Adaptive neural consensus tracking control for multi-agent systems with unknown state and input hysteresis

An indirect adaptive consensus control method is presented for multi-agent systems (MASs) with unknown hysteresis states and input. All system states that can be utilized to design the controller are measured by the sensors subjected to hysteresis, and thus, the system state values are inaccurate. Meanwhile, it is difficult to compensate the input hysteresis for it is coupled with the state hysteresis. The unknown function from agent’s neighbors also increases the difficulty of controller design. To eliminate the influence of unknown input hysteresis, an inverse adaptive compensated method is presented. The problem of state hysteresis is addressed by designing two adaptive laws to approximate the upper and lower bounds of unknown hysteresis coefficient. Neural networks are introduced to handle the unknown dynamics of agent and its neighbors. The proposed control scheme can guarantee that the consensus errors of followers converge to a predefined interval of zero asymptotically. In addition, the transient performance of MASs can be further ensured. The simulation examples are included to verify the effectiveness of the presented control approach.


Introduction
Multi-agent systems (MASs) have an extensive application in military, satellite clusters and so on [1][2][3][4][5], and therefore, numerous researchers focus on it. Therein, there are many significant results for linear or low-order nonlinear MASs which have achieved the ideal tracking consensus [6][7][8]. Moreover, these interesting works were applied in the high-order MASs [9][10][11]. However, the dynamics of MASs in the important works need to be completely known.
Considering that unknown nonlinearities universally exist in the dynamics of MASs, some novel corresponding methods, such as neural networks (NNs) and fuzzy-logic systems (FLSs), are presented to handle the unknown dynamics. In [12], an innovative consensus tracking control scheme was proposed for MASs with unknown dynamics. In [13], by employing NNs, the consensus control method of MASs with unknown control sign and dynamic uncertainties was presented. In [14], a tracking control scheme for MASs with unknown function existing in each subsystem was pre-sented. However, as the scale of the MAS expands, the number of the estimated parameters in [14] becomes unbearable, which requires lots of computation. Such a problem was properly solved in [15] for nonlinear system and then in [16,17] for MASs by treating the norm of the ideal weight vector as an unknown constant. This novel scheme is widespreadly adopted in [18][19][20][21] for the neural or fuzzy controller design. However, as a result of estimating the norm, control schemes in the aforementioned papers [16][17][18][19][20][21] only achieve the system stability with unknown tracking error. These schemes introduce some additional terms, and thus, the derivative of Lyapunov function is difficult to be designed negatively. To produce the asymptotic tracking control of nonlinear system, in [22], based on an innovative Lyapunov function design, a performanceoriented fuzzy control approach was proposed. The method in [22] not only ensures the fixed tracking accuracy and transient performance, but also does not increase the computational burden. In [23], a neural control method is presented for uncertain MASs with predefined accuracy. However, there are few results for MASs with fixed tracking performance in the presence of hysteresis.
Adaptive control for nonlinear systems and MASs with input hysteresis has attracted lots of attention, because the hysteresis problem exists in extensive devices and actual systems, such as electromagnetism and mechanical actuators [24][25][26][27]. To eliminate the effect of input hysteresis, in [25], by constructing an inverse compensation function, an output-feedback control scheme was proposed. Even so, the perfect compensation requires that the hysteresis model is completely known. To solve the issue of unknown hysteresis, a promising adaptive method was developed in [28]. In [29], an adaptive control method was proposed for stochastic nonlinear systems with input hysteresis and unknown control gain. Moreover, a fuzzy event-triggered-based control scheme was proposed for stochastic MASs with Bouc-Wen hysteresis input in [30].
Nevertheless, the mentioned works are states feedback control methods or output feedback control methods. They require accurate values of the states or the actual system output. In practical application, the system variables are usually measured by sensors with errors, which makes the above methods impractical. It is reported in [31][32][33][34] that various sensors suffer from hysteresis and in such case the exact values of sys-tem states become unknown. For nonlinear system, the use of inaccurate state value may cause performance degradation and even system instability [35]. There is a fundamental difference between the nonlinearities in actuator and in sensor, because the controller design only can use the inaccurate value. It is a challenging task to relieve the impacts caused by hysteresis existing in sensors. In [24,36], two adaptive control approaches for linear system with uncertain hysteresis in both actuator and sensor were presented. It is assumed that hysteresis only exists in the sensor measuring the output. To remove this obstacle, Liu et al. [37] discussed an adaptive control for nonlinear systems in the presence of both input and state hysteresis with prescribed accuracy. So far, there are few researches on MASs with state hysteresis and input hysteresis. The control scheme in [37] cannot be applied in MASs because the distributed consensus controller for agent in MAS is designed using information from itself and its neighbor. The states from different agents are subjected to different unknown hysteresis. How to design the controller for MASs with state hysteresis is an interesting work.
Driven by the above discussion, this article explores the predefined precision consensus control of MASs with hysteresis in actuators and measuring sensors. The controllers are fully distributed requiring partial information of MASs. The adaptive compensation scheme is presented to mitigate the effects of state hysteresis. Since the parameters of input hysteresis are unknown, NNs are utilized to estimate the input hysteresis and adaptive laws are given to approximate the weight matrix. The contributions of this article are epitomized as follows.
• Compared with the issue that hysteresis only exists in the actuator [25,[28][29][30], state hysteresis causes more technical difficulties in back-stepping design. Unlike [24,36,37], for the MASs with state hysteresis, the state of agent's neighbor is subjected to different hysteresis. Hysteresis is related to the states of agent and its neighbors which cannot be obtained in the current step. Compared with [37], the compensation of hysteresis is more difficult because we need to handle several different hysteresis nonlinearities from agent's neighbors in the same step.
To handle the unknown dynamics caused by state hysteresis, adaptive laws are designed at each step to estimate the upper and lower bounds of the time-varying term. The unavailable states from anget's neighbors are approximated by NNs. In addition, an inverse compensation is proposed for input hysteresis. • Most control methods [13][14][15][16][17][18][19][20][21] achieved system stability with unknown error, while in this paper by using the smooth sign function to construct Lyapunov function, all followers can track the leader asymptotically. Compared with the control algorithm for nonlinear system in [22], the uncertainties from agent's neighbor are handled by an altered tuning strategy. Unlike [23], the actual system states of agent and its neighbor are unavailable and the controllers are developed using the value measured by sensors. The proposed method can ensure a predefined steady-state error and improve the transient performance of MASs. Moreover, MASs can still maintain stability and performance even in the presence of interference.

