Retrofit Control with Approximate Environment Modeling \thanksreffootnoteinfo

Retrofit Control with Approximate Environment Modeling \thanksreffootnoteinfo

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Abstract

In this paper, we develop a retrofit control method with approximate environment modeling. Retrofit control is a modular control approach for a general stable network system whose subsystems are supposed to be managed by their corresponding subsystem operators. From the standpoint of a single subsystem operator who performs the design of a retrofit controller, the subsystems managed by all other operators can be regarded as an environment, the complete system model of which is assumed not to be available. The proposed retrofit control with approximate environment modeling has an advantage that the stability of the resultant control system is robustly assured regardless of not only the stability of approximate environment models, but also the magnitude of modeling errors, provided that the network system before implementing retrofit control is originally stable. This robustness property is practically significant to incorporate existing identification methods of unknown environments, because the accuracy of identified models may neither be reliable nor assurable in reality. Furthermore, we conduct a control performance analysis to show that the resultant performance can be regulated by adjusting the accuracy of approximate environment modeling. The efficiency of the proposed retrofit control is shown by numerical experiments on a network of second-order oscillators.

TIT]Takayuki Ishizaki, TIT]Takahiro Kawaguchi, TIT]Hampei Sasahara, and TIT]Jun-ichi Imura

Tokyo Institute of Technology; 2-12-1, Ookayama, Meguro, Tokyo, 152-8552, Japan. 


Key words:   Retrofit control, Approximate modeling, Modular design, Network systems, Decentralized control.

 

11footnotetext: This work was supported by JST CREST Grant Number JP-MJCR15K1, Japan. Corresponding author: T. Ishizaki, Tel. & Fax: +81-3-5734-2646.

1 Introduction

A module is one of semi-independent parts or subsystems in an integrated system of components. For example, in software development, a unit of programs that can be handled by an individual developer is called a software module [1]. As pointed out in a broad range of literature [2, 3, 4], increasing the modularity in design is the key to developing large-scale complex systems with flexibility to meet heterogeneous demands. Such “modular design” enables multiple entities or subsystem operators to individually develop, modify, and replace respective modules or subsystems, serving for significant reduction of efforts to adjust and coordinate a family of integrated components. This is a strong advantage as compared to “integral design,” where each component has strong interdependence among others [5, 6].

For dynamical network systems, a modular design method of decentralized controllers has been introduced in the context of retrofit control [7, 8, 9, 10]. The retrofit control can be applied to a general stable network system whose subsystems are supposed to be managed by their corresponding subsystem operators. From the standpoint of a single subsystem operator who performs the design of a retrofit controller, the subsystems managed by all other operators can be regarded as an environment, the system model of which is assumed not to be available. This reflects a practical situation where subsystem models, control policies, and demands of the other subsystem operators may not be public and stationary.

Most existing decentralized and distributed control methods, such as in [11, 12, 13, 14, 15, 16], can be classified as an integral design approach of structured controllers, where a single authority with availability of an entire system model is premised for simultaneous design of all subcontrollers constituting a decentralized or distributed controller. In contrast, the retrofit control is classified as a modular design approach, where multiple subsystem operators are supposed to parallelly design individual retrofit controllers with accessibility only to respective subsystem models. A retrofit controller is defined as a plug-in-type robust controller such that the stability of the resultant control system can be robustly assured for any possible variation of environments such that the original network system before implementing retrofit control is stable. We aim at improving the resultant control performance while preserving the entire network stability.

In the line of our previous work, it is shown that such a retrofit controller from the standpoint of each subsystem operator can be designed without requiring any model of its environment; see [7, 8, 9, 10] and references therein. However, this, at the same time, implies that no information of an actual environment is used for retrofit controller design. Therefore, the resultant control performance is generally dependent on the possible variation of environments. Such a low degree of freedom in the existing retrofit control could make a possible bottleneck for performance regulation, as will be demonstrated in this paper. In fact, there remains a possibility to make use of some available information of environments to further improve the resultant control performance.

