On the Negative Imaginary Time-Delay Systems With Its Application to Flexible Systems Control Open Access

Passivity as a fundamental concept of circuit theory has become significant for systems and control theory after its generalization to dissipativity by Willems [1]. Both the passivity and dissipativity properties of linear and nonlinear systems have been extensively studied; see, e.g., Refs. [2] and [3]. Its implications are important in a wide range of control concepts with applications in many different fields. For instance, it is used in the robust control design of chemical processes [4] and applied to the adaptive control of robot manipulators [5]. The standout property of the dissipativity theorem is preserving the dissipative property in parallel or feedback connections of two dissipative systems. This fact provides applicability to the control of complex systems such as multi-agent system control, e.g., see Refs. [6] and [7]. Passivity and dissipativity theories involve the energy-based analysis of the input–output characteristics of systems. Passive systems basically cannot store more energy than they receive from the environment. The stored energy and the energy supplied to the system are generalized for dissipative systems as the storage function and the supply rate function, respectively [2,8]. In the context of time-delay systems (TDS), which is the main concern of this paper, the general approach to investigate the passivity of TDS is to utilize an appropriate Lyapunov–Krasovskii functional (LKF) as a storage function. In Ref. [9], sufficient conditions in terms of linear matrix inequalities (LMIs) are given for the passivity of linear TDS of retarded and neutral types. Passivity analysis based on LKF construction and the passivity-based stabilization of linear TDS are addressed in Ref. [10]. Dissipativity-based state feedback and dynamic output feedback control of TDS is studied in Ref. [11]. One can see Ref. [12] reducing conservativeness on passivity conditions. Furthermore, dissipativity conditions for the class of singular TDS are determined over LMIs in Ref. [13].

The energy-based approach of dissipativity provides a unified framework for the design and stability analysis of a wide class of complex systems. However, there are classes of systems which cannot be comprehensively analyzed within the conventional dissipation-based framework, despite exhibiting energy dissipation and finite energy storage. One of these class of systems is referred to negative imaginary (NI) systems since their dissipation property cannot be analyzed via static supply rate functions. NI systems theory is inspired by vibration suppression control of flexible systems with positive position feedback (PPF) where position sensors and force actuators are used [14]. First studies on NI systems were conducted on the frequency domain properties of transfer function models [15,16] where the stability condition of positive feedback interconnection of NI systems is defined simply over the open loop system DC gain. NI system theory has been preferably used in various fields because of this simple stability rule, e.g., see Ref. [17]. A recent study [18] proposed an algorithm to design static output feedback control which makes the closed-loop system NI with a predetermined $H∞$-norm. Moreover, NI theory is utilized to unify the vibration suppression techniques using PPF [19], and applied, for instance, to nanopositioning [20,21], to highly resonant structures [22], and to multi-agent system control [23–25].

Negative imaginary system properties are also related to the energy storage and the dissipation properties of systems. Reference [26] has shown the correspondence between NI and passive systems based on their dissipation characteristics. Passive systems are dissipative with respect to the supply rate defined by the input to output signal relation $(u,y)$⁠, whereas the NI systems are dissipative with respect to the dynamic supply rate defined for the pair $(u,y˙)$⁠. Consequently, NI, namely, dynamic passive systems, can be considered complementary to passive systems, i.e., to positive real systems. Contrary to passive systems, the dissipativity analysis of NI systems has not been thoroughly explored in Ref. [17]. References [27–30] primarily focus on the dissipative properties of linear NI systems, while nonlinear systems are considered in Refs. [26] and [31]. The frequency domain properties of NI-TDS are studied in Refs. [32] and [33]. To the best of our knowledge, the dissipation-based analysis of the NI characteristics of TDS has not been studied yet.

In this paper, we examine the NI (dynamic passivity) property of linear TDS of neutral and retarded type based on dissipativity framework and propose novel sufficient conditions. As an application of the studied problem, we consider the vibration suppression of a flexible cantilever beam which is known to be an NI system [19]. We present a new time-delay controller of retarded type used within positive feedback, which is applied to the collocated piezoelectric sensor (PES) and piezoelectric actuator (PEA) pair to control the beam. The vibration suppression performance of the proposed controller is shown to be better via simulations as compared to a state-of-the-art work [34]. The proposed controller is parametrized with the formulated norm-based optimization problem solved by the standard genetic algorithms and the recent tds-control toolbox [35]. The parametrized controller is shown to be NI by the derived NI condition for linear TDS. Then, a complete stability analysis of the system with the proposed controller is performed by applying a dissipativity-based stability theorem for interconnected NI systems.

