Stabilization of Time-Varying System by Controllers with Internal Loop

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 [1–5]). 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 [9–16]). 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 ℋ be the complex infinite dimensional Hilbert sequence space: ℓ²=𝑥₀,𝑥₁,𝑥₂,…∶𝑥_𝑖∈ℂ,^∞_𝑖=0||𝑥_𝑖||²<∞,(2.1) where |⋅| denotes the standard Euclidean norm on ℂ with inner product (𝑥,𝑦)=∑^∞𝑖=0𝑥_𝑖𝑦_𝑖. ℋ_𝑒 will denote the extended space: ℋ_𝑒=𝑥₀,𝑥₁,𝑥₂,…∶𝑥_𝑖∈ℂ.(2.2) For each 𝑛≥0, let 𝑃_𝑛 denote the standard truncation projection defined on ℋ and ℋ_𝑒 by 𝑃_𝑛𝑥₀,𝑥₁,…,𝑥_𝑛,𝑥_𝑛+1,…=𝑥₀,𝑥₁,…,𝑥_𝑛,0,0,….(2.3) A continuous linear transformation 𝑇 on ℋ_𝑒 with the standard seminorm topology ([8, Chapter 5]) is a causal linear system (or a linear system) if for each 𝑛≥0, 𝑃_𝑛𝑇=𝑃_𝑛𝑇𝑃_𝑛. Let ℒ be the set of all linear systems on ℋ_𝑒. Then any element of ℒ 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 ℋ 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∈ℒ, we consider the standard feedback configuration with plant 𝑃 and controller 𝐶 shown in Figure 1.

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⎤⎥⎥⎦=⎡⎢⎢⎣−𝐼𝑃𝐶𝐼⎤⎥⎥⎦⎡⎢⎢⎣𝑒₁𝑒₂⎤⎥⎥⎦.(2.4) The system is well posed if the internal input 𝑒=^𝑒1_𝑒2 can be expressed as a causal function of the external input 𝑢=^𝑢1_𝑢2. This is equivalent to requiring that ^{𝐼𝐶−𝑃𝐼} be invertible. This inverse can be easily computed and is given by the transfer matrix 𝐻(𝑃,𝐶)=⎡⎢⎢⎣(𝐼+𝐶𝑃)⁻¹−𝐶(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 ℋ. 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)={𝑥∈ℋ∶𝑃𝑥∈ℋ} is 𝒢(𝑃)={^𝑥𝑃𝑥∶𝑥∈𝒟(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)=Ran^𝑀𝑁,
(2) there exist 𝑋,𝑌∈𝒮 such that [𝑌𝑋]^𝑀𝑁=𝐼.
A plant 𝑃 has a strong left representation [−𝑁𝑀] with 𝑀 and 𝑁 stable if
(1)𝒢(𝑃)=Ker[−𝑁𝑀],
(2) there exist 𝑋,𝑌∈𝒮 such that [−𝑁𝑀]⁻𝑌𝑋=𝐼.

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

Theorem 2.3 (see [8]). Suppose 𝑀,𝑁∈𝒮. Then ^𝑀𝑁 is a strong right representation of 𝑃∈ℒ if and only if
(1) there exist 𝑋,𝑌∈𝒮 such that [𝑌𝑋]^𝑀𝑁=𝐼,
(2) M is invertible in ℒ.

We say that a plant 𝑃 has a right coprime factorization if there exist 𝑀, 𝑁, 𝑋, 𝑌∈𝒮 such that 𝑃=𝑁𝑀⁻¹ 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 𝑁𝑀⁻¹ is a right coprime factorization of 𝑃. Similarly, [−𝑁𝑀] is a strong left representation of 𝑃 if and only if 𝑀⁻¹𝑁 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∈ℒ 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 ⎡⎢⎢⎣−𝑌𝑁𝑀𝑋⎤⎥⎥⎦⎡⎢⎢⎣𝑀𝑁−𝑌𝑋⎤⎥⎥⎦=⎡⎢⎢⎣𝑀𝑁−𝑌𝑋⎤⎥⎥⎦⎡⎢⎢⎣−𝑌𝑁𝑀𝑋⎤⎥⎥⎦=⎡⎢⎢⎣𝐼0𝐼0⎤⎥⎥⎦.(2.6) A causal linear system 𝐶 stabilizes 𝑃 if and only if it has a strong right representation 𝑋𝑌^−𝑁𝑄_+𝑀𝑄 and a strong left representation [−(𝑋+𝑄𝑀)𝑌−𝑄𝑁] for some 𝑄∈𝒮.