With this background, to reduce such a bottleneck in the existing retrofit control, we aim at developing a novel design method of retrofit controllers such that the resultant control performance can be regulated by adjusting the accuracy of approximate environment modeling. In particular, we show that the stability of the resultant control system is robustly assured regardless of not only the stability of approximate environment models, but also the magnitude of modeling errors. This robustness property of the proposed retrofit control is practically significant to incorporate existing identification methods of unknown environments, such as in [17, 18], because the accuracy of identified models may neither be reliable nor assurable in reality. Furthermore, we conduct a control performance analysis to show that the foregoing bottleneck in the existing retrofit control can be reduced as improving the accuracy of environment modeling. It should be noted that we do not explicitly discuss how to produce an approximate environment model in this paper, but we discuss how to effectively utilize an approximate environment model found by some offline identification before implementing retrofit control.

A distributed design method of decentralized controllers is developed in [19], where the authors discuss the performance limitation of a linear quadratic regulator designed in a modular fashion. This result is then generalized to the case of a network of multi-dimensional subsystems, the states of which are assumed to be fully controlled [20]. As an approach to modular design of decentralized controllers, a system decomposition method based on an integral quadratic constraint is developed in [21]. Though their formulation can actually frame a broad class of systems, conditions required for decomposed subsystems are generally conservative as remarked there. As compared to these related approaches, the retrofit control has the advantage of applicability to more general stable network systems, for which we just assume the measurability of interconnection signals among subsystems.

We remark also that the retrofit control has a clear distinction from plug-and-play control [22, 23], in which incremental addition of new devices, such as controllers, is considered for a working control system. In general, the existing design schemes for plug-and-play control are not modular, meaning that an entire system model or its estimation is required for controller design. From the viewpoint of modularity in design, we can also find a similarity with control system design based on passivity, or, more generally, dissipativity and passivity shortage [24, 25, 26]. It is well known that negative feedback of passive subsystems retains the passivity. This means that the stability of the entire network system can be ensured if individual subsystems are designed to be passive. However, though a theoretically grounded procedure with modularity can be developed, the applicability of such a passivity-based approach is restrictive as compared to the retrofit control. This is simply because a network system of interest is not always decomposable into passive or passifiable subsystems.

The proposed retrofit control with approximate environment modeling is relevant to low-dimensional controller design based on model reduction [27, 28, 29]. In particular, we can consider first applying model reduction to a system of interest, and then perform controller design based on the resultant approximate model, where an approximation error due to model reduction can be handled as a model uncertainty in robust control [30]. However, such a model reduction method may not be applicable for a practical network system managed by multiple subsystem operators because a “complete” system model, to which model reduction is applied, is generally difficult to obtain. In view of this, there are practical difficulties not only to find an approximate model, but also to assure the accuracy of approximate models. The proposed retrofit control is a promising approach based on approximate modeling that can assure the control system stability without requiring the assurance of approximation accuracy. We remark that such an unassured modeling error is not considered in a standard robust control setting.

The remainder of this paper is organized as follows. In Section 2, we first review several existing results of retrofit control as a preliminary. In particular, we give a motivating example that demonstrates a bottleneck for performance regulation in the existing retrofit control. In Section 3, we develop a novel retrofit control method with approximate environment modeling. We first give a characterization of the new retrofit controllers by a frequency-domain analysis, based on the premise of a stability assumption of approximate environment models. This stability assumption, making the Youla parameterization of retrofit controllers simpler, is made just for improving the visibility of a particular structure inside the retrofit controller. Then, we provide a tractable state-space representation of approximate environment models the retrofit controllers by a time-domain analysis, which shows that the stability assumption of premised in the frequency-domain analysis is not essential to prove the internal stability of the resultant control system. Section 4 revisits the motivating example to show practical significance of the proposed retrofit control. Finally, concluding remarks are provided in Section 5.

Notation  The notation in this paper is generally standard: The identity matrix with an appropriate size is denoted by . The set of stable, proper, real rational transfer matrices is denoted by . For simplicity, all transfer matrices in the following are assumed to be proper and real rational. The -norm of a transfer matrix with no singularities on the imaginary axis is denoted by , which coincides with the -norm if is stable. A transfer matrix is said to be a stabilizing controller for if the feedback system of and is internally stable in the standard sense [30].