Communication graph
The interaction between nodes is described by a directed graph G = (V, E), in which V = {0, 1, . . . , K } means a set of agents and E ⊆ V × V denotes a set of communicated edge. Generally, the node labeled 0 is treated as the leader and K is the number of followers. The adjacency matrix is defined as A = [a ik ] K ×K . If the ith node can receive message from the kth node, a ik > 0, otherwise a ik = 0. Moreover, L = D − A denotes Laplacian matrix where D = diag(d 1 , . . . , d K ) and d i = K p=1 a ik . b i shows the communication from leader to the ith follower.

Radial basis function NNs
It has been proven that radial basis function (RBF) NNs are very influential in dealing with unknown dynamics [38]. This article combines NNs and a novel Lyapunov function to ensure the asymptotic stability of MASs with unknown dynamics. The RBF NNs are employed to approximate the unknown function.
wherex represents the input vector composed of independent variable. (x) is an unknown but bounded approximation error. W * denotes the ideal weight vector defined as is the Gaussian function defined as where p and q p , respectively, denote the width and the center of basis function.
The following lemma describes a useful property of neural networks which will be helpful in reducing the number of variable inputs.

Problem formulation
Consider a MAS with hysteresis phenomenon presenting in both actuators and sensors. The dynamics of the followers are modeled as: where π i, p , y i andx i, p = [x i,1 , . . . , x i, p ] T represent unknown smooth function, system output and state vector, respectively. v i is the control signal we actually designed and u i is the output of actuator acting on the plant. They are not equal when hysteresis exists in actuator. The hysteresis phenomenon can be described by the most popular Bouc-Wen model H i . While the actuator is not affected by hysteresis, While hysteresis exists in actuator, it can be described by Bouc-Wen model as where μ i1 and μ i2 are unknown constants and sign (μ i1 ) = sign (μ i2 ). ξ i is given bẏ and with h i > |χ i | and l i ≥ 1. h i , ξ i and l i are the unknown parameters representing the shape, the amplitude and the smoothness, respectively. The Bouc-Wen model is shown in Fig. 1. The proposed control scheme will discuss these two cases separately.

Remark 1
The important works in [6][7][8][9][10][11] require that the system dynamics are completely known because their controller design uses all the information of system dynamics. Specially, in [9], the system dynamics are described asẋ i = A i x i + B i u i , and the controller design uses the matrices A i and B i . The system model in [11] is similar to the model (4) in this paper. However, the controller design in [10] directly uses the function π i, p , and therefore, [10] requires that the system dynamics are completely known. By employing RBFNNs, this restriction is removed in this article.
The dynamics of the leader are described as: Assumption 1 [39] The leader's output y 0 and its derivativeẋ 0 are bounded. The reference signal of the leader is known to some agents.
In this paper, the exact system states are not available and only the hysteresis states received by sensorŝ x i,1 ,x i,2 , . . . ,x i,n i can be used in our control design. According to [25,40], the relationships between the received signals and the genuine system states are given as the Bouc-Wen model: where μ i, p1 and μ i, p2 are unknown and sign μ i, p1 = sign μ i, p2 . Without loss of generality, the parameters μ i, p1 and μ i, p2 are assumed to be positive. ξ i, p is given byξ where h i, p > |χ i, p | and l i, p ≥ 1.
Remark 2 New types of sensors and actuators are widely used in industry due to their small size and low energy consumption. However, due to the use of smart materials, nonlinear hysteresis inevitably exists in these sensors and actuators. Some important compensated schemes are proposed to address this issue [32,36,[41][42][43][44]. The model of hysteresis is the key to the controller design. Since the hysteresis phenomenon is caused by smart materials, the hysteresis models of actuator and sensor can be similar. In [42,43], the Prandtl-Ishlinskii model that is used to describe the hysteresis phenomenon in the actuator is extended to describe the phenomenon in sensors. Moreover, Prandtl-Ishlinskii model can be regarded as a special case of the Bouc-Wen hysteresis model [25]. Therefore, this paper uses Bouc-Wen Model to describe the hysteresis in sensors.