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.

Figure 2 

The plant 𝑃 connected to a controller 𝐾𝐼 with internal loop.

The intuitively interpretation of Figure 2: 𝑃∈ℒ is the plant and 𝐾_𝐼 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 𝐾_𝐼 with internal loop is denoted by {𝑃,𝐾_𝐼}.

Partitioning 𝐾_𝐼 into ⎡⎢⎢⎣𝐶₁₁𝐶₁₂𝐶₂₁𝐶₂₂⎤⎥⎥⎦,(3.1) where 𝐶_𝑖𝑗∈ℒ,   𝑖,𝑗=1,2, the closed loop system equations are ⎡⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣𝐼𝐶₁₁𝐶₁₂−0𝑃−𝐶𝐼₂₁0𝐼−𝐶₂₂⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣𝑒₁𝑒₂𝑒₃⎤⎥⎥⎥⎥⎦.(3.2)

We say that the system is well posed if ^{𝐼𝐶11𝐶12−𝑃𝐼00−𝐶}_{21𝐼−𝐶22} is invertible and We denote this inverse by 𝐻(𝑃,𝐾_𝐼).

Definition 3.1. The closed loop system {𝑃,𝐾_𝐼} determined by the plant 𝑃∈ℒ and the controller with internal loop 𝐾_𝐼 is stable if all the entries of 𝐻(𝑃,𝐾_𝐼) are stable. The plant 𝑃 is stabilizable by a controller with internal loop if there exists a 𝐾_𝐼 such that 𝐻(𝑃,𝐾_𝐼) is stable. In this case, 𝐾_𝐼 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 𝐾_𝐼 are both well-posed transfer functions (bounded and analytic on some right half plane). And in all applications, the controller 𝐾_𝐼 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 𝐶_𝑖𝑗∈ℒ, 𝑖,𝑗=1,2 and 𝐾_𝐼 need not to satisfy the two conditions proposed in [1].
Suppose 𝐼−𝐶₂₂ is invertible in ℒ, then we have that 𝐻𝑃,𝐾_𝐼=⎡⎢⎢⎢⎢⎣(𝐼+𝐶𝑃)⁻¹−𝐶(𝐼+𝑃𝐶)⁻¹𝑇₁₃𝑃(𝐼+𝐶𝑃)⁻¹(𝐼+𝑃𝐶)⁻¹𝑇₂₃𝑇₃₁𝑇₃₂𝑇₃₃⎤⎥⎥⎥⎥⎦,(3.3) where 𝑇𝐶₁₃=𝐶₁₁+𝐶₁₂𝐼−𝐶₂₂⁻¹𝐶₂₁,𝑇₂₃=−(𝐼+𝐶𝑃)⁻¹𝐶₁₂𝐼−𝐶₂₂⁻¹,𝑇₃₁=−𝑃(𝐼+𝐶𝑃)⁻¹𝐶₁₂𝐼−𝐶₂₂⁻¹,𝑇₃₂=𝐼−𝐶₂₂⁻¹𝐶₂₁𝑃(𝐼+𝐶𝑃)⁻¹,𝑇₃₃=𝐼−𝐶₂₂⁻¹𝐶₂₁(𝐼+𝑃𝐶)⁻¹,=𝐼−𝐼−𝐶₂₂⁻¹𝐶₂₁𝑃(𝐼+𝐶𝑃)⁻¹𝐶₁₂𝐼−𝐶²²⁻¹.(3.4)

Remark 3.2. Notice that the upper left 2×2 corner of the above transfer matrix 𝐻(𝑃,𝐾_𝐼) is just the transfer matrix 𝐻(𝑃,𝐶) of the standard feedback system with the plant 𝑃 and the controller 𝐶=𝐶₁₁+𝐶₁₂(𝐼−𝐶₂₂)⁻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 𝐼−𝐶₂₂ invertible in ℒ.

