## 1. Introduction

It is well known that control systems can be found in abundance in all sectors of industry such as robotics, power systems, transportation systems space technologies, and many others, and thus control theory has been well studied. In order to design control systems, designers have to derive mathematical models for dynamical systems, and there are mainly two types of representations for mathematical models, that is, transfer functions and state equations. In other words, control theory is divided into “classical control” and “modern control” (e.g., see [12]).

Classical control means an analytical theory based on transfer function representations and frequency responses, and for classical control theory, we can find a large number of useful and typical results such as Routh-Hurwitz stability criterion [20] based on characteristic equations in the nineteenth century, Nyquist criterion [28] in the 1930s, and so on. Moreover, by using classical control ideas, some design methods of controllers such as proportional, derivative, and integral (PID) controllers and phase lead-lag compensators have also been presented [21]. In classical control, controlled systems are mainly linear and time-invariant and have a single input and a single output only. Furthermore, it is well known that design approaches based on classical control theory need experiences and trial and error. On the other hand, in the 1960s, state variables and state equations (i.e., state-space representations) have been introduced by Kalman as system representations, and he has proposed an optimal regulator theory [14, 15, 16] and an optimal filtering one [17]. Namely, controlled systems are represented by state equations, and controller design problems are reduced to optimization problems based on the concept of state variables. Such controller design approach based on the state-space representation has been established as “modern control theory.” Modern control is a theory of time domain, and whereas the transfer function and the frequency response are of limited applicability to nonlinear systems, state equations and state variables are equally appropriate to linear multi-input and multi-output systems or nonlinear one. Therefore, many existing results based on the state-space representation for controller design problems have been suggested (e.g., [7, 43]).

Now, as mentioned above, in order to design control systems, the derivation of a mathematical model for controlled system based on state-space representation is needed. If the mathematical model describes the controlled system with sufficient accuracy, a satisfactory control performance is achievable by using various controller design methods. However, there inevitably exists some gaps between the controlled system and its mathematical model, and the gaps are referred to as “uncertainties.” The uncertainties in the mathematical model may cause deterioration of control performance or instability of the control system. From this viewpoint, robust control for dynamical systems with uncertainties has been well studied, and a large number of existing results for robust stability analysis and robust stabilization have been obtained [34, 36, 47, 48]. One can see that quadratic stabilization based on Lyapunov stability criterion and

In recent years, a great number of control systems are brought about by present technologies and environmental and societal processes which are highly complex and large in dimension, and such systems are referred to as “large-scale complex systems” or “large-scale interconnected systems.” Namely, large-scale and complex systems are progressing due to the rapid development of industry, and large-scale interconnected systems can be seen in diverse fields such as economic systems, electrical systems, and so on. For such large-scale interconnected systems, it is difficult to apply centralized control strategies because of calculation amount, physical communication constraints, and so on. Namely, a notable characteristic of the most large-scale interconnected systems is that centrality fails to hold due to either the lack of centralized computing capability of or centralized information. Moreover, large-scale interconnected systems are controlled by more than one controller or decision-maker involving decentralized computation. In the decentralized control strategy, large-scale interconnected systems are divided into several subsystems, and various types of decentralized control problems have been widely studied [13, 38, 44]. The major problem of large-scale interconnected systems is how to deal with the interactions among subsystems. A large number of results in decentralized control systems can be seen in the work of Šijjak [38]. Moreover, a framework for decentralized fault-tolerant control has also been studied [44]. Additionally, decentralized robust control strategies for uncertain large-scale interconnected systems have also attracted the attention of many researchers (e.g., [3, 4, 5, 11]). Moreover, in the work of Mao and Lin [24], for large-scale interconnected systems with unmodeled interaction, the aggregative derivation is tracked by using a model following the technique with online improvement, and a sufficient condition for which the overall system when controlled by the completely decentralized control is asymptotically stable has been established. Furthermore, decentralized guaranteed cost controllers for uncertain large-scale interconnected systems have also been suggested [26, 27].

