Stability Analysis of Sampled-data Control System and Its Application to Electric Power Market *

: The stability of sampled-data systems is investigated using a new type of Lyapunov functional. The interval from the sampling point k t to 1 k t + is assumed to be a sampling interval. By fully utilizing the characteristic information on the whole sampling interval, a new two-sided closed-loop Lyapunov functional is proposed, which utilizes the information on both the intervals from the sampling point t to k t and from t to 1 k t + . Based on the two-sided closed-loop Lyapunov functional and modified free-matrix-based inequality, a less conservative stability criterion is derived for a sampled-data control system, and three numerical examples are provided to verify the effectiveness and reduced conservativeness of the proposed method. Furthermore, the proposed method is applied to solve the stability problem of electric power markets, and the practical significance of reducing the conservativeness is discussed.


Introduction
With the rapid development of communication and computer technology, sampled-data control has become a core technology in the field of automatic control. Since sampled-data control only needs the state information of a system at the sampling time, the transmission of information is greatly reduced, and the control efficiency improved. Therefore, sampled-data control has been widely used in digital systems, network systems, and other fields, and has received extensive attention from many researchers [1][2][3][4] . It is known that the sampling period is an important indicator for measuring the conservativeness of the stability criterion in a sampled-data control system. The larger the sampling period for a stable system, the lower the conservativeness, and the lower the requirements for communication rate, capacity, and bandwidth of the system. This indicates that the larger the allowed sampling period, the lower the requirements for the system's hardware, and sampling performance, and in turn, lower the cost of sampleddata control equipment. Therefore, it is of great scientific and practical significance to investigate the stability of sampled-data control systems and derive the conditions for low conservativeness, thereby obtaining the largest sampling period.
A sampled-data control system is a hybrid system with both continuous and discrete signals, which cannot be directly analyzed and designed using the theory of continuous systems. Currently, the methods of analyzing and designing sampled-data control systems mainly include the following methods: discretiza-tion, pulse control, and input time-delay. In Refs. [5][6], the discretization method was used to analyze the stability of sampled-data control systems. However, the discretization method had significant limitations in dealing with the systems with uncertainties or variable periods. In Refs. [7][8], the stability problem of the variable-period sampled-data control system was discussed using the pulse control method. This method transformed the sampled-data control system into a pulse system and used pulse control theory to analyze the transformed system. In Ref. [9], the input time-delay method was proposed, which transformed the input signal of the sampled-data controller into a time-delay signal input by the system. This indicates that the state quantity at the time of sampling was regarded as the system state with a time delay, so that the sampled-data control system was modeled as a linear time-varying time-delay system. The Lyapunov functional and integral inequality techniques were used to analyze and design the transformed system. This method overcomes the difficulty of dealing with uncertain and variable-period systems faced by the discretization method.
Currently, the input time-delay method is widely used to deal with various types of sampled-data control problems. For example, the stability and stabilization of sampled-data control systems with data lost was discussed in Ref. [10]. In Ref. [11], the sampling synchronization control problem of neural networks was analyzed. In Ref. [12], the H ∞ output tracking control problem of a T-S fuzzy network control system was researched. In early studies, the sawtooth structure features of a sampled-data control system were not fully considered, and therefore the conclusions obtained were still largely conservative. To overcome this disadvantage, some researchers have proposed new types of Lyapunov functional, based on the input time-delay method, such as time-dependent Lyapunov functional [13] , time-dependent discontinuous Lyapunov functional [14][15] and closed-loop function methods [16][17][18] . These methods lower conservativeness as they considered the sawtooth structure features. Recently, a time-dependent discontinuous Lyapunov functional by modified free-matrix integral inequality [19][20][21] was established in Ref. [19], and the stability problem of sampled-data control systems was discussed, concluding with less conservativeness. However, only the information of the interval between the current sampling point t and a previous sampling point k t was considered, and the information of the interval between the current sampling point t and a future sampling point 1 k t + was ignored. Additionally, in Ref. [22], the state information of a future sampling point was integrated into the closed-loop function for the first time, and a new closed-loop function was proposed that effectively reduced the conservativeness of the stable results of the sampled-data control system. However, this study still ignored a large amount of future sampling information, and needs to be further improved. Simultaneously, it also had some conservativeness to the processing of functional derivative. Therefore, this paper aims to construct a new Lyapunov functional, for reducing the conservativeness by effectively mining future sampling information.
The theoretical research on sampled-data control systems has attracted extensive attention [1-6, 10-11, 13-19] . In the above research, the conservativeness of the obtained criterion was usually compared by simple numerical examples, which is difficult to use to explain the practical significance of reducing conservativeness effectively. If an actual system is used to demonstrate the conservativeness of the obtained criterion, it will be more helpful to explain the practical significance of reducing conservativeness. Therefore, this study takes the electric power market as a practical example to demonstrate the significance of reducing conservativeness, with the aim of maintaining the supply and demand balance of electric power in the grid. Considering the market clearing time (i.e., the time of updating power price), the electric power market is modeled as a linear sampleddata control system, and then the conservativeness of the obtained criterion is discussed.
For variable-period sampled-data control systems, this study proposes a new stability analysis method and applies it to the electric power market. Compared with the existing research, the main contributions of this study are summarized as follows: (1) The information of the whole sampling interval is fully utilized. Further, a two-sided closedloop Lyapunov functional is proposed, which not only considers the information of the intervals from the current state ( ) x t to the previous state ( ) k t t + . Additionally, the cross-information of the states mentioned, is incorporated in the two-sided closed-loop Lyapunov functional. By using the two-sided closed-loop Lyapunov functional and the modified free-matrix inequality, the stability criterion of the sampled-data control system is derived. The advantages and low conservativeness of the proposed method are verified by two examples.
(2) The proposed method is employed to solve the stability problem of the electric power market. This study not only investigates the impact of clearing time on power system stability but also discusses the practical significance of reducing the conservativeness of the stability criterion.
In this paper, n R and n m × R represent the n-dimensional vector space and the n×m matrix space of the real number field, respectively. 1 − Q and T Q represent the inverse and transpose operations of a matrix, respectively. "0" and "I" represent the zero matrix and unit matrix of the appropriate dimension, respectively. 0 > R represents a positive definite symmetric matrix. T sym{ } = + M M M represent the sum of matrix X and transposed matrix X. "*" represents the symmetric terms in a matrix.

