𝑋=2200330004.(5.1) From Theorem 2.4, we obtain the parametrization of the stabilizing controller 𝐶=(𝑌𝑄𝑁)1(𝑋+𝑄𝑀) for some stable 𝑄. Take 𝑄=0, then 𝐶=𝑌10𝑋stabilizes𝑃and𝐶=010020003(5.2) is unstable. In order to satisfy the practicing control engineers' requirement, we set 𝐶11=0, 𝐶12=𝐼, 𝐶21=𝑋 and 𝐶22=𝐼𝑌. Then we obtain a stable controller 𝐾𝐼=0𝐼𝑋𝐼𝑌 with internal loop which stabilizes the plant 𝑃.

6. Conclusion

In this paper, the closed-loop system whose stability is achieved by the controller with internal loop is studied in the framework of nest algebra. The controllers with internal loop considered here are more general than those in the previous paper and they are not necessarily stable and need not to satisfy the two conditions proposed in [1]. We give a parametrization for all stabilizing controllers with internal loop which has never been studied before. Then we show that this parametrization covers the parametrization for canonical or dual canonical controllers obtained in [6]. By taking the sensitivity minimization problem as an example, we show that, in the practical application, the controller with internal loop solves the difficulty brought by the invertibility condition in the parametrization of the conventional controller. In the framework of nest algebra, the strong stabilization problem is still an open problem, and no sufficient and necessary condition was found to characterize the plant which can be strongly stabilized. While, with the help of the concept of stabilization with internal loop, we show that any stabilizable plant can be strongly stabilized by the controller with internal loop. This addresses an advantage of the controller with internal loop in the framework of nest algebra. By using the parametrization of the controller with internal loop, we are considering other questions in the control theory for the this model of closed-loop systems with internal loop.

Acknowledgments

The authors would like to thank Doctor Liu Liu for pointing out Lemma 3.1 and Doctor Gong Ting for her invaluable suggestions. This research is supported by NSFC, item number: 10971020.

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