In this chapter, for a class of uncertain linear systems, we show LMI-based design strategies for adaptive gain robust controllers for a class of uncertain dynamical systems. The adaptive gain robust controllers consist of fixed gains and adaptive gains which are tuned by time-varying adjustable parameters. The proposed adaptive gain robust controller can achieve asymptotical stability but also improving transient behavior of the resulting closed-loop system. Moreover, by adjusting design parameters, the excessive control input is avoided [32]. In this chapter, firstly, a design method of the centralized adaptive gain robust stabilizing controllers for a class of uncertain linear systems has been shown, and the maximum allowable perturbation region of uncertainties is discussed. Namely, the proposed adaptive gain robust controllers can achieve robustness for the derived perturbation regions for unknown parameters. Additionally, the result for the centralized adaptive gain robust stabilizing controllers is extended to the design problem of decentralized robust control systems.

The contents of this chapter are as follows, where the item numbers in the list accord with the section numbers:

2. Synthesis of centralized adaptive gain robust controllers.

3. Synthesis of decentralized adaptive gain robust controllers.

4. Conclusions and future works.

The basic symbols are listed below.

The set of the real number | |

The set of | |

The set of | |

The set of complex numbers |

Other than the above, we use the following notation and terms: For a matrix

Furthermore, the following useful lemmas are used in this chapter.

Lemma 1.1. *For arbitrary vectors* *and* *and the matrices* *and* *which have appropriate dimensions, the following relation holds:*

where

*Proof.* The above relation can be easily obtained by Schwartz’s inequality (see [9]).

**Lemma 1.2**. *(Schur complement) For a given constant real symmetric matrix* *, the following arguments are equivalent:*

*Proof.* See Boyd et al. [2].

## 2. Synthesis of centralized adaptive gain robust controllers

A centralized adaptive gain robust state feedback control scheme for a class of uncertain linear systems is proposed in this section. The adaptive gain robust controller under consideration is composed of a state feedback with a fixed gain matrix and a time-varying adjustable parameter. In this section, we show an LMI-based design method of the adaptive gain robust state feedback controller, and the allowable perturbation region of unknown parameters is discussed.

### 2.1. Problem statement

Consider the uncertain linear system described by the following state-space representation:

where

In Eq. (2),

First of all, we design the state feedback control for the nominal system of Eq. (2) so as to generate the desirable transient behavior in time response for the uncertain linear system of Eq. (1). Namely, the nominal control input is given as

and thus the following nominal closed-loop system is obtained:

where

where

Now, we introduce complementary matrices

Additionally, applying **Lemma 1.2** (Schur complement) to Eq. (7), one can easily see that the matrix inequality condition of Eq. (7) is equivalent to

Thus, the control gain matrix

Now, for the uncertain linear system of Eq. (1), we define the following control input [37]:

where

From the above, the control objective in this section is to design the adaptive gain robust control which achieves satisfactory transient behavior. Namely, the control problem is to derive the adjustable time-varying parameter

### 2.2. Synthesis of centralized adaptive gain robust state feedback controllers

In this subsection, we deal with design problems for the adjustable time-varying parameter

**Theorem 1:** *Consider the uncertain linear system of* *Eq. (1)* *and the adaptive gain robust control of* *Eq. (9)* *with the adjustable time-varying parameter*

*For a given design parameter* *and the known upper bound* *for the unknown parameter* *, if the scalar parameter* *exists satisfying*

*the adjustable time-varying parameter* *is determined as*

*Then, the uncertain closed-loop system of* *Eq. (10)* *is asymptotically stable*.