Theorem 3.3. Suppose 𝑃∈ℒ and there exist 𝑀, 𝑁, 𝑋, 𝑌, 𝑀, 𝑁, 𝑋, 𝑌∈𝒮 such that ^𝑀𝑁 and [−𝑁𝑀] are, respectively, strong right and left representation for 𝑃 that satisfy the double Bezout identity ⎡⎢⎢⎣−𝑌𝑁𝑀𝑋⎤⎥⎥⎦⎡⎢⎢⎣𝑀𝑁−𝑌𝑋⎤⎥⎥⎦=⎡⎢⎢⎣𝑀𝑁−𝑌𝑋⎤⎥⎥⎦⎡⎢⎢⎣−𝑌𝑁𝑀𝑋⎤⎥⎥⎦=⎡⎢⎢⎣𝐼0𝐼0⎤⎥⎥⎦.(3.5) Then all stabilizing controllers with internal loop 𝐾_𝐼=^𝐶11𝐶12_𝐶21𝐶22 are parameterized by 𝐶₁₁=𝑋+𝑀𝑄𝑌−𝑁𝑄⁻¹−𝑌−𝑄𝑁⁻¹𝑅₁𝑅³+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹𝐶₁₂𝑅₂𝑌−𝑁𝑄⁻¹,=𝑌−𝑄𝑁⁻¹𝑅₁𝑅³+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹𝐶₂₁,=𝑅₃+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹𝐶₂₂𝑅₂𝑌−𝑁𝑄⁻¹,=𝐼−𝑅₃+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹,(3.6) for some 𝑄,𝑅₁,𝑅₂,𝑅₃∈𝒮.

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=(𝑌−𝑄𝑁)⁻¹(𝑋+𝑄𝑀)=(𝑋+𝑀𝑄)(𝑌−𝑁𝑄)⁻¹ for some 𝑄∈𝒮.

Proof. Suppose 𝐶 stabilizes 𝑃. By Theorem 2.4, we have that 𝐶 has a right coprime factorization 𝐶=(𝑋+𝑀𝑄)(𝑌−𝑁𝑄)⁻¹ for some 𝑄∈𝒮. It is easy to check that ⎡⎢⎢⎣−𝑋𝑀+𝑄𝑀𝑌−𝑁𝑄𝑁⎤⎥⎥⎦⎡⎢⎢⎣−𝑀𝑁𝑋𝑌+−𝑀𝑁𝑄𝑄⎤⎥⎥⎦==⎡⎢⎢⎣⎡⎢⎢⎣−𝐼0𝑀𝑁𝐼0⎤⎥⎥⎦𝑋𝑌.+−𝑀𝑁Q𝑄⎤⎥⎥⎦⎡⎢⎢⎣−𝑋𝑀+𝑄𝑀𝑌−𝑁𝑄𝑁⎤⎥⎥⎦(3.7) Thus, −𝑋+𝑄𝑀𝑌−𝑄𝑁⎡⎢⎢⎣𝑋𝑌+−𝑀𝑁𝑄𝑄⎤⎥⎥⎦=0.(3.8) This implies that 𝒢(𝐶)=Ran⎡⎢⎢⎣𝑋𝑌+−𝑀𝑁𝑄𝑄⎤⎥⎥⎦⊆Ker−𝑋+𝑄𝑀𝑌−𝑄𝑁.(3.9) On the other hand, for any ^𝑥𝑦∈Ker[−(𝑋+𝑄𝑀)𝑌−𝑄𝑁], we have ⎡⎢⎢⎣𝑥𝑦⎤⎥⎥⎦===⎡⎢⎢⎣⎛⎜⎜⎝⎡⎢⎢⎣⎡⎢⎢⎣−𝑋𝑌𝑀−𝑁𝑀+−𝑁𝑀𝑁⎤⎥⎥⎦𝑋𝑌𝑄𝑄+−−⎤⎥⎥⎦𝑀𝑁𝑋𝑀𝑄𝑄+⎤⎥⎥⎦𝑁⎡⎢⎢⎣𝑄−𝑀⎡⎢⎢⎣𝑋𝑥𝑦⎤⎥⎥⎦𝑀+𝑌∈−𝑄𝒢𝑀𝑄(𝐶𝑁),𝑌+−𝑁⎡⎢⎢⎣𝑄𝑋𝑌𝑁+−⎤⎥⎥⎦𝑀𝑁⎡⎢⎢⎣𝑥𝑦𝑄𝑄⎤⎥⎥⎦⎤⎥⎥⎦𝑀𝑁⎞⎟⎟⎠⎡⎢⎢⎣𝑥𝑦⎤⎥⎥⎦(3.10) that is, Ker−𝑋+𝑄𝑀𝑌−𝑄𝑁⊆𝒢(𝐶).(3.11) Thus, 𝒢(𝐶)=Ker−𝑋+𝑄𝑀𝑌−𝑄𝑁.(3.12) Since −𝑋+𝑄𝑀𝑌−𝑄𝑁⎡⎢⎢⎣−𝑀𝑁⎤⎥⎥⎦=𝐼,(3.13) we obtain that [−(𝑋+𝑄𝑀)𝑌−𝑄𝑁] is a strong left representation of 𝐶 and 𝐶=(𝑌−𝑄𝑁)⁻¹(𝑋+𝑄𝑀). This completes the proof.