Problem formulation
The following linear control system is considered Here, ( ) n x t ∈ R is the system state and , n ∈ A R n I ∈ B R are known as real constant matrices. The system control input u(t) satisfies the following form is the control gain matrix, ( ) k t k ∈ N is the sequence of sampling time, and the control signal between adjacent sampling points is maintained by a zero-order hold (ZOH). The variable sampling time k t satisfies where k h is a certain sampling period and has different values for different k values, which reflects the variable periodicity of sampling, and 2 h is the maximum sampling period.
Bringing formula (2) into formula (1), a closedloop control system is obtained, which is given by where .
c I = B B K For a given control gain matrix K , this study aims to derive the condition to guarantee the stability of the system represented by formula (4), thereby obtaining the largest sampling period.
To analyze the stability of the sampling system shown in formula (4), the following lemmas are introduced.
Lemma 1 [16] : then, the following two conditions are equivalent , the Lyapunov function is strictly a negative definite.
Hence, for ( , , ) Furthermore, the sampled-data control system shown in formula (4) is asymptotically stable if one of the above two conditions is met.
Lemma 2 [21] : For a given positive definite matrix

Two-sided looped Lyapunov functional
To simplify the process of deriving the stability condition for the system shown in formula (4), the following symbols are defined For the stability of the sampled-data control system shown in formula (4), the following two-sided closed-loop Lyapunov functional is constructed.
is an augmented closed-loop function expressed as The definition of each matrix in the two-sided closed-loop Lyapunov functional in formula (10) can be found in Lemma 1. It is noted that is the boundary condition that satisfies formula (7) in Theorem 1, that is, [16], the function that satisfies such conditions is called a closed-loop function.
Note 1: Refs. [15][16][17][18][19] only consider the sampling characteristics between the current sampling point t and the previous sampling point k t (the one-sided sampling characteristic). By contrast, the two-sided closed-loop Lyapunov functional in formula (10) considers the sampling characteristics of the other side by adding a new term 4 ) , that is, the sampling characteristics between the current sampling point t and the future sampling point 1 k t + . Additionally, compared with Refs. [15][16][17][18][19], the functional in formula (10) incorporates more information about the sampled-data control system in formula (4)  α as well as their respective states. However, this information was not considered in Refs. [15][16][17][18][19]. The stability conditions derived from the functional in formula (10) are, thus, less conservative than those in Refs. [15][16][17][18][19].
satisfies the condition of closed-loop function and considers the two-sided sampling characteristics of the system, and 0 ( ) V t is a standard Lyapunov functional, then ( ) V t is called a two-sided closed-loop Lyapunov functional. This function can be applied not only to sampled-data control systems, but also to the sampled-data control of other complex dynamic systems such as fuzzy control systems [12,23] , neural network systems [11,21,24] , and multi-agent systems [25] .