*Proof.* In order to prove Theorem 1, by using symmetric positive definite matrix

as a Lyapunov function candidate. Let

Firstly, the case of **Lemma 1.1** that the following inequality holds:

Moreover, since the relation

Substituting the adjustable time-varying parameter

If the solution of the LMI of Eq. (11) exists, then the inequality

is satisfied. Thus, one can see that the following relation holds:

Next, we consider the case of

we see from Eq. (14) that some algebraic manipulations give

where

Let us consider the last term of the right-hand side of Eq. (21). We see from Eq. (12) and the relation

then the following relation for the quadratic function

Furthermore, applying **Lemma 1.2** (Schur complement) to Eq. (22), we find that the matrix inequality condition of Eq. (22) can be transformed into the LMI of Eq. (11). Namely, the quadratic function

From the above, we show an LMI-based design strategy for the proposed adaptive gain robust control. Namely, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of Eq. (11). Note that the LMI of Eq. (11) defines a convex solution set of

where

and the constraint of Eq. (24) can be transformed into

From the above, we consider the following constrained optimization problem:

If the optimal solution of the constrained optimization problem of Eq. (27) exists, in which are denoted by

Consequently, the following theorem for the proposed adaptive gain robust control with guaranteed allowable region of unknown parameter

**Theorem 2:** *Consider the uncertain linear system of* *Eq. (1)* *and the adaptive gain robust control of* *Eq. (8)* *with the adjustable time-varying parameter*

*If the optimal solution* *of the constrained optimization problem of* *Eq. (27)* *exists, then the adjustable time-varying parameter* *is designed as* *Eq. (12)**, and asymptotical stability of the uncertain closed-loop system of* *Eq. (10)* *is ensured. Moreover, the upper bound* *for the unknown parameter* *is given by* *Eq. (28)*.

**Remark 1:** *In this section, the uncertain linear dynamical system of* *Eq. (1)* *is considered, and the centralized adaptive gain robust controller has been proposed. Although the uncertain linear system of* *Eq. (1)* *has uncertainties in the state matrix only, the proposed adaptive gain robust controller can also be applied to the case that the uncertainties are included in both the system matrix and the input one. Namely, by introducing additional actuator dynamics and constituting an augmented system, unknown parameters in the input matrix are embedded in the system matrix of the augmented system [**45**]. As a result, the proposed controller design procedure can be applied to such case.*

**Remark 2:** *In* Theorem 1*, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of* *Eq. (11)**. Namely, in order to design the proposed robust control system, designers have to solve the LMI of* *Eq. (11)**. If the LMI of* *Eq. (11)* *is feasible for* *, then one can easily see that the LMI of* *Eq. (11)* *is always satisfied for the positive scalar* *. Moreover, if a positive scalar* *exists satisfying the LMI of* *Eq. (11)* *for* *, then the proposed adaptive gain robust controller can also be designed, and note that the adaptive gain robust controller for* *coincides exactly with the one for* *. Furthermore, one can see from* Theorem 2 *that the resultant adaptive gain robust controller derived by solving the constrained convex optimization problem of* *Eq. (27)* *is same, because the solution of LMI of* *Eq. (8)* *or one of the constrained convex optimization problem of* *Eq. (27)* *cannot be reflected the resultant controller. Note that in the general controller design strategies for the conventional fixed gain robust control, the solution of the some constraints can be applied to the resultant robust controller. This is a fascinating fact for the proposed controller design strategy.*

**Remark 3:** *The proposed adaptive gain robust controller with the adjustable time-varying parameter has some advantages as follows: the proposed controller design approach is very simple, and by selecting the design parameter, the proposed adaptive gain robust control system can achieve good transient performance which is close to the nominal one or avoid the excessive control input (see [**32**]). Besides, the structure of the proposed control system is also simple compared with the existing results for robust controllers with adjustable parameters (*e.g., *[**29*, *30**]). However, the online adjustment strategy for the design parameter* *has not been established, and this problem is one of our future research subjects.*

**Remark 4:** *In this section, firstly the nominal control input is designed by adopting pole placement constraints, and the fixed gain* *can be derived by using the solution of the LMI of* *Eq. (8)**. Note that the quadratic function* *is a Lyapunov function for both the uncertain linear system of* *Eq. (1)* *and the nominal system of* *Eq. (2)**, that is, the Lyapunov function for the uncertain linear system of* *Eq. (1)* *and one for the nominal system of* *Eq. (2)* *have same level set. Therefore, by selecting the design parameter* *, the proposed adaptive gain robust control system can achieve good transient performance which is close to the nominal one or avoid the excessive control input.*

*On the other hand, if the design problem for a state feedback control *

### 2.3. Illustrative examples

In order to demonstrate the efficiency of the proposed control strategy, we have run a simple example.