2 Review of Existing Retrofit Control

2.1 General Formulation

In this section, we first review several existing results of retrofit control reported in [7, 8, 9, 10]. Consider an interconnected linear system depicted in Fig. 1(a) where

(1a)
is referred to as a subsystem of interest for retrofit control, and
(1b)
is referred to as its environment.

From the viewpoint of controlling a general network system composed of multiple subsystems, corresponds to a particular subsystem for which a retrofit controller is designed by a subsystem operator, while the environment corresponds to a lumped representation of all other subsystems, which can be high-dimensional. In this formulation, the subsystem model of is assumed to be available for the retrofit controller design, while that of , which can be affected by other subsystem operators, is assumed to be unknown.

We denote the interconnection signals between the subsystem and its environment by and , the evaluation output and disturbance input by and , and the measurement output and control input by and , respectively. For the subsequent discussion, we use symbols denoting the submatrices of , for example, as

(2)

Then, we introduce the transfer matrix defined by the feedback system of and as

(3)

We refer to as a preexisting system, described as the dotted box in Fig. 1(a). Based on this system description, the notion of retrofit controllers is defined as follows.

Fig. 1: (a) Preexisting system composed of a subsystem of interest and its environment. (b) Resultant control system.
Definition 2.1

For the preexisting system in (2.1), define the set of all admissible environments as

(4)

An output feedback controller

(5)

is said to be a retrofit controller if the resultant control system in Fig. 1(b) is internally stable for any possible environment .

The retrofit controller is defined as a plug-in-type robust controller such that the stability of the resultant control system can be robustly assured for any possible variation of environments such that the preexisting system is stable. We remark that the norm bound of the environment is not premised. Instead, we just premise the internal stability of the preexisting system. Based on this definition, we first consider giving a parameterization of retrofit controllers. To avoid unnecessary complication of controller parameterization based on the Youla parameterization [31, 32], we make the following assumption.

Assumption 2.1

The subsystem in (1a) is stable.

Assumption 2.1 is made just for improving the visibility of a particular structure inside the retrofit controller, but it is not crucial to prove the resultant control system stability, as will be shown in Theorem 3.3. Then, we can derive the following parameterization of retrofit controllers.

Proposition 2.1

Let Assumption 2.1 hold. Consider the Youla parameterization of in (2.1) as

(6)

where is the Youla parameter. If

(7)

then is a retrofit controller.

Proposition 2.1 shows that the constrained version of the Youla parameterization in (2.1) gives the parametrization of retrofit controllers. We remark that, more generally, “all” retrofit controllers can also be parameterized by the Youla parameterization such that

(8)

which is more general than (2.1). This general parameterization, derived in [9, 10], further shows that the retrofit controller in Definition 2.1 can be characterized as a controller such that the transfer matrix from to of the local control system isolated from the environment is kept invariant. This can be seen as follows. Let denote the transfer matrix from to in Fig. 1(b) as removing the block of . Then, we have

(9)

which implies that for any retrofit controller because of (2.1).

In the rest of this paper, as following the terminology used in [7, 8, 9, 10], we refer to a retrofit controller parameterized in Proposition 2.1 as an output-rectifying retrofit controller. Note that such retrofit controllers are conditioned by the transfer matrices and that are relevant to the subsystem , but not to the environment . In other words, an output-rectifying retrofit controller can be designed only with the model information of isolated from .

to       \vsplit      

2.2 Review of Output-Rectifying Retrofit Control with Interconnection Signal Measurement

In this subsection, as a preliminary for the main theoretical developments in Section 3, we describe specific results on output-rectifying retrofit control in Proposition 2.1. These results can be derived as a simple generalization of results in [7, 8, 9, 10], but are not exactly the same as those. In the rest of this paper, we consider the following situation.

Assumption 2.2

The interconnection signal and are measurable in addition to the measurement output in (2.1).

