New results on passivity-based ${\mathcal{H}}_{\infty }$ control for networked cascade control systems with application to power plant boiler–turbine system

Abstract

This paper is concerned with the problem of passivity-based ${\mathcal{H}}_{\infty }$ controller design for a class of networked cascade control systems (NCCSs) with random packet dropouts. The NCCS under consideration is modeled by using state feedback controllers and the network-induced imperfections like packet dropouts and time-varying delays. The model is defined with a stochastic packet-dropout case by using the Bernoulli distributed white sequence with time-varying probability measures. The probability-dependent conditions for stabilization of NCCSs are established to guarantee the resulting closed-loop system to be stochastically stable and achieve a prescribed mixed ${\mathcal{H}}_{\infty }$ and passivity performance. The Lyapunov stability theory and linear matrix inequality (LMI) approach are used to derive criteria for the existence of the state feedback controllers. The proposed probability-dependent gain scheduled controller can be designed by solving the convex optimization problem by means of a set of LMIs, which can be easily solved by using some standard numerical packages. Finally, a practical application is presented to illustrate the effectiveness and potential of the proposed results.

Introduction

Networked control systems (NCSs) are feedback control systems, in which the control loops are closed through a real-time network. In recent years, NCSs are becoming progressively more significant and have many outstanding advantages compared to traditional point-to-point control systems  [1], [2], [3]. Here, it should be noted that the network-induced delay and packet loss are the main factors which sources the performance weakening and potential system instability of NCSs. The time delays occur when the data are exchanged between the connected devices to a shared medium. Moreover, during the transmission, the data packets can be lost in the network itself, in literature there are some interesting results found in  [4], [5], [6]. In NCSs, the data packets sent through the communication channels might encounter failures in a random way due to the sharing of a common network medium. However, in literature, there are two main approaches followed to study the packet dropouts in NCSs: One is to model NCSs with packet dropouts as switched systems  [7], [8] and the other is the usage of Markov chain and independent Bernoulli random process  [9], [10]. It is noted that the model studied in a deterministic framework in the first approach, while the NCSs are transformed into a system with stochastic parameters which has known expectation and variance in the second approach.

On the other hand, cascade control systems in which the control loops are closed through a real-time network are called NCCSs. Cascade control was originated by Franks and Worley  [11]. After that, it became a useful approach to develop the performance of the control system; particularly, it has the ability to rapidly reduce the disturbances in the inner loop. The NCCSs have been extensively used in practical industrial process due to their excellent control performances and also have the advantages similar to the NCSs model such as cost reduction, maintenance and so on. Further, like NCSs, the NCCSs also has some unavoidable problems such as network-induced delays, data packet dropout, and so on. Transmission topology and traffic cause transmission delay which can degrade the control performance of NCCSs  [12]. Therefore, it is of useful to study that how to handle these network induced imperfections in NCCSs has attracted much research interest.

In the existing literature, packet dropouts and time delays have received substantial interests from many researchers and some remarkable results have been proposed, for example, see  [13], [14], [15] and the references therein. Moreover, many interesting approaches has been proposed and applied for the industrial and real world systems see  [16], [17] and the references therein. Exponential stability problem for a class of networked control systems with time delays and packet dropouts has been proposed in  [18]. The problem of input-to-state stability for networked control systems with random delays and packet dropouts appearing simultaneously in both feedback and forward channels has been investigated in  [10]. Millan et al.  [19] proposed a novel distributed estimation technique for linear time-invariant systems with network-induced delays and packet dropouts. A modeling method for networked control systems with random time delays and communication constraints based on quasi-TS fuzzy models has been presented in  [20]. In  [21], the authors studied the event-triggered communication scheme and an ${\mathcal{H}}_{\infty }$ control co-design method for networked control systems with communication delay and packet loss. The problem of predictive output feedback control for networked control systems with random communication delays has been investigated in  [22].

But, most of the papers in the existing literature are concerned about using single-loop control systems or multi-input–multi-output control systems. It should be noted that only few papers have studied the multi-loop control systems, particularly on cascade control systems by considering various parameters which degrades the performance of NCCSs. For example, Guo et al.  [23] proposed a cascade control strategy for temperature control of air handling units. The problem of stabilization and ${\mathcal{H}}_{\infty }$ control problems for a class of singular networked cascade control systems with state delay and disturbance is studied in  [24]. Huang et al.  [25] studied the ${\mathcal{H}}_{\infty }$ state feedback control for a class of networked cascade control systems with uncertain delay. Luo et al.  [26] presented a robust cascaded control strategy to underwater robot thrust. Predictive compensation for networked cascade control systems with uncertainties has been presented in  [27].

In NCCSs, the primary sensor, the secondary sensor and the actuator are the intelligent field devices. The primary and secondary controllers can be placed in any one of the three devices or in any other independent intelligent node connected to the same network  [27], [28]. So, as combinations, there may be at least eleven different NCCSs configurations which can be modeled theoretically. However, based on the real world industrial applications, four common configurations of NCCSs are explicitly proposed and often employed. In  [28], the authors studied the advantages and disadvantages of each of these four classic configurations of NCCSs model and comparison study over each other also presented. As mentioned earlier, NCCSs have issues such as the network-induced delays, single-packet transmission, multiple-packet transmission and data packet dropout due to the existence of real-time network in the feedback control loop. This makes analysis and design of an NCCSs more complex  [20]. But, a realistic NCCSs design must take all of them into account in the analysis and modeling.