Consider the following linear system with unknown parameter

Firstly, we design the nominal control input

Thus, the following fixed gain matrix can be computed:

Next, we solve the constrained optimization problem of Eq. (27), then the solutions

can be derived, and therefore the allowable upper bound of unknown parameter is given as

In this example, we consider the following two cases for the unknown parameter

• Case 1)

• Case 2)

Note that the unknown parameter of Case 1 satisfies the matching condition [45]. In addition, for the design parameter **Figures 1**–**4** and **Table 1**. In these figures, “Case 1)” and “Case 2)” represent the time histories of the state variables **Table 1**,

where

From **Figures 1**–**4** the proposed adaptive gain robust state feedback controller stabilizes the uncertain linear system of Eq. (29) in spite of uncertainties. Furthermore, we also find that the proposed adaptive gain robust controller achieves the good transient performance close to the nominal system.

For Case 1 in this example, one can see from **Table 1** that the adaptive gain robust controller for **Figures 2(a)** and **4(a)** that by selecting the design parameter

Therefore, the effectiveness of the proposed adaptive gain robust controller is shown.

### 2.4. Summary

In this section, an LMI-based design scheme of the centralized adaptive gain robust state feedback controller for a class of uncertain linear systems has been proposed, and by simple numerical simulations, the effectiveness of the proposed robust control strategy has been presented. Since the proposed adaptive gain robust controller can easily be obtained by solving the constrained convex optimization problem, the proposed design approach is simple. Moreover, by selecting the design parameter, the proposed adaptive gain robust controller can achieve good transient performance and/or avoid excessive control input. Note that there are trade-offs between achieving good transient performance and avoiding excessive control input.

The future research subject is the extension of proposed robust control scheme to such a broad class of systems as linear systems with state delays, uncertain systems with some constraints, and so on. Additionally, we will discuss the online adjustment for the design parameter

## 3. Synthesis of decentralized adaptive gain robust controllers

In this section, on the basis of the result derived in Section 2, an LMI-based design method of decentralized adaptive gain robust state feedback controllers for a class of uncertain large-scale interconnected systems is suggested. The design problem of the decentralized adaptive gain robust controller under consideration can also be reduced to the feasibility of LMIs, and the allowable perturbation region of uncertainties is also discussed.

### 3.1. Problem statement

Consider the uncertain large-scale interconnected system composed of

where

In Eqs. (35) and (36), the matrices

where

where

where

Now, by using symmetric positive definite matrices

as a Lyapunov function candidate. For the quadratic function

For the second term on the right side of Eq. (42), by using the well-known relation of Eq. (20), we can obtain the following relation:

(43) |

From Eqs. (40) and (43), we have

(44) |

The inequality of Eq. (44) can also be rewritten as

Therefore, if the matrix inequality

holds, then the following relation for the time derivative of

Now, as with Section 2, we derive a decentralized controller with pole placement constraints for the nominal subsystem of Eq. (37). Namely, from Eq. (46), the matrix inequality

is considered. In Eq. (48),

We introduce symmetric positive definite matrices

Thus, by applying **Lemma 1.2** (Schur complement) to Eq. (49), we find that the matrix inequality of Eq. (49) is equivalent to the following LMI:

In Eq. (50), matrices

(51) |

Therefore, if matrices

Now, by using the fixed gain matrix

where

From the above discussion, the designed objective in this section is to determine the decentralized robust control of Eq. (52) such that the resultant overall system achieves robust stability. That is to design the adjustable time-varying parameter

### 3.2. Decentralized variable gain controllers

The following theorem shows sufficient conditions for the existence of the proposed decentralized adaptive gain robust control system.