From a symbolic viewpoint, Assumption 2.2 corresponds to the situation where every symbol in the discussion of Section 2.1 is to be replaced with the new measurement output . Based on this premise, the transfer matrices in (1a) relevant to are also redefined. For example, and are redefined as

Furthermore, the controller in (2.1) is also redefined as

(10)

the Youla parameterization of which can be written as

A simple but notable fact to be used is that

(11)

is a basis of the left kernel of in , i.e.,

(12)

Using this left kernel basis , we can rewrite (2.1) and (2.1) as

where we have used the fact that

(13)

We refer to as an output rectifier, the name of which is based on the fact that the measurement output is rectified in such a way that

(14)

The rectified output is used as the input of , which is referred to as a module controller. This discussion leads to an “explicit” representation of all output-rectifying retrofit controllers as follows.

Proposition 2.2

Let Assumptions 2.1 and 2.2 hold. Then, in (2.2) is an output-rectifying retrofit controller if and only if

(15)

where is a stabilizing controller for and is defined as in (2.2). Furthermore, the block diagram of the resultant control system is depicted as in Fig. 2, i.e., the entire map is given as

(16)

where and denote the transfer matrices compatible with Fig. 2, given as

(17)
Fig. 2: Block diagram of existing retrofit control.

Proposition 2.2 can be proven as a special case of Theorem 3.1 below. It is shown that, if the module controller is designed as a stabilizing controller for , which is isolated from , then the stability of the resultant control system is always assured by in (2.2). This clearly shows the modularity of retrofit controller design; the model information of is not required to assure, at least, the stability of the resultant control system.

Next, we analyze the resultant control performance. From the block diagram in Fig. 2, we see that can be decomposed as where

The triangular inequality for the induced norm of leads to the following upper and lower bounds of the resultant control performance.

Proposition 2.3

With the same notation as that in Proposition 2.2, the resultant control performance is bounded as

(18)

where the induced gains of and are given as

We remark that is “directly regulatable” by a suitable choice of , but is not because the term dependent on is involved. For explanation, let us consider a situation where is made sufficiently small, but is not, i.e., . Then, (2.3) implies that . This means that actual control performance may not be satisfactory, even if is regulated desirably. Such an undesirable situation possibly arises when the magnitude of is large.

From the observation above, we can see that a large value of makes a “bottleneck” to perform satisfactory regulation based on the existing retrofit control. We can say that the value of evaluates a gap between and , each of which corresponds to the “actual performance level” of the resultant control system and the “assumed performance level” of the modular control system. The simplest but not realistic situation for the minimum gap is , which leads to the ideal situation where , or equivalently, , i.e., the actual performance level is equal to the assumed performance level.

2.3 Motivating Example

Fig. 3: Second-order oscillator network composed of 36 nodes. The subnetwork of the first six nodes corresponds to a subsystem of interest, and the remaining part is its unknown environment.

We give a motivating example that demonstrates the bottleneck of the existing retrofit control described in Section 2.2, towards highlighting the main contribution of this paper. Consider a network system composed of 36 nodes depicted in Fig 3. The subnetwork of the first six nodes is supposed to be a subsystem of interest, and the remaining part is supposed to be its unknown environment . Let and denote the label sets corresponding to and , i.e.,

Furthermore, let denote the label set corresponding to the set of nodes such that they are adjacent to the th node and involved in . In a similar fashion, let denote the label set of the other adjacent nodes involved in . With this notation, for each , the node dynamics is given as

(19)

where denotes the angular state, denotes the control input, denotes the disturbance input, and

(20)

denotes the interconnection signal from the environment. The node dynamics of the environment is given in the same fashion without the terms of and . The second-order oscillator network (2.3) can be regarded as a mechanical analog of synchronous generators [35]. In the context of power system modeling, the interconnection signal in (2.3) corresponds to the power flow between the subsystem and its environment. The three interconnection links are depicted by the dotted lines in Fig 3.

In the following simulation, we set all the inertia constants and damping constants as and . Furthermore, we set the coupling constants inside the subsystem and inside the environment uniformly as

The coupling constants between the subsystem and environment are to be varied as a parameter , i.e.,

(21)

For simplicity, we assume the symmetry .