The delayed feedback control is used in different concepts to reduce the vibration of flexible cantilever beams with PES–PEA pairs. The cantilever beam is a flexible system with distributed parameters and therefore naturally modeled by a partial differential equation (PDE) [36,37], which results in infinite modes. It is often to model such a system by a transfer function with a finite number of modes utilizing an appropriate truncation method [38]. The stability analysis and control design are then performed on this approximate transfer function model with a small number of modes [39,40]. In Ref. [41], the authors apply an inverse shaper based controller to suppress the vibratory modes of the beam using collocated PES–PEA pairs. This controller design and performance analysis are conducted through the finite mode system model simulation results, where the closed-loop stability is not guaranteed theoretically but shown via simulation results. In Ref. [42], the delayed feedback combined with a modal filter is applied to suppress the vibratory multimodes where the stability analysis is fulfilled via Nyquist technique based on a nonminimum phase beam model with a small number of modes. An important concept using delays for control is the delayed resonator, which has been used for both vibration suppression and energy harvesting for flexible beams with PES–PEA pairs [43,44]. Therein, the resonator is designed over a truncated finite mode system model which is also used for stability analysis applying the D-subdivision method to the transcendental characteristic polynomial. A nonlinear flexible beam model is considered in Refs. [45] and [46], and the delayed feedback using a multiscale method is applied for control where the closed-loop system stability is ensured through the eigenvalues of the linearized finite-mode model. As compared to the mentioned literature, we address the stability problem of the considered TDS differently over the flexible system model with infinite modes. The stability conditions are derived for the infinite-mode system model and depend mainly on the dissipative properties of the time-delayed controller. The analysis over the finite mode system model is avoided for the accuracy.

The paper is organized as follows. In Sec. 2, the existing definitions related to linear time-invariant (LTI) dissipative and dynamic passive systems are provided along with the stability theorem for dynamic passive systems. Section 3 presents the problem description, including the system model and controller structure. In Sec. 4, we introduce a new dissipativity approach for TDS of neutral and retarded type and a new time-delay controller for vibration suppression. In Sec. 5, we present a simple motivational example to show advantages of the proposed time-delay controller and provide a case study that presents the usage of the proposed stability conditions on NI-TDS to the active vibration control of a flexible beam via simulations. The paper concludes in Sec. 6.

Notation: $Rn$ denotes the n dimensional Euclidean space, and the set of non-negative real numbers is represented as $R+$⁠. $Rm×n$ denotes the set of real matrices of dimension $m×n$⁠. $P>0$ is referred to as symmetric and positive definite matrix for $P∈Rn×n$⁠. $C1$ stands for all differentiable class of functions whose derivative is continuous. The scalar valued storage function and the supply rate function are denoted by $S(·)$ and $s(·)$⁠, respectively.

where $x(t)∈Rn$ is the system state, $u(t)∈Rm$ is the system input, and $y(t)∈Rm$ is the system output. The definitions of dissipativity and dynamic passivity are given below.

for all input signals$u(t)$, where$x(t)$exists for all$t∈[0,T]$⁠.

which is called as differential dissipation inequality [2]. The choice of supply rate holds significant importance in characterization of the dissipation properties of systems. There are different options for selecting the supply rate. In particular, the system given by Eq. (1) is passive if it is dissipative for supply rate $s(u(t),y(t))=u(t)Ty(t)$⁠.

for all input signals$u(t)$, where$s(u(t),y˙(t))=u(t)Ty˙(t)$is the dynamical supply rate function and$x(t)$exists for all$t∈[0,T]$⁠.

Remark 1. Note that the NI systems are the dual of the positive real (passive) systems [26]. Also instead of the term dynamic passive, the term NI is used throughout the paper.

An LMI-based condition for NI-LTI systems is given in the following lemma.

A more strict dissipation-based characterization of NI systems is presented in the following definition.

is satisfied for all$t>0$, then$limt→∞u(t)=0$⁠.

A dissipation-based analysis of the dynamic passivity of nonlinear systems along with a corresponding stability theorem is proposed in Ref. [26]. Since we are concerned only with linear systems in this paper, an equivalent version of this stability theorem for linear systems is represented in the following.

Theorem 1. [26] Consider two linear dynamical systems G and C each represented byEq.(1)and interconnected as in Fig. 1. Assume that G and C are NI and SNI, respectively. If the DC gain of C is positive and the open-loop DC gain of the system in Fig. 1 lies within the interval$[−1,1]$, then the positive feedback interconnection of G and C is asymptotically stable.