In  [24], [25], the authors studied the second configuration of NCCSs which is also known as Type II NCCSs (as mentioned in  [28]). In this paper, we consider the Type I NCCSs as presented in Fig. 1. Also, it should be noticed that there are no existing method works on the problem of stabilization for NCCSs when the packet loss happens in a random way. Moreover, in existing literature, the random packet dropout conditions are defined by using a Bernoulli distributed stochastic variable with a known constant expectation and variance. Recently, Wei et al.  [29] proposed a probability-dependent gain-scheduled control for a discrete-time stochastic systems by assuming that the sector nonlinearities occur according to a Bernoulli distribution with measurable time-varying probability. In this paper, we utilized some new techniques for the case of random data packet dropouts.

In literature, two important control schemes have been studied, namely ${\mathcal{H}}_{\infty }$ control and passivity based control. The ${\mathcal{H}}_{\infty }$ control problem is used to design an appropriate controller such that the energy-to-energy gain from exogenous disturbances may be restricted less than a prescribed level. It is also noted that ${\mathcal{H}}_{\infty }$ setting is robust with respect to the disturbances because they do not use any statistical information. The design problem of ${\mathcal{H}}_{\infty }$ controller is designed with the minimum disturbance attenuation level by means of the following energy function described by $J_1(z\left(k\right),v\left(k\right))=z^T\left(k\right)z\left(k\right)-{\gamma }^2{v}^T\left(k\right)v\left(k\right)$, where $v\left(k\right)$ is the disturbance input, $z\left(k\right)$ is the output and $\gamma$ is the nonnegative constant. On the other hand, the passivity framework has its origins in the circuit theory and plays a vital role in electrical networks as well as in control systems. The problem of passivity-based controller can guarantee that the system is strictly-output passive with the energy function $J_2(z\left(k\right),v\left(k\right))=-2z^T\left(k\right)v\left(k\right)-\gamma v^T\left(k\right)v\left(k\right)$. As it is well known that the individual ${\mathcal{H}}_{\infty }$ and passivity-based controller design have readily computable results and more useful in practice, see  [30], [31] and the references therein. So, combining these two objective (energy) functions can systematically make the design tradeoffs between these design procedures more useful. In this paper, we make an attempt to capture the benefits of the both ${\mathcal{H}}_{\infty }$ controller and passivity-based controller for NCCSs. For this, we define the following energy function:

$$J(z\left(k\right),v\left(k\right))={\gamma }^{-1}\delta z^T\left(k\right)z\left(k\right)-2(1-\delta )z^T\left(k\right)v\left(k\right)-\gamma v^T\left(k\right)v\left(k\right)\text{,}$$

where $J(\cdot )$ be the supply rate of system and is a real-valued functional in terms of input $v\left(k\right)$ and output $z\left(k\right)$ and $\delta \in [0,1]$ represents a weighting parameter that defines the trade-off between ${\mathcal{H}}_{\infty }$ and passivity performances.

Based on the above discussions, in this paper, the probability-dependent conditions are obtained for the passivity-based ${\mathcal{H}}_{\infty }$ control design. In many cases, performance of the simple control structures will not be satisfactory. In this paper, we propose a probability-dependent gain-scheduled cascade control strategy for NCCSs. The main aim of this work is to design the gain-scheduled control with a disturbance attenuation level such that the resulting closed loop system is stochastically stable. In the proposed NCCS model, the accessible status of the actuator is governed by a Bernoulli distributed white sequence, that is, the communication scheduling with a stochastic distribution scheme using the time-varying probability values for the random variables. This is an advantage here because, in most cases, the fixed probability values for the random variables are used in literature. Also, it should be noted that the controller is mostly designed based on either ${\mathcal{H}}_{\infty }$ control or passivity based control. The objective of this paper is to overcome this issue by implementing a common control strategy for using these two controller design. So, the proposed results are more general one because the existence of controller gains are expressed with a trade-off parameter between the ${\mathcal{H}}_{\infty }$ and passivity performance criterions. Sufficient probability-dependent conditions are derived for the existence of time-varying feedback controller to achieve a prescribed mixed ${\mathcal{H}}_{\infty }$ and passivity performance level. The algorithms for the proposed cascade control strategy are constructed by means of two control loops: an inner loop with fast dynamic to eliminate the input disturbances and an outer loop to regulate the output performance.

This paper is organized as follows. The configuration of NCCSs is proposed and the problem is formulated in the next section. In Section  3, main results for the controller design in terms of probability-dependent conditions are presented. In Section  4, a power plant boiler–turbine system as a practical example is considered to show that how the network-induced delays and data packet dropouts affect the performance of the system. Finally, the paper is concluded in Section  5.

Notations: Throughout this paper, the superscripts “$T$” and “$(-1)$” stand for matrix transposition and matrix inverse respectively. ${\mathbb{R}}^n$ denotes the $n$-dimensional Euclidean space. ${\mathbb{R}}^{n\times n}$ denotes the set of all $n\times n$ real matrices. $P>0$ (respectively $P<0$) means that $P$ is positive definite (respectively, negative definite). $I$ and 0 represent the identity matrix and the zero matrix with compatible dimension. In symmetric block matrices or long matrix expressions, we use an asterisk ($\ast$) to represent a term that is induced by symmetry. $\mathrm{diag}\{.\}$ stands for a block-diagonal matrix. $\Vert \cdot \Vert$ refers to the Euclidean vector norm. $\mathbb{E}[\cdot ]$ stands for the mathematical expectation operator with respect to the given probability measure $\mathcal{P}$.