For retrofit controller design, the control input and the disturbance input are assigned as

respectively. The measurement output and the evaluation output are assigned as

respectively. In addition to , the interconnection signals

are assumed to be measurable. We remark that only the local model parameters for and for are assumed to be available. The entire network system is originally stable for any nonnegative value of in (2.3). Though the system has a single zero eigenvalue, it does not matter because the corresponding eigenspace is unobservable from the evaluation output .

Fig. 4: Resultant control system behavior in response to impulsive disturbance at the first node. (a) Case of weak coupling between and . (b) Case of moderate coupling between and . (c) Case of strong coupling between and .

For the design of the module controller in (2.2), we apply the standard -control synthesis technique to , isolated from , such that

(22)

is minimized where is a weighting constant for the control input. Setting , we plot the impulse response of the resultant control system in Fig. 4, where Figs. 4(a)-(c) correspond to the cases of , , and , i.e., weak coupling, moderate coupling, and strong coupling, respectively. In each top subfigure of Figs. 4(a)-(c), the blue solid lines show the trajectory of when the retrofit controller in (2.2) is used, the red solid lines show the case where no controller is used, and the magenta dotted lines show the case where the output rectifier is not involved in the controller, i.e., the module controller is directly implemented as a simple decentralized controller

(23)

From these top subfigures, we see that the direct implementation of induces the instability of the resultant control systems even though is designed to be a stabilizing controller for . In contrast, the retrofit controller can actually guarantee the stability of the resultant control system for all the values of the coupling constant .

However, we can also see that the amplitude of becomes larger as the coupling between and becomes stronger. This outcome can be explained as follows. The decomposed outputs and in Fig. 2 are plotted in the middle and the bottom of Figs. 4(a)-(c), where the blue and red lines correspond to the cases with and without the retrofit controller, respectively. Note that the actual output is equal to the sum of and , the induced gains of which are denoted by and in (2.3), respectively. In fact, the behavior of is well controlled, and it is identical for all the values of because the module controller is designed only with the information of , which is not dependent on . In contrast, the magnitude of is amplified as increases, i.e., as the gain of increases. In accordance with this amplification, the magnitude of the resultant is also amplified.

Fig. 5: Block diagram of extended retrofit control.

As demonstrated here, a small-gain property for may be required for satisfactory regulation, though the internal stability of the resultant control system can be assured for any possible . This is mainly because only an “identical” retrofit controller designed with is used regardless of the variation of . To overcome this drawback, in the following sections, we will develop a new retrofit control method that can produce the block diagram in Fig. 5, where we make use of an approximate model of the environment . An important difference between Fig. 2 and Fig. 5 is that the block of in the middle of Fig. 2 is replaced with the block of

(24)

in Fig. 5. Note that represents a modeling error because represents an approximate model of . Intuitively, as making the modeling error small, we can generally reduce the amplitude of . We remark that the norm bound of is assumed not to be assurable because is assumed to be unknown. In the next section, we will develop such an extended version of the retrofit control method that assures the entire system stability without assuming any assurance of environment modeling accuracy.

3 Theoretical Developments

3.1 Frequency-Domain Analysis: Characterization of Extended Retrofit Controllers

In this section, we premise that an approximate environment model has been found in some way, though its modeling accuracy is not assured for retrofit controller design. Our basic strategy to incorporate such unassured environment modeling is to regard the feedback of the subsystem and the approximate environment model as a new subsystem of interest. This corresponds to the situation where the original preexisting system Fig. 1(a) is equivalently regarded as the feedback system in Fig. 6. In particular, we regard

(25)

as a new subsystem of interest and the modeling error in (2.3) as a new environment. In this formulation, it is interesting to note that the modeling error can be viewed as a dynamical component that stabilizes the new subsystem . Clearly, holds if .