Now we can give the proof of Theorem 3.3.

Proof of Theorem 3.3. Suppose {𝑃,𝐾_𝐼} is stable, then every entry of the matrix ⎡⎢⎢⎣(𝐼+𝐶𝑃)⁻¹−𝐶(𝐼+𝑃𝐶)⁻¹𝑃(𝐼+𝐶𝑃)⁻¹(𝐼+𝑃𝐶)⁻¹⎤⎥⎥⎦(3.14) is in 𝒮 and 𝑇₁₃, 𝑇₂₃, 𝑇₃₁, 𝑇₃₂, 𝑇₃₃∈𝒮. 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: 𝐶=𝑌−𝑄𝑁⁻¹𝑋+𝑄𝑀=𝑋+𝑀𝑄𝑌−𝑁𝑄⁻¹,(3.15) for some 𝑄∈𝒮. In this case, 𝑇₁₃=−(𝐼+𝐶𝑃)⁻¹𝐶₁₂𝐼−𝐶₂₂⁻¹𝑇=−𝑀𝑌−𝑄𝑁𝐶₁₂𝐼−C₂₂⁻¹∈𝒮,₂₃=−𝑃(𝐼+𝐶𝑃)⁻¹𝐶₁₂𝐼−𝐶₂₂⁻¹=−𝑁𝑌−𝑄𝑁𝐶₁₂𝐼−𝐶₂₂⁻¹∈𝒮,(3.16) if and only if 𝑌−𝑄𝑁𝐶¹²𝐼−𝐶₂₂⁻¹∈𝒮.(3.17) It follows that 𝐶₁₂𝐼−𝐶₂₂⁻¹=𝑌−𝑄𝑁⁻¹𝑅₁(3.18) for some 𝑅₁∈𝒮.
In the same way, we obtain that 𝐼−𝐶₂₂⁻¹𝐶₂₁=𝑅₂𝑌−𝑁𝑄⁻¹,(3.19) for some 𝑅₂∈𝒮. So we get 𝐶𝐶₁₂₂₁=𝑌−𝑄𝑁⁻¹𝑅₁𝐼−𝐶₂₂,=𝐼−𝐶₂₂𝑅₂𝑌−𝑁𝑄⁻¹.(3.20) Since 𝑇₃₃=𝐼−𝐼−𝐶₂₂⁻¹𝐶₂₁𝑃(𝐼+𝐶𝑃)⁻¹𝐶₁₂𝐼−𝐶²²⁻¹=𝐼−𝐶₂₂⁻¹−𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑌−𝑄𝑁𝑌−𝑄𝑁⁻¹𝑅₁=𝐼−𝐶₂₂⁻¹−𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁∈𝒮,(3.21) we have 𝐼−𝐶₂₂⁻¹=𝑅₃+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁,(3.22) for some 𝑅₃∈𝒮. Thus, 𝐶₂₂=𝐼−𝑅₃+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹.(3.23) Then we can obtain the following representations for 𝐶₁₂ and 𝐶₂₁: 𝐶₁₂=𝑌−𝑄𝑁⁻¹𝑅₁𝑅³+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹𝐶₂₁,=𝑅₃+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹𝑅₂𝑌−𝑁𝑄⁻¹.(3.24) Substituting the representations of 𝐶, 𝐶₁₂, 𝐶₂₁, and 𝐶₂₂ into 𝐶₁₁=𝐶−𝐶₁₂(𝐼−𝐶₂₂)⁻¹𝐶₂₁, we obtain 𝐶₁₁=𝑋+𝑀𝑄𝑌−𝑁𝑄⁻¹−𝑌−𝑄𝑁⁻¹×𝑅₁𝑅³+𝑅₂𝑌−𝑁𝑄⁻¹𝑁𝑅₁⁻¹𝑅₂𝑌−𝑁𝑄⁻¹.(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 𝐾_𝐼=^𝐶0𝐼_21𝐶22 with 𝐶₂₁,𝐶₂₂∈𝒮. Analogously, a controller with internal loop is called a dual canonical controller for the plant 𝑃 if it is of the form ^{0𝐶12𝐼𝐶}₂₂ with 𝐶₁₂, 𝐶₂₂∈𝒮.
For canonical controllers, we have the following results.