Stability analysis of sampled-data control system
In this section, the proposed closed-loop Lyapunov functional in formula (10) is used to derive the stability criterion of the sampled-data control system in formula (4), and the effectiveness and low conservativeness of the proposed method are verified by two examples.

Stability criteria
Theorem 1: For a given scalar there is a symmetric matrix of the appropriate dimension, , , M Z Z and a matrix and Here, { T   T  T  T  1  1  2  1  4  2  3  1 2 5   T  T  T  1  1  3 4 5  6  7  1  9  2 10   T  T  T  1 1 1  3 3   The sampled-data control system in formula (4) is then considered asymptotically stable.
Proof The two-sided closed-loop Lyapunov functional ( ) V t is derived along with the sampled-data control system in formula (4) with respect to time t. .

( ) C t V
is processed by the modified free-matrix integral inequality in Lemma 1. Then Similarly, by processing the integral term of the functional derivative The integral operation is performed on the time interval [ ) on the two sides of the sampled-data control system in formula (4).
Thus, there is a matrix 1 22) [ ] Additionally, since ) 0 (t Γ = ξ , there is a matrix N of suitable dimension that satisfies Summarizing formulas (10) to (24), are equivalent to linear matrix inequalities in formulas (11) and (12), respectively. If the conditions of formulas (11) and (12) are satisfied, then, according to Lemma 1, the sampleddata control system in formula (4) is asymptotically stable, and the proof is completed.
Note 3: In the process of deriving Theorem 1, the sampled-data control system in formula (4) (22) and (23), respectively. Its connection with the system states is established by the free weight matrices of 1 Y and 2 Y , that is, more system information is used. The zero-value formulas (22) and (23), thus, help reduce the conservativeness of the system further.
Note 4: Theorem 1 provides a stability criterion for a variable-period sampled-data control system. Obviously, if 1 2 h h = in Theorem 1, the same criterion applies to the constant-period sampled-data control system. Additionally, Theorem 1 is also applicable to the sampled-data control systems of A and c B with polyhedral uncertainties through appropriate transformations.

Example simulation
This subsection illustrates the effectiveness of the proposed method through two numerical examples.
Example 1: The sampled-data control system in formula (5) is assumed to satisfy the following matrix parameters For the stable sampled-data control system, when 5 1 10 h − = , according to the stability criterion in Theorem 1 and Refs. [16-17, 19, 22], the upper bound 2 h of the maximum sampling period is listed in Tab. 1. It can be seen that, Theorem 1 obtains a larger sampling period compared with the results obtained in Refs. [16-17, 19, 22], indicating that the proposed method has a lower conservativeness, also verifying the effectiveness of Note 1. Additionally, in Theorem 1,

Tab. 1 The maximum allowable upper bound of h
, respectively (that is, to eliminate the effect of formulas (22) or (23), respectively), and the upper bound 2 h of the maximum sampling interval for the stable sampled-data control system in formula (4) is calculated, as shown in Tab. 1. Tab. 1 shows that, formulas (18), and (19) are helpful in reducing the conservativeness of the system, verifying the effectiveness of Note 3.
Example 2: The sampled-data control system in formula (4) is assumed to satisfy the following matrix parameters For the stable sampled-data control system in formula (4), when 1 h has different values, the maximum sampling interval is calculated using Theorem 1 and Refs. [6,8,[17][18][19]. The results are shown in Tab. 2. It can be seen from Tab. 2 that the proposed method can obtain a larger sampling interval compared with [6,8,[17][18][19], indicating that the proposed method is better and less conservative, further verifying the effectiveness of Note 1.

Application to electric power markets
A prerequisite for the reliable and stable operation of the grid is that the output and the load of the electric power (i.e., supply and demand) are balanced at all times. To maintain power balance, the scheme of load frequency control is usually adopted [26] . In recent years, researchers have proposed that the electric power market can use power prices to maintain the balance between supply and demand, which has attracted extensive attention [15,[27][28][29] . This section applies the proposed method to the stability analysis of the electric power market (mainly including modeling the electric power market) and theoretical calculation (for example simulation and discussion).