In the following discussion, in a manner similar to (2.1), we denote submatrices of , e.g., by

One may think that the existing results in Section 2.2 can be directly applied as simply replacing with , and with . However, it is not very clear to see if such a simple replacement is valid or not because the interconnection signals between and are found to be and , which are clearly different from the original interconnection signals and between and . Therefore, we need to carefully discuss how in the form of (2.2) should be modified or generalized in this new formulation of retrofit control. As an answer to this question, we will show that the set of all retrofit controllers with environment modeling actually coincides with the set of all retrofit controllers in Proposition 2.2, but has a much complicated structure.

Fig. 6: An equivalent representation of preexisting system.

In the derivation of Proposition 2.2, we started the discussion from the fact that the Youla parameter can be factorized as in (2.2), and then we showed that in (2.2) is found to be a stabilizing controller for . In what follows, as a converse direction, we first suppose that is given as a stabilizing controller for , and then we will derive a compatible factorization of . To make the Youla parameterization tractable, we make the following assumption.

Assumption 3.1

The approximate model belongs to , i.e., in (3.1) is internally stable.

Assumption 3.1 is in fact not crucial to prove the resultant control system stability, as shown in Theorem 3.3 below. Owing to this assumption, the Youla parameterization of can be simply written as

(26)

This means that is a stabilizing controller for . As a generalization of (2.2), we notice that

(27)

where , being invertible in , is defined as

(28)

Note that for any . In addition, if . Substituting (3.1) into (3.1) and multiplying it by from the right side, we have

(29)

which gives the Youla parameterization of such that (2.2) holds. We remark that can be seen as an extended output rectifier that performs the output rectification of

(30)

which is a generalization of (2.2). This derivation enables to generalize Proposition 2.2 as follows.

Theorem 3.1

Let Assumptions 2.1, 2.2, and 3.1 hold. Then, in (2.2) is an output-rectifying retrofit controller if and only if

(31)

where is a stabilizing controller for , and and are defined as in (2.2) and (3.1), respectively. Furthermore, the block diagram of the resultant control system is depicted as in Fig. 5, i.e., the entire map is given as

(32)

where and denote the transfer matrices compatible with Fig. 5, given as

(33)

and is defined as in (2.3).

Proof

We see that (3.1) is the Youla parameterization of in (3.1). Note that (3.1) is equivalent to (3.1) because is invertible and is right invertible. Thus, all output-rectifying retrofit controllers in the form of (2.2) can be written as (3.1).

Next, let us prove (3.1). As shown in [10], for any output-rectifying retrofit controller, the entire map is given as

In a similar manner to (3.1), we have

Thus, we see that

On the other hand, the input-to-output map of Fig. 5, denoted here by , is given as

For the identity of , it suffices to show that

(34)

and

Because both equalities can be proven in a similar manner, we only prove (3.1). Subtracting the left-hand side of (3.1) from the right-hand side, we have

The relations indicated by the underbraces are proven by

(35)

Hence, the claim is proven.

Theorem 3.1 provides another representation of all output-rectifying retrofit controllers in which the approximate environment model is involved as a tuning parameter. In particular, in (3.1) is shown to be an output-rectifying retrofit controller if the module controller

(36a)
is a stabilizing controller for the new subsystem of interest
(36b)

The resultant retrofit controller is specifically found as

It is not trivial to see that the control system in Fig. 1(b) with such a complicated controller can be equivalently expressed as the cascade block diagram in Fig. 5.

The feedback structure in the retrofit controller is encapsulated as the invertible transfer matrix “” involved in (3.1), which gives a clear bridge between the new retrofit controller in Theorem 3.1 and the existing one in Proposition 2.2. In fact, those retrofit controllers have a one-to-one correspondence, i.e., is a stabilizing controller for in the new retrofit control formulation if and only if is a stabilizing controller for in the existing formulation. We remark that such an idea of factorizing a stabilizing controller for as in the particular form of “” is generally difficult to devise in the framework of the existing retrofit control.

