Output feedback negative imaginary synthesis under structural constraints

Abstract

The negative imaginary property is a property that many practical systems exhibit. This paper is concerned with the negative imaginary synthesis problem for linear time-invariant systems by output feedback control. Sufficient conditions are developed for the design of static output feedback controllers, dynamic output feedback controllers and observer-based feedback controllers. Based on the design conditions, a numerical algorithm is suggested to find the desired controllers. Structural constraints can be imposed on the controllers to reflect the practical system constraints. Also, the separation principle is shown to be valid for the observer-based design. Finally, three numerical examples are presented to illustrate the efficiency of the developed theory.

Introduction

The study of negative imaginary systems has attracted much attention in recent years (Cai and Hagen, 2010, Ferrante and Ntogramatzidis, 2013, Lanzon and Petersen, 2008, Mabrok et al., 2015, Mabrok et al., 2014, Wang et al., 2015a, Wang et al., 2015b, Xiong et al., 2012). By appropriately choosing the system input and output, many practical dynamic systems can be modelled as negative imaginary systems. Examples could be found in active vibration control systems (Das et al., 2015, Fanson and Caughey, 1990, Moheimani et al., 2006, Petersen and Lanzon, 2010) and circuit systems (Petersen & Lanzon, 2010). An important class of results for negative imaginary systems is the stability results developed in  Lanzon and Petersen (2008), Xiong, Petersen, and Lanzon (2010), Mabrok et al. (2014), Liu and Xiong (2015). For positive-feedback interconnected negative imaginary systems, necessary and sufficient conditions are established to test the system stability. These results can be considered as a generalization of the positive position control results in  Moheimani et al. (2006), Fanson and Caughey (1990), and depend only on the system gains at zero and infinite frequencies. In other words, the interconnected systems might have large control gains over other frequencies. In contrast, the small gain theorem requires that the control gains be small over all frequencies. An important application of the results is to robust control problems, where system uncertainty can be modelled by a negative imaginary system. Then, the closed-loop system will be stable as long as the controller is negative imaginary and satisfies the gain conditions. An illustrative example can be found in  Xiong et al. (2010), where the uncertainty parameter is allowed to be arbitrarily large. Therefore, the results in  Lanzon and Petersen (2008), Xiong et al. (2010), Mabrok et al. (2014), Liu and Xiong (2015) provide an attractive tool for robust control.

The underlying motivation of this study is to extend the application areas of negative imaginary systems theory. Consider the case that the uncertainty part in a system is negative imaginary while the remaining part of the system is not. The stability results in  Lanzon and Petersen (2008), Xiong et al. (2010), Mabrok et al. (2014), Liu and Xiong (2015) will not be applicable. To make them applicable, one has to design controllers such that the remaining part of the system is negative imaginary; see examples in  Petersen and Lanzon (2010), Song, Lanzon, Patra, and Petersen (2012), Mabrok et al. (2015). The problem of designing controllers for non-negative imaginary systems such that the resulting closed-loop systems become negative imaginary is called the negative imaginary synthesis problem, and the designed controllers are called negative imaginary controllers. When the full system state is available, state feedback negative imaginary controllers can be designed, and the corresponding design conditions have been established in  Petersen and Lanzon (2010), Song et al. (2012), Mabrok et al. (2015) for both minimal and nonminimal state-space realizations. However, in practice, the system state is often not available and only measurement output can be used when designing controllers. Also in many cases, the desired controllers have to meet structural constraints in the system design (Lin et al., 2011, Rubió-Massegú et al., 2013, Siljak, 1991, Zečević and Šiljak, 2008). For example, the controllers have to be of a block diagonal structure in the decentralized control of large-scale systems. Therefore, the design of output feedback negative imaginary controllers with structural constraints is an appealing practice use of the stability results in negative imaginary systems theory. The design of output feedback controllers has been recognized as a hard problem in general (Abbaszadeh and Marquez, 2009, Dinh et al., 2012, Shu et al., 2010, Syrmos et al., 1997).

This paper studies the negative imaginary synthesis problem when designing output feedback controllers. Firstly, the design of static output feedback controllers is considered. A sufficient condition is established in terms of a linear matrix inequality and a linear matrix equality. For the established design condition, an arbitrarily structural constraint can be readily imposed on the controller to meet practical requirements. It deserves mentioning that the solvability of the condition depends on the choice of the right inverse of the measurement matrix. An iterative algorithm is proposed to search for an approximate right inverse. When the measurement output equals to the system state, our result recovers the available ones in  Petersen and Lanzon (2010), Song et al. (2012). Then, the design condition is extended to design dynamic output feedback controllers and observer-based state feedback controllers. In particular, for observer-based control, the separation principle is shown to hold. Finally, three numerical examples are used to demonstrate the developed design theory. The first example demonstrates the application of the results to a robust stabilization problem where the uncertainty is modelled by a strictly negative imaginary system. The conservatism of the developed design condition is studied via the first example. The second example validates the applicability of the results to MIMO systems. The third example illustrates how structural constraints are imposed on the designed controllers. The designed controller is a decentralized reduced-order dynamic output feedback controller. In all, the contribution of this paper is that a systematic design theory for output feedback negative imaginary controllers is developed and controller structural constraints can be enforced.

Notation: Let ${\mathbb{R}}^{m\times n}$ and ${\mathcal{R}}^{m\times n}$ denote the set of $m\times n$ real matrices and real-rational proper transfer function matrices, respectively. $A^T$ and $A^{\ast }$ denotes the transpose and the complex conjugate transpose of a complex matrix $A$, respectively. $\Re [\cdot ]$ is the real part of a complex number. The notation $X>0$ or $X\ge 0$, where $X$ is a real symmetric matrix, means that the matrix $X$ is positive definite or positive semidefinite, respectively.