Models of the electric power market
A simple electric power market consists of power suppliers, power users, and real-time market.
(1) Market supply model. The marginal production cost of power generation and the power supply satisfy the following relationship Here, g λ is the marginal cost, g b is the fixed cost of the supplier, g c is the fixed coefficient, and g P is the amount of power supplied. If the power supplier observes that the power price on the market is higher than its production cost, the power supplier will expand production, and the degree of expansion will be proportional to the difference between the observed market power price and the actual production cost. Furthermore, additional costs will be incurred based on an oversupply of power at some point in the past. Therefore, the model of the electric power market supply can be described as where, g τ is the time constant representing the response speed of the power output, λ is the observed power price on the market, k is the fixed coefficient, and E is the integral of the difference between the power supply and the demand.
(2) Market consumption model. The marginal benefit and the demand for electrical power satisfy the following relationship where, d λ is the marginal income, d b is the fixed income, d c is the fixed coefficient, and d P is the power demand. When marginal returns are higher than marginal prices, consumers will expand their consump-tion, and the speed of expansion is related to consumers. Therefore, the market consumption model can be described as where, d τ is the time constant representing the expansion speed of consumer demand.
(3) Unbalanced power supply and demand as well as market price response.
The imbalance between the supply and the demand can be expressed by the integral of the difference between the power supply and demand as The change in the power price on the market is determined by observing the demand and supply of the grid. This is given by where, λ τ is the time constant representing the response speed to the power price at the time of market disturbance. In the actual electric power market, the discrete power price signal received at a certain interval is equivalent to the market-clearing time, meaning the period of updating the power price [28] . According to formula (27) and formulas (29) to (31), the following linear control model based on sampled-data control is obtained.
Here, k t is the clearing time for the power price and it satisfies is the balance point of the system in formula (34), to facilitate the processing, the equilibrium point * y is moved to the origin through coordinate transformation, and then the system in formula (34) can be transformed into This system has been discussed in Section 4. The key parameters A and c B of the system in formula (35) have the same physical meanings as those defined by the system in formula (34). Therefore, the proposed stability analysis method of the sampled-data control system can also be used to discuss the stability of the electric power market.

Simulation verification
The model parameters of the system in formula (34) are 0.2, 0.1, 2, .
According to the methods in Theorem 1 and Refs. [14][15] The state trajectory of the system is shown in Fig.  1. The figure shows that when the update period of the price signal is 10.64 mct T = , the system is stable. The discrete control signal going through the zero-order holder (ZOH) is shown in Fig. 2.

Significance of conservativeness reduction
The comparison and analysis were performed on the conservativeness between the proposed method and the existing methods by two numerical examples. This subsection will discuss the practical significance of reducing conservativeness through the example of the electric power market.
The maximum update period, M mct T , of the acceptable power price signal calculated by the stability criterion, meaning the sampling period in the sampled-data control system can be used as a reference for selecting a sampling device, such as the capacity and type of communication channel, and the sampling frequency of signal collector. Tab. 3 shows that the M mct T value obtained by Theorem 1 is larger than that in Refs. [14][15], indicating that the conservativeness, and a lower requirement for hardware devices for the proposed method. That is, based on the conditions for maintaining the stability of the electric power market (i.e., the supply and demand balance of power), the sampling device can be selected with a lower sampling performance (and consequent lower price) through the proposed method compared to that used in Refs. [14][15], which also reduces cost.
It can be seen from Tab. 3 that both the methods in Ref. [15] and Theorem 1 (the sampling frequency of the collector satisfies 0 1 / 8.59 f ≥ ), can maintain the stability of the electric power market. When 8.59 10.64 M mct T ≤ ≤ , Ref. [15] cannot maintain the stability of the electric power market, while Theorem 1 can still maintain the stability of the system, that is, for Theorem 1, the sampling frequency of 1 1/ 10.64 f ≥ is enough. Here, 1 0 f f < , highlighting that the proposed method allows for the selection of a signal collector with a lower frequency, hence reducing the cost.

Conclusions
This study provides a method for analyzing the stability of sampled-data control systems, and applies it to the electric power market. First, a new two-sided closed-loop Lyapunov functional is constructed by considering the information of the system over the entire sampling interval. Second, the modified free-matrix integral inequality is used to estimate the integral term in the functional derivative, and then derived the stability condition of the system. The effectiveness and low conservativeness of the proposed method are verified through two examples. Finally, through the examples, the proposed method is applied to analyze the stability of the electric power market, and to assess the practical significance of reducing the conservativeness.