Simultaneous stabilization of discrete-time delay systems and bounds on delay margin

Abstract

This paper studies the delay margin problem of linear time-invariant (LTI) systems. For discrete-time systems, this problem can also be posed as one of simultaneous stabilization of multiple unstable systems with different lengths of delay. Our contribution is threefold. First, for general LTI plants with a number of distinct unstable poles and nonminimum phase zeros, we employ analytic function interpolation and rational approximation techniques to derive bounds on the delay margin. We show that readily computable and explicit lower bounds can be found by computing the real eigenvalues of a constant matrix, and LTI controllers, potentially of a low order, can be synthesized to achieve the bounds based on the ${\mathcal{H}}_{\infty }$ control theory. Second, we show that these results can be coherently extended to systems with time-varying delays, also resulting in bounds on the delay range and delay variation that ensure simultaneous stabilization. Finally, we investigate the delay margin problem in the context of PID control. For first-order unstable plants, we obtain bounds achievable by PID controllers. It is worth noting that unlike its continuous-time counterpart, the discrete-time delay margin problem is fundamentally more difficult, due to the intrinsic difficulty in simultaneously stabilizing several systems. Nevertheless, while previous works on the discrete-time delay margin led to largely negative results, the bounds developed in this paper provide instead conditions that guarantee the simultaneous stabilization of multiple delay plants, or ranges within which the delay plants can be robustly stabilized.

Introduction

Time delays are found in many engineering systems, especially in modern interconnected networks, which may result from communication delays, measurement delays, and computational delays. The presence of time delays can degrade the performance and robustness of control systems, and in the extreme lead to instability. There has been a large body of literature documenting the advances on time-delay control systems in the recent years; see, e.g., Gu, Chen, and Kharitonov (2003), Loiseau, Michiels, Niculescu, and Sipahi (2009), Michiels and Niculescu (2014), Niculescu and Gu (2012) and Richard (2003).

The delay margin furnishes a fundamental robustness measure in stabilizing a system against unknown, uncertain, and possibly time-varying delays, which concerns the question (Blondel & Megretski, 2009): For a fixed finite-dimensional LTI plant, is there an upper bound on the uncertain delay that can be tolerated by a LTI stabilizing controller? The problem dwells on one of the fundamental limitations of LTI feedback controllers, and it bears close similarity to the classical measures of gain and phase margin. Unlike the gain and phase margin problems (Doyle, Francis, & Tannenbaum, 1992), however, the delay margin problem proves to be fundamentally more challenging due to the difficulty of controlling infinite-dimensional systems; delay systems constitute a subclass of infinite-dimensional systems and are inherently more difficult to control.

Most of the existing work on the delay margin problem has been focused on continuous-time systems. In Middleton and Miller (2007), explicit upper bounds are derived for single-input single-output (SISO) systems determined by the unstable poles and nonminimum phase zeros of the plant. The bounds become tight for plants containing one unstable pole and one nonminimum phase zero. Subsequent improvements are made in Ju and Zhang (2016), for plants with two or more real unstable poles. On the other hand, Qi, Zhu, and Chen (2017) obtained lower bounds for systems with an arbitrary number of unstable poles and nonminimum phase zeros, and developed an operator-interpolation approach applicable to SISO and multi-input multi-output systems. Other works show that by using more sophisticated control laws, such as linear periodic controllers (Miller & Davison, 2005), nonlinear periodic controllers (Gaudette & Miller, 2014), and nonlinear adaptive controllers (Bekiaris-Liberis and Krstic, 2013, Bresch-Pietri et al., 2012), the delay margin can be made infinite. Truncated predictor feedback was also employed in a series of work (Wei and Lin, 2017, Yoon and Lin, 2013, Zhou et al., 2009, Zhou et al., 2012) that established bounds on the delay margin under a number of specialized circumstances, which in most of the cases involve infinite-dimensional, delayed linear controllers. Similarly, parallel results for discrete-time systems were obtained in Wei and Lin (2016).

In this paper we examine the delay margin problem for LTI discrete-time plants with LTI controllers, which, alternatively, can be posed as a simultaneous stabilization problem. Surprisingly, unlike the parallelism one typically expects between continuous-time and discrete-time results, the discrete-time delay margin problem turns out to be fundamentally more difficult. Indeed, for continuous-time systems, if a LTI plant can be stabilized by a LTI controller free of delay, then the system can always tolerate a finite amount of delay; in other words, the delay margin is always greater than zero. This is no longer true for discrete-time systems. In stark contrast, it is shown in Gaudette and Miller (2011) that a discrete-time plant has a zero delay margin whenever it contains a negative real unstable pole. Fundamentally, unlike its continuous-time counterpart, the discrete-time delay margin problem amounts to stabilizing simultaneously multiple plants, each differing from another by its length of delay. Problems in this class are generically difficult; in general, simultaneous stabilization of multiple systems poses a NP-hard decision problem, and is considered intractable due to its prohibitive computational complexity (Blondel & Megretski, 2009). The difficulty in solving the discrete-time delay margin problem, despite being a special class of simultaneous stabilization problems, can be attributed to this complexity.