Theorem 4.2. The canonical controller 𝐾_𝐼=^𝐶0𝐼_21𝐶22 stabilizes 𝑃∈ℒ with internal loop if and only if Δ=𝐼−𝐶₂₂+𝐶₂₁𝑃 is invertible in ℒ and Δ⁻¹, 𝑃Δ⁻¹∈𝒮.
If 𝑃 has a strong right representation ^𝑀𝑁, then the canonical controller 𝐾_𝐼 stabilizes 𝑃 if and only if 𝐷=𝑀−𝐶₂₂𝑀+𝐶₂₁𝑁 is invertible in 𝒮.

Proof. According to the system equations in (3.2), we have that, for the canonical controllers 𝐾_𝐼=^𝐶0𝐼_21𝐶22, the transfer matrix 𝐻(𝑃,𝐾_𝐼) can be given by 𝐻𝑃,𝐾_𝐼=⎡⎢⎢⎢⎢⎣𝐼−Δ⁻¹𝐶₂₁𝑃−Δ⁻¹𝐶₂₁−Δ⁻¹𝑃𝐼−Δ⁻¹𝐶₂₁𝑃𝐼−𝑃Δ⁻¹𝐶₂₁−𝑃Δ⁻¹Δ⁻¹𝐶₂₁𝑃Δ⁻¹𝐶₂₁Δ⁻¹⎤⎥⎥⎥⎥⎦.(4.1)
Thus, 𝐻(𝑃,𝐾_𝐼)∈𝑀₃(𝒮) if and only if Δ⁻¹, 𝑃Δ⁻¹, Δ⁻¹𝐶₂₁𝑃, and 𝑃(𝐼−Δ⁻¹𝐶₂₁𝑃) are all in 𝒮. Since Δ⁻¹𝐶₂₁𝑃=Δ⁻¹(Δ+𝐶₂₁𝑃)−𝐼=Δ⁻¹(𝐼−𝐶₂₂)−𝐼 and 𝑃(𝐼−Δ⁻¹𝐶₂₁𝑃)=𝑃Δ⁻¹(𝐼−𝐶₂₂). We have that all the entries of 𝐻(𝑃,𝐾_𝐼) are in 𝒮 if and only if Δ⁻¹ and 𝑃Δ⁻¹ are in 𝒮. Thus the first statement is proved.
Let us prove the second assertion in the theorem. If 𝑃=𝑁𝑀⁻¹ and 𝐷⁻¹∈𝒮, we have that Δ⁻¹=𝑀(𝑀−𝐶₂₂𝑀+𝐶₂₁𝑁)⁻¹=𝑀𝐷⁻¹∈𝒮 and 𝑃Δ⁻¹=𝑁𝐷⁻¹∈𝒮. By using the first result, we have that 𝐾_𝐼=^𝐶0𝐼_21𝐶22 stabilizes 𝑃. Conversely, if 𝐾_𝐼=^𝐶0𝐼_21𝐶22 stabilizes 𝑃. By the first result, we have that Δ⁻¹, 𝑃Δ⁻¹ are both in 𝒮. Suppose that 𝑀, 𝑁, 𝑋, and 𝑌 are as in Definition 2.2, then 𝑌Δ⁻¹+𝑋𝑃Δ⁻¹=(𝑌𝑀+𝑋𝑁)𝑀−𝐶₂₂𝑀+𝐶₂₁𝑁=𝐷⁻¹.(4.2) Since 𝑋 and 𝑌 are in 𝒮, we see that 𝐷⁻¹∈𝒮. This completes the proof.