The stability of the positive feedback interconnection of the NI and SNI systems is defined in terms of simple and trivial gain conditions above. This paper considers the problem of analyzing the NI characteristics of linear TDS. Moreover, we provide conditions similar to Theorem 1 for the stability of positive feedback interconnection of infinite-dimensional SNI systems, in particular, flexible Euler–Bernoulli beams, with an NI-TDS which is the proposed delayed controller in this paper.

where $x(t)∈Rn$⁠, $u(t)∈Rm$⁠, $z(t)∈Rm$⁠, $A0, A1∈Rn×n$⁠, $B1∈Rn×m$⁠, $C∈Rm×n$⁠, and $g, h∈R+$ are constant delays. As for passive systems, the NI property holds for dynamical systems only with the same number of inputs and outputs [2]. Here, we aim to establish a criterion for the system (8) to be NI independent of the neutral-type delay g, i.e., being NI for all g values. To achieve this, the system first needs to be robustly stable against small variations in g, which implies that the absolute values of all the eigenvalues of F are strictly less than one, i.e., F is a Schur–Cohn stable matrix; see Refs. [9] and [48]. Thus, the following assumption holds for the rest of the paper.

Assumption 1. All the eigenvalues of F are inside the unit circle.

for all $x(0)=x0$⁠. Following this approach, we conclude that Definitions 2 and 3 are valid also for the delay system (8) to be NI or SNI. On the other hand, dissipative properties change regarding the delay type or the occurrence of delay meaning which signal is delayed in the system. For instance, it is shown in Refs. [49] and [50] that the systems with input and/or output delays cannot be passive or dynamically passive. The delays in Eq. (8) occur at the state vector and its derivative, which is one of our motivations to investigate the NI properties of the system.

Based on Definition 2 for NI systems and Remark 2, the following NI lemma is proposed for the systems represented by Eq. (8) utilizing an appropriate LKF as the storage function.

then the system(8)under Assumption 1 is NI for all$g≥0$⁠.

Note that LMI (10) is affine in $h2$⁠. Thus, the upper bound of the delay h where the system (8) is NI can be computed using an iterative line search on h keeping Eq. (10) feasible [54]. The computational complexity of LMIs is commonly given in terms of the number of decision variables and order. The order and the number of decision variables of LMI (10) is 5n and $3n2+3n$⁠, respectively, which is one of the least among other studies on TDS; see, e.g., Table 1 in Ref. [55] for the LMI analyses of key TDS studies. Indeed, the computational complexity rises as the system dimension n increases. We demonstrate the number of decision variables and the consumed CPU time to solve LMI (10) in Sec. 6 for the proposed time-delay controller with increasing dimensions.

The other objective of this study is to demonstrate the usage and verification of the proposed approach above for LTI–TDS through a suitable case study. For this purpose, we first present a flexible beam system in Sec. 4 and then propose a time-delay controller to suppress the vibrations on the beam.

In this section, we present the flexible beam system fixed at one end and released at the other so that it can move freely at one end as shown in Fig. 2. A collocated PES–PEA pair is attached to the beam for vibration suppression. The Euler–Bernoulli beam model is used to characterize the beam. The amplitude of oscillation, denoted as $y(r,t)$ along the $y‐axis$ direction at any point r on the beam, is measured by the induced voltage $V0(t)$ of the PES. The applied voltage $Vi(t)$ by the PEA generates a bending moment $M(r,t)$ to counteract and mitigate the vibration.

where $ϕi(r)$ is the $ith$ mode shape, $ωsi$ is the $ith$ natural frequency, $ζsi$ is the $ith$ damping ratio, L is the length of the beam, $Ca$ and $Cs$ are the gain values calculated with the beam parameters, and $r1$ and $r2$ are the starting and ending positions of the piezoelectric material on the beam, respectively. An important feature of the infinite dimensional beam model, which is utilized in this study, is given in the following property.

Property 1. Second-order single-input single-output systems characterized by positive parameters and their infinite sums are shown to be SNI inRef. [19]. Consequently, the beam model G(s) given byEq.(21)is SNI since the parameter$Ca$is negative whereas all the other parameters are positive; seeRef. [56].

where usually the high frequency modes are neglected. The constant feed-through term D in Eq. (22) is to correct the undesired fact of the truncation, namely, the alteration of the actual zeros of Eq. (21) with neglecting the high frequency modes. The term D is determined via various approaches such as in Refs. [37] and [38] to improve the accuracy of the truncated model.