Our purpose in this paper is to establish lower bounds on the delay margin, which consequently provide a guaranteed range of delays ensuring the robust stabilization of the systems within that range, or equivalently, conditions that guarantee the simultaneous stabilization of multiple delay systems. Based on the small-gain stability condition and rational approximation techniques, we cast the delay margin problem as one of parameter-dependent ${\mathcal{H}}_{\infty }$ optimization problems. The problem is then tackled and solved by employing analytic interpolation theory (Ball, Gohberg, & Rodman, 1990). Our main contributions can be summarized as follows:

• A crucial step in our solution approach is to construct low-order rational function approximations of certain delay transfer functions. Required to meet several specifications, these constructions by themselves are nontrivial and require careful numerical experiments. The approximations not only lead to bounds on the delay margin, but also enable us to synthesize potentially low-order optimal ${\mathcal{H}}_{\infty }$ controllers that in fact attain the bounds. We note that the approximations do not readily translate from their continuous-time counterparts, but are new contributions and likely of independent interest in their own right. Such rational functions are delay-dependent and are constructed for both constant and time-varying delays.

• With the rational approximations so constructed, we then develop a computational formula and explicit bounds on the delay margin. This is accomplished by solving a delay-dependent ${\mathcal{H}}_{\infty }$ optimal control problem using Nevanlinna–Pick interpolation techniques, whose solution in turn can be obtained explicitly by solving an eigenvalue problem. The computational formula requires solving only the real eigenvalues of a constant matrix, and is applicable to plants with arbitrarily many distinct unstable poles and nonminimum phase zeros. The explicit bounds demonstrate, by a closer examination of more specific cases, how the plant unstable poles and zeros may limit the delay margin achievable. Such formulas and bounds are obtained for systems with both constant and time-varying delays.

• In yet another contribution, we consider controllers with fixed order and structure. Specifically, we derive bounds on the delay margin of low-order unstable systems achievable by PID controllers. Instead of employing the interpolation techniques, we analyze the system frequency response directly. These bounds are similar in spirit to their continuous-time counterparts developed in Silva et al., 2002, Silva et al., 2005, and are motivated by the considerable interest long held in PID control (Åström and Hägglund, 2006, Xue et al., 2007), which is prevalent in the design and implementation of industrial control systems. In many industrial control applications, it is typical to model physical processes by first-order dynamics. We follow the suit by addressing first-order delay systems. The results consequently shed lights into the limitation of PID controllers in controlling delay systems and in simultaneous stabilization.

The notation used in this paper is standard. Let $\mathbb{N}$ denote the set of nonnegative integers. Let $\mathbb{D}≔\{z:\left|z\right|<1\}$, $\overline{\mathbb{D}}≔\{z:\left|z\right|\le 1\}$, and ${\mathbb{D}}^C≔\{z:\left|z\right|\ge 1\}$. For any complex number $z$, we denote its conjugate by $\bar{z}$. For a matrix $A$, we denote by $A^{\mathrm{H}}$ its conjugate transpose. If $A$ is a Hermitian matrix, its largest eigenvalue will be denoted as $\bar{\lambda }\left(A\right)$. At times, where no confusion may arise, we also denote by $\bar{\lambda }\left(A\right)$ the largest real eigenvalue provided a real eigenvalue exists, even though $A$ may not be symmetric. The matrix inequality $A\ge 0(A\le 0)$ indicates that $A$ is nonnegative (nonpositive) definite, and $A>0(A<0)$ indicates that $A$ is positive (negative) definite. $\lfloor x\rfloor$ denotes the floor function, i.e., the largest integer less than $x$. The notation $m\left(mod\right)n$ represents the remainder $m∕n$ of two integers $m$ and $n$. Let ${\mathcal{L}}_2$ (see, e.g., Zhou, Doyle, and Glover (1996)) be the space of square summable sequences $u=\{u\left(0\right),u\left(1\right),\dots \}$ with the norm

$${\Vert u\Vert }_2=\sqrt{\sum _{k=0}^{\infty }{\Vert u\left(k\right)\Vert }^2},$$

where $\Vert u\left(k\right)\Vert$ is the Hölder ${\ell }_2$ norm of the vector $u\left(k\right)$. For any stable linear system $G$, we may define the ${\mathcal{L}}_2$ induced system norm as

$${\Vert G\Vert }_{2,2}=\underset{u\ne 0}{sup}\frac{{\Vert Gu\Vert }_2}{{\Vert u\Vert }_2}.$$

If $G$ is a stable LTI system, then it can be represented by its transfer function $G\left(z\right)$, and ${\Vert G\Vert }_{2,2}$ reduces to the ${\mathcal{H}}_{\infty }$ norm of $G\left(z\right)$:

$${\Vert G\left(z\right)\Vert }_{\infty }=\underset{z\in \overline{\mathbb{D}}}{sup}\left|G\left(z\right)\right|.$$

The set of all stable transfer functions with ${\mathcal{H}}_{\infty }$ norm bounded by a certain $\gamma >0$ is denoted as

$$B{\mathcal{H}}_{\infty }\left(\gamma \right)≔\{G\left(z\right):{\Vert G\left(z\right)\Vert }_{\infty }\le \gamma \}.$$

Finally, the identity operator is denoted by $\mathcal{I}$.