There is a similar result for the dual canonical controller.

Theorem 4.3. The dual canonical controller 𝐾_𝐼=^0𝐶12_𝐼𝐶22 stabilizes 𝑃∈ℒ with internal loop if and only if Δ=𝐼−𝐶₂₂+𝑃𝐶₁₂ is invertible in ℒ and Δ⁻¹, Δ⁻¹𝑃∈𝒮.
If 𝑃 has a strong left representation [−𝑁𝑀], then the dual canonical controller 𝐾_𝐼 stabilizes 𝑃 if and only if 𝐷=𝑀−𝑀𝐶₂₂+𝑁𝐶₁₂ 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∈ℒ satisfies the assumption of Theorem 3.3. Then all canonical controllers that stabilize 𝑃 are parameterized by ⎡⎢⎢⎣𝐸𝑋+0𝑄𝑀𝐼−𝐸𝑌𝐼−𝑄𝑁⎤⎥⎥⎦,(4.3) where 𝑄∈𝒮 and 𝐸 is invertible in 𝒮.
Analogously, all dual canonical controllers that stabilize 𝑃 are parameterized by ⎡⎢⎢⎣𝐼0𝐼−𝑋𝑌+−𝑀𝑁𝑄𝑄𝑅𝑅⎤⎥⎥⎦,(4.4) where 𝑄∈𝒮 and 𝑅 is invertible in 𝒮.

Remark 4.5. Indeed, if we choose the parameters in Theorem 3.3 such that 𝑅₁=𝐸⁻¹, 𝑅₂=𝑋+𝑀𝑄, and 𝑅₃=𝑀𝐸⁻¹, 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 𝐼−𝐶₂₂ is invertible in ℒ, 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 𝐾_𝐼=^0𝐶12_𝐼𝐶22 with 𝐼−𝐶₂₂ invertible in ℒ. Then all entries of 𝐻(𝑃,𝐾_𝐼) 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,𝐾_𝐼) is just the transfer matrix 𝐻(𝑃,C) in the standard feedback system with the plant 𝑃 and the controller 𝐶=(𝐼−𝐶₂₂)⁻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=(𝑌−𝑄𝑁)⁻¹(𝑋+𝑄𝑀) with some 𝑄∈𝒮. Let 𝐶₂₁=𝑋+𝑄𝑀, 𝐶₂₂=𝑌−𝑄𝑁, then we obtain a canonical controller 𝐾_𝐼=^{0𝐼𝑋+𝑄}𝑀_𝑌−𝑄𝑁 and it is easy to verify that 𝐾_𝐼 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 𝑆_𝑊=(𝐼+𝑃𝐶)⁻¹𝑊 and the weighted sensitivity minimization problem is to find inf‖‖𝑆_𝑊‖‖∶𝐶stabilizes𝑃.(4.5)

Figure 3 

Standard feedback system with outside disturbance 𝑑 and 𝑊 invertible.