Various control methods have been developed for vibration suppression of smart flexible systems. PPF control is a pioneering method designed specifically for flexible systems with collocated PES–PEA pairs. The block diagram of PPF is given in Fig. 1, where $G(s)$ and $C(s)$ represent the flexible system model and the PPF controller, respectively. Moreover, $w(t)$ and $y(t)$ represent the disturbance signal and the transverse vibration (displacement) of the flexible system, respectively.

and named as the multimode modified positive position feedback (MMPPF). This extension provides a performance enhancement especially for the steady-state disturbances acting on the system, see Refs. [34] and [58]. Note that N in Eq. (23) is the number of targeted system modes that is commonly chosen $N≤M$ to reduce the computational load in design. Nevertheless, the increment in the number of resonant modes to be damped corresponds to an increase in the number of both the second-order and the first-order filters in Eq. (23), which means an increase in the order of MMPPF controller.

To avoid the mentioned rise in the controller order and to achieve better vibration suppression for the considered flexible structure, we take advantage of the time delays as control parameters. Namely, we propose a time-delay controller in Sec. 5 where also an optimization-based parametrization is presented. Furthermore, the stability of the time-delayed closed-loop system is analyzed via the proposed NI lemma in Sec. 3.

used within positive position feedback for vibration suppression on flexible systems like the considered beam in Sec. 4. We refer to Eq. (24) as the PPFR controller, where R stands for retarded, and apply it to the flexible system as shown in Fig. 1 to suppress vibrations. Compared to the MMPPF in Eq. (23), we replace the multiple first-order low-pass filters interconnected in parallel with a single time-delayed low-pass filter in Eq. (24). Note that, to suppress N modes of the system, the MMPPF requires N low-pass filters in parallel increasing the controller order, whereas the PPFR uses a single filter regardless of the number of targeted modes to be suppressed. Thus, PPFR has an advantage of having lower order in implementation for practical applications when $N>1$⁠.

In the following motivational example, we explain and demonstrate the advantages of the proposed PPFR controller compared to the MMPPF controller for vibration suppression problem over a simple vibratory system model.

where $k1−k2,ζ1−ζ2$⁠, and $ω1−ω2$ are the gains, damping ratios, and natural frequencies for the first and second vibration modes, respectively, which are given in Table 1.

In order to suppress the vibration of the first resonant mode of the system above, we use PPFR (24) and MMPPF (23) controllers for $N=1$ in a positive feedback interconnection, as shown in Fig. 1. The damping and frequency parameters of both controllers are chosen as $ζc1=ζ1$⁠, $ζc2=ζ2$⁠, $ωc1=ω1$⁠, and $ωc2=ω2$ as explained in Ref. [34]4A. The rest of the both controllers' parameters are determined as in Table 2 using an $H∞$-norm based optimization method, which is detailed in Sec. 5.3. Note that for a fair comparison of the controller performances, the same optimization procedure with the same cost functions is applied to parametrize the both controllers. The frequency responses of the two-mode system model and closed-loop system with the PPFR and MMPPF controllers are illustrated in Fig. 3, and the followings are observed:

• PPFR controller provides lower magnitude at the targeted first mode frequency which means better vibration suppression compared to MMPPF.

• PPFR controller provides also better vibration suppression for the second mode which is not considered as a design objective, compared to MMPPF.

The reason of observed advantages of PPFR above can be explained by the frequency responses of the controllers in Fig. 4. PPFR controller has periodic peaks in frequency response due to the periodic characteristic of the delay term $e−hs$⁠, which results in magnitudes higher than MMPPF. Consequently, better performance in suppressing vibrations for both the targeted and nontargeted modes in design is achieved. Another fact seen from the closed-loop frequency responses in Fig. 3 is that the PPFR controlled system has lower magnitudes for low frequencies which is a desired behavior to avoid spillover effect of the frequencies away from the targeted resonance frequency [34,58].

The closed-loop system with PPFR has a retarded distribution in poles which depart from imaginary axis with increasing moduli. Rightmost pole within a spectral distribution is often the dominant pole of TDS which characterizes stability and dynamic response. Regarding that, the rightmost pole's damping ratio is defined as the equivalent damping factor where a higher equivalent damping factor corresponds to a better-damped system. As suggested in Ref. [64], we calculate the equivalent damping factors of the dominant poles of the closed-loop systems as given in Table 3 where the PPFR controlled system has a higher equivalent damping. Thus, PPFR damps the flexible system better than MMPPF.