**Theorem 3:** *Consider the uncertain large-scale interconnected system of* *Eq. (35)* *and the control input of* *Eq. (52)*.

*For a given positive constant* *, if positive constants* *, and* *exist which satisfy the LMIs*

*the time-varying adjustable parameters* *are determined as*

(55) |

*where matrices* *, and* *are given by*

(56) |

*Then, the overall close-loop system composed of* *closed-loop subsystems is asymptotically stable*.

*Proof.* In order to prove Theorem 3, the following Lyapunov function candidate is introduced by using symmetric positive definite matrices

where

We can obtain the following relation for the time derivative of the quadratic function

(59) |

Firstly, we consider the case of **Lemma 1.1** that the following relation for the quadratic function

(60) |

Substituting the adjustable time-varying parameter

(61) |

and, thus, we have the following inequality for the function

(62) |

Furthermore, the inequality of Eq. (62) can be rewritten as

(63) |

Therefore, if the matrix inequality

holds, then the following relation for the time derivative of

Next, we consider the case of

(66) |

The last term on the right side of Eq. (66) is less than 0 because the matrix

(67) |

Therefore, we see from Eqs. (57) and (67) that the following inequality:

(68) |

can be derived. Moreover, one can easily see that the inequality of Eq. (68) can be rewritten as

(69) |

Therefore, if the matrix inequality

(70) |

holds, then the relation of Eq. (65) for the time derivative of the function

For the matrix inequality of Eq. (70), by applying **Lemma 1.2** (Schur complement), one can find that the matrix inequalities of Eq. (70) are equivalent to the LMIs of Eq. (54). Therefore, by solving the LMIs of Eq. (54), the adjustable time-varying parameter is given by Eq. (55), and proposed control input of Eq. (52) stabilizes the overall system of Eq. (35). Thus, the proof of Theorem 3 is completed.

Next, as mentioned in Section 2, we discuss the allowable region of the unknown parameters

From the relations of Eq. (71), one can find that the maximization of

(73) |

Furthermore, we introduce a positive scalar

and consider the following additional condition:

Namely, we can replace the maximization problem of **Lemma 1.2** (Schur complement), one can easily see that the constraint of Eq. (75) can be transformed into

Thus, in order to design the proposed decentralized adaptive gain robust controller, the constrained convex optimization problem

should be solved.

As a result, the following theorem can be obtained:

**Theorem 4:** *Consider the uncertain large-scale interconnected system of* *Eq. (35)* *and the control input of* *Eq. (52)*.

*If positive constants* *, and* *exist which satisfy the constrained convex optimization problem of* *Eq. (77)**, the adjustable time-varying parameter* *is designed as* *Eq. (55)**. Then, the overall uncertain closed-loop system of* *Eq. (53)* *is asymptotically stable. Furthermore, by using the optimal solution* *and* *for* *Eq. (77)**, the upper bound of unknown parameters* *and* *is given by*

### 3.3. Illustrative examples

To demonstrate the efficiency of the proposed decentralized robust controller, an illustrative example is provided. In this example, we consider the uncertain large-scale interconnected system consisting of three two-dimensional subsystems, that is,

(79) |

Firstly, by selecting the design parameters

(80) |

Thus, the symmetric positive definite matrices

(81) |

Next, by solving the constrained convex optimization problem of Eq. (77), the following solution can be obtained:

(82) |

Therefore, the allowable upper bound of unknown parameters is given as

(83) |

In this example, unknown parameters

Moreover, the design parameters

The result of this example is shown in **Figures 5** and **6**. In these figures,

From these figures, the proposed decentralized adaptive gain robust controller stabilizes the uncertain large-scale interconnected system with system parameters of Eq. (79). Furthermore, one can see that each subsystem achieves good transient behavior close to nominal subsystems by the proposed decentralized robust controller. Thus, the effectiveness of the proposed robust control strategy is shown.