Suppose 𝑃∈ℒ satisfy the condition in Theorem 2.4, then all stabilizing controllers can be given by 𝐶=(𝑋+𝑀𝑄)(𝑌−𝑁𝑄)⁻¹ with 𝑄∈𝒮 and 𝑌−𝑁𝑄 is invertible in ℒ. By simple computation, we can obtain that the weighted sensitivity minimization problem is to find inf‖‖𝑌𝑀𝑊−𝑁𝑄𝑀𝑊‖‖∶𝑄∈𝒮,𝑌−𝑁𝑄isinvertibleinℒ.(4.6) Obviously, the extra condition that 𝑌−𝑁𝑄 is invertible in ℒ 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 𝐾_𝐼=^0𝐶12_𝐼𝐶22 with 𝐶₁₂, 𝐶₂₂∈𝒮. When 𝑢₁=𝑢₂=𝑢₃=0, it is easy to check that the weighted sensitivity operator for this system is 𝑆_𝑊=𝐼−𝑃𝐶₁₂𝐼−𝐶₂₂+𝑃𝐶₁₂⁻¹𝑊,(4.7) and the weighted sensitivity minimization problem is to find inf⎧⎪⎨⎪⎩‖‖𝑆_𝑊‖‖∶𝐾_𝐼=⎡⎢⎢⎣0𝐶¹²𝐼𝐶₂₂⎤⎥⎥⎦stabilizes𝑃⎫⎪⎬⎪⎭.(4.8) By using the parametrization of the dual canonical controller given in Theorem 4.4, the weighted sensitivity minimization problem is to find inf‖‖𝑌𝑀𝑊−𝑁𝑄𝑀𝑊‖‖∶𝑄∈𝒮.(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.

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∈ℒ is said to be strongly stabilizable with internal loop if it can be stabilized by the controller 𝐾_𝐼=^𝐶11𝐶12_𝐶21𝐶22 with 𝐶_𝑖𝑗∈𝒮, 𝑖,𝑗=1,2. This controller 𝐾_𝐼 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 𝑌−𝑁𝑄 and 𝑅₃+𝑅₂(𝑌−𝑁𝑄)⁻¹𝑁𝑅₁ are invertible in 𝒮. The following theorem shows the existence of the strongly stabilizing controller with internal loop.

Theorem 5.2. Suppose 𝑃∈ℒ 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=(𝑌−𝑄𝑁)⁻¹(𝑋+𝑄𝑀) for some stable 𝑄. Set 𝐶₁₁=0, 𝐶₁₂=𝐼, 𝐶₂₁=(𝑋+𝑄𝑀) and 𝐶₂₂=𝐼−(𝑌−𝑄𝑁), then 𝐾_𝐼=^{0𝐼𝑋+𝑄}𝑀_{𝐼−(𝑌−𝑄}𝑁₎ 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 𝑃=𝑀⁻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 𝑌=𝑌=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣101𝑁=𝑁=𝑀=𝑀=𝐼,002100031⋮⋮⋮⋮4⋱⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝑋=𝑋=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣001002200033⋮⋮⋮⋮4⋱⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(5.1) From Theorem 2.4, we obtain the parametrization of the stabilizing controller 𝐶=(𝑌−𝑄𝑁)⁻¹(𝑋+𝑄𝑀) for some stable 𝑄. Take 𝑄=0, then 𝐶=𝑌⁻¹𝑋stabilizes𝑃and𝐶=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⋮0000⋮100⋮20⋮3⋱⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(5.2) is unstable. In order to satisfy the practicing control engineers' requirement, we set 𝐶₁₁=0, 𝐶₁₂=𝐼, 𝐶₂₁=𝑋 and 𝐶₂₂=𝐼−𝑌. Then we obtain a stable controller 𝐾_𝐼=^{0𝐼𝑋𝐼−𝑌} 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.