Stability analysis of the infinite-mode model for cantilever beam (21) controlled with a time-delay controller as in Fig. 1 constitutes a challenging problem. In Refs. [43–46] where delays are utilized as control parameters, stability is analyzed considering the truncated system model (22) in closed-loop architecture which may cause inaccuracy. It is because of the approximation of high-frequency modes by the constant term D in Eq. (22). To overcome this fact, dissipation-based stability analysis stands out as a particularly suitable approach due to SNI property of the infinite-mode beam model (21). In Ref. [65], the passivity definitions for nonlinear systems without delay and linear time-delay systems are naturally extended to retarded types of nonlinear time-delay systems. The stability analysis of the negative feedback interconnection of passive nonlinear time-delay systems is performed through an inherent generalization of the dissipation-based stability theorem for nonlinear systems. For linear NI systems with rational transfer functions, the positive feedback interconnection is guaranteed under a straightforward open-loop gain condition. A Lyapunov-based stability condition equivalent to the open-loop DC gain criterion is proposed, and the stability condition for the positive feedback interconnection of NI and SNI systems is provided [66]. In Ref. [26], the definitions of linear NI systems are generalized to nonlinear dynamic passive systems. Using Lyapunov stability theory and certain gain conditions which can be equivalently formulated as the DC gain criterion for linear systems, stability conditions for nonlinear dynamic passive systems are derived. Following a similar approach, we demonstrate that an infinite-dimensional linear system interconnected with a time-delay system via positive feedback is stable under specific conditions (cf. Corollary 1).

Note that Eq. (26) is in the form of the system (8) with $F=0$⁠. Therefore, Lemma 2 formulated in terms of the LMI (10) can be employed to analyze that the PPFR controller is NI or not. Then, we provide the following corollary for the stability of the considered system with the proposed delayed control.

where$K∈R+$is the upper limit of the DC gain of the flexible beam(21). Then, the system is asymptotically stable if the LMI(10)with the matrices(27)is satisfied for any$P>0, R>0, U>0, S>0, P2, P3$, and$F=0$⁠.

In this subsection, we present an optimization-based approach to parametrize the proposed controller PPFR for efficient suppression of the resonant modes. Also, a sensitivity function is presented to evaluate the controller robustness against the deviation of the natural frequency of flexible beam.

where $bi$ is the predefined weighting constant for $ith$ resonant mode, and $Hi(s)$ is the predefined weighting bandpass filter around $ith$ system resonant frequency $ωsi$⁠. The term $b0 T(0)$⁠, representing the weighted DC gain, is also considered to mitigate deterioration around the DC frequency and enhance the controller's performance at lower frequencies. The closed-loop system transfer function $T(s)$ in Eq. (36) includes internal time delays. Due to the nonconvex nature of the optimization problem, matlabgeneticalgorithm (ga) is used to search parameters, and the weighted infinite norm of the cost function (36) is calculated via tds-control toolbox [35].

The flowchart in Fig. 5 shows the integrated use of the ga and tds-control toolboxes. The initial value of the cost function is calculated for the initial parameters determined by ga. Standard genetic operations are then used to generate new parameters for optimization problem based on the best parameters determined in the previous iteration. These iterations are continued until the maximum number of iteration is reached, then the search is terminated. The parameters that minimize the cost function over all iterations are the solution to the optimization problem.

where $Δω$ is the uncertainty of $ith$ system resonant frequency utilized to analyze the controller robustness to uncertainties in the system's resonant frequency [67,68]. The smaller the slope of $κi$ means the lower sensitivity of the controlled system to deviations in the resonance frequency to be suppressed.

In Sec. 6, the proposed $H∞$-norm based design is performed for PPFR and MMPPF controllers, where MMPPF controller as a state-of-the-art work is chosen for comparison of the performance with the PPFR controller.

The first model is with low number of modes where $M=4$⁠, which is used for the controller parametrization to reduce the computational burden. The second one is with higher number of modes where $M=20$ used for simulations to verify the stability ensured by Corollary 1 and the controller performance. The model with higher number of modes is to represent the infinite system model more accurately.

We first consider the beam model (22) derived from the parameters given in Table 4 with $M=4$ resonant modes. The natural frequencies and damping ratios for each mode are provided in Table 5. We target two of these modes, i.e., choose $N=2$ in the controllers PPFR (24) and MMPPF (23), and parametrize them by solving optimization problem (36) following the procedure in Fig. 5. Note that the parameters to be optimized for MMPPF in Eq. (36) are $kci$ and $γci$⁠. The obtained controller parameters are given in Table 6. Then, the parametrized controller is shown to be NI via using the proposed Lemma 2.