Owing to this special controller factorization, the extended retrofit controller gains higher flexibility in design. In the existing formulation, the Youla parameter of all output-rectifying retrofit controllers is expressed as where we can choose as “any” stable transfer matrices. This means that even the dimension of can be arbitrary in general. However, a standard controller design technique, such as the -control synthesis, generally produces a stabilizing controller , or equivalently , only with a dimension comparable to that of . This can be seen as an implicit limitation to find a possibly better controller. In contrast, the new retrofit control formulation provides an additional degree of freedom to find out a higher-dimensional by tuning the approximate environment model , whose dimension can be selected arbitrarily.

Another practical insight gained from Theorem 3.1 is the fact that the gap between the actual performance level of the resultant control system and the assumed performance level of the modular control system can be reduced if accurate environment modeling is performed. In particular, we can easily have a generalization of Proposition 2.3 as follows.

Theorem 3.2

With the same notation as that in Theorem 3.1, the resultant control performance is bounded as

(37)

where the induced gains of and are given as

As a generalization of Proposition 2.3, again corresponds to the assumed performance level, and evaluates the gap between and . Because the modeling error is linearly involved in , we can expect that decreases if the magnitude of is made small. Clearly, , or equivalently, if . Therefore, as improving the accuracy of environment modeling, we can generally reduce the “bottleneck” of the existing retrofit control described in Section 2.2.

to       \vsplit      

3.2 Time-Domain Analysis: State-Space Realization of Extended Retrofit Controllers

For simplicity of the Youla parameterization, we have assumed in Theorem 3.1 that the subsystem is stable, and the approximate environment model belongs to . However, these assumptions are, in fact, not crucial to prove the internal stability of the resultant control system as shown in this subsection. To prove this, we derive a tractable state-space realization of in (3.1). Furthermore, we show that the block diagram in Fig. 5 can be understood as a particular state-space realization of the entire control system obtained by a coordinate transformation.

We describe a time-domain representation of the preexisting system (2.1) by

(38a)
For simplicity of description, we suppose here that is a static map, i.e., a matrix. We remark that the subsequent discussion can be easily extended to the case of dynamical environments in such a way that is regarded as the convolution operator associated with , i.e.,
where is the impulse response of . The bold face symbols that will appear in the subsequent discussion, such as , are also supposed to be static, just for simplicity of description.
The premise of , i.e., the internal stability of (2.1), in Section 3.1 can be rephrased as the stability of
(38b)
which is a combined representation of the subsystem and environment in (38a).

As a time-domain analog of Definition 2.1, we introduce the following terminology.

Definition 3.1

For the preexisting system in (38b), define the set of all admissible environments as

(39)

Under Assumption 2.2, an output feedback controller

(40)

where denotes a dynamical map, is said to be a retrofit controller if the resultant control system that is composed of (3.2) and (3.1) is internally stable for any possible environment .

On the basis of this definition, a state-space realization of the extended retrofit controller is given as follows. We again remark that Assumptions 2.1 and 3.1, i.e., the assumptions on the stability of and , are not required to prove the internal stability of the resultant control system.

Theorem 3.3

Let Assumption 2.2 hold. For any approximate environment model and any feedback gains and such that

(41)

is stable, an output feedback controller

(42)

is a retrofit controller.

Proof

We first prove that (3.3) is a state-space realization of in (3.1). The time-domain representation of in (3.1) is now given as

(43)

The stability of (3.3) corresponds to the fact that the feedback gains and are chosen such that (3.2) stabilizes

(44)

which is a time-domain representation of . What remains to show is that

(45)

is a realization of , i.e., the output rectifier given in the frequency domain as

The block diagram of (3.2) can be depicted as in Fig. 7. From this diagram, we see that

where we have used the fact that

(46)

Therefore, we have

where we have used (3.1) for the calculation of . Thus, (3.3) is proven to be a state-space realization of in (3.1).

In what follows, we show that the resultant control system composed of (3.2) and (3.3) is internally stable for any . To this end, we apply the coordinate transformation of

(47)

Then, we see that the entire control system is given as the cascade connection of the upstream system

the stability of which is equivalent to that of the closed-loop system of (3.2) and (3.2), and the downstream system

(48)

the stability of which is equivalent to . Thus, the control system composed of (3.2) and (3.3) is equivalent to

(49)

Hence, the internal stability is proven for any