### 3.4. Summary

In this section, on the basis of the result of Section 2, we have suggested the decentralized adaptive gain robust controller for the large-scale interconnected system with uncertainties. Furthermore, the effectiveness of the proposed controller has been shown via an illustrative example. The proposed adaptive gain robust controller can be easily designed by solving a constrained convex optimization problem and adjust the magnitude of the control input for each subsystem. Therefore, we find that the proposed decentralized robust controller design method is very useful.

Future research subjects include analysis of conservatism for the proposed controller design approach and extension of the proposed adaptive gain robust control strategies to uncertain systems with time delay, decentralized output/observer-based control systems, and so on.

## 4. Conclusions and future works

In this chapter, firstly the centralized adaptive gain robust controller for a class of uncertain linear systems has been proposed, and through a simple numerical example, we have shown the effectiveness/usefulness for the proposed adaptive gain robust control strategy. Next, for a class of uncertain large-scale interconnected systems, we have presented an LMI-based design method of decentralized adaptive gain robust controllers. In the proposed controller robust synthesis, advantages are as follows: the proposed adaptive gain robust controller can achieve satisfactory transient behavior and/or avoid the excessive control input, that is, the proposed robust controller with adjustable time-varying parameters is more flexible and adaptive than the conventional robust controller with a fixed gain which is derived by the worst-case design for the unknown parameter variations. Moreover, in this chapter we have derived the allowable perturbation region of unknown parameters, and the proposed robust controller can be obtained by solving constrained convex optimization problems. Although the solution of the some matrix inequalities can be applied to the resultant robust controller in the general controller design strategies for the conventional fixed gain robust control, the solutions of the constrained convex optimization problem derived in this chapter cannot be reflected to the resultant robust controller. Note that the proposed controller design strategy includes this fascinating fact.

In Section 2 for a class of uncertain linear systems, we have dealt with a design problem of centralized adaptive gain robust state feedback controllers. Although the standard LQ regulator theory for the purpose of generating the desired response is adopted in the existing result [32], the nominal control input is designed by using pole placement constraints. By using the controller gain for the nominal system, the proposed robust control with adjustable time-varying parameter has been designed by solving LMIs. Additionally, based on the derived LMI-based conditions, the constrained convex optimization problem has been obtained for the purpose of the maximization of the allowable perturbation region of uncertainties included in the controlled system. Section 3 extends the result for the centralized adaptive gain robust state feedback controller given in Section 2 to decentralized adaptive gain robust state feedback controllers for a class of uncertain large-scale interconnected systems. In this section, an LMI-based controller synthesis of decentralized adaptive gain robust state feedback control has also been presented. Furthermore, in order to maximize the allowable region of uncertainties, the design problem of the decentralized adaptive gain robust controller for the uncertain large-scale interconnected system has been reduced to the constrained convex optimization problem.

In the future research, an extension of the proposed adaptive gain robust state feedback controller to output feedback control systems or observer-based control ones is considered. Moreover, the problem for the extension to such a broad class of systems as uncertain time-delay systems, uncertain discrete-time systems, and so on should be tackled. Furthermore, we will examine the conservativeness of the proposed adaptive gain robust control strategy and online adjustment way of the design parameter which plays important roles such as avoiding the excessive control input.

On the other hand, it is well known that the design of control systems is often complicated by the presence of physical constraints: temperatures, pressures, saturating actuators, within safety margins, and so on. If such constraints are violated, serious consequences may ensue. For example, physical components will suffer damage from violating some constraints, or saturations for state/input constraints may cause a loss of closed-loop stability. In particular, input saturation is a common feature of control systems, and the stabilization problems of linear systems with control input saturation have been studied (e.g., [33, 40]). Additionally, some researchers have investigated analysis of constrained systems and reference managing for linear systems subject to input and state constraints (e.g., [10, 19]). Therefore, the future research subjects include the constrained robust controller design reducing the effect of unknown parameters.