Remark 3
In [25], an output feedback control method is proposed for nonlinear system with unknown input hysteresis. The controller design only requires the information of system output and the unknown hysteresis is perfect compensated by constructing an inverse function. However, the method in [25] still requires the exact value of system output. If the sensor that measures system output is also subjected to hysteresis, the system stability cannot be guaranteed. Moreover, the method in [25] does not take into account information from other agents, such that it cannot handle the unknown dynamics from agent's neighbor. The inaccurate system states from other agent also make the problem more difficult.
The following lemma gives an important property which will help our method development.

Lemma 2h i, p is nonnegative along with time.
Proof From [25], we have Substituting into the definition ofh i, p (11), one has that Remark 4 It is a huge technical challenge that all exact states of agents are unavailable in this article. Different from [25,[28][29][30], hysteresis affects each measuring sensor, and in the meanwhile, the characteristics of hysteresis are totally unknown. Compared with [37], the hysteresis information from agent's neighbor is more difficult to handle because it is related to another different unknown hysteresis nonlinearity.
Using the measured states instead of actual system states, the consensus tracking errors of the ith agent are defined as: The following lemma is needed. 1 ] T and P = L + B, and then one has: where denotes the minimum singular value of P.

Consensus control
In this section, the consensus control schemes are presented for MASs with states hysteresis and input hysteresis. RBFNNs are utilized to approximate unknown function at each step. Before our beginning, for the sake of convenience the unknown function π i, p can be expressed as where and Define To continue our work, two smooth functions are introduced [45].
Remark 5 The function sg i,k plays the same role in this article as in [45]. As shown in the proof of Theorem 1, by employing the functions sg i,k and f i,k to construct Lyapunov function, the tracking accuracy can be designed in advance, that is, lim t→∞ |z i,1 | ≤ i,1 . However, the method in [45] requires that the function input is the exact tracking error z i, p = x i, p − α i, p not the z i, p we define in Eq. (12), such that it cannot be applied in this paper to address the state hysteresis. It is more challenging since the real system states are unavailable in this paper. Moreover, [45] does not consider the communication between agents, so that the unknown terms from agent's neighbor cannot be handled. In addition, as we mentioned in Remark 8, this paper has no need to calculate the derivatives of virtual control laws repeatedly, so that the function sg i,k is only required to be first-order differentiable. While in [45], it should be (n i − p + 1)th-order differentiable. In addition, an alternative reduced-order function is provided: The relationship (9) between actual system state and sensor output is rewritten as Now we can develop the control scheme and the details are presented in the following two cases.

Case 1: only states hysteresis
In this case, Step 1 From the definition of consensus errors (12) and Eq. (21),ż i1 can be derived aṡ 21 and r j,1 = (μ j,11 + μ j,12h j,1 )/μ j, 21 . From the definition of r i,1 , we can directly conclude that r i,1 is positive and bounded. Define η i, p representing the upper bound of 1 r i, p .η i, p is the estimation of η i, p and the errorη i, p = η i, p −η i, p .
Choose a Lyapunov function candidate as where i,1 is a predefined tracking accuracy and γ i is a positive constant. Subsequently, we havė where the unknown continuous functionḡ i, Since π i,1 and π j1 are unknown and contain the exact value of system sates, , and according to Lemma 1,with Z where with d representing a design parameter. Define i, p as the upper bound of r i, p ,ˆ i, p is an estimate of i, p and˜ i, p = i, p −ˆ i, p . To ensure thatV i,1 is negative, the tuning function, virtual control law and adaptive law are given as follows: where c i,1 is a positive design parameter.
From (60), one hasθ i (0) > 0, and therefore, if θ i (0) > 0 is chosen,θ i > 0 holds. Similarly, we can prove thatˆ i, p > 0, and thus, we have Hence, By combining (28)-(33),V i,1 (26) can be calculated aṡ Remark 6 If the quadratic Lyapunov function V = 1 2 z 2 is chosen and the NN W T R + is utilized to approximate the unknown function, we can only achieve thatV ≤ cV +σ where c > 0 is a constant and σ is unknown [13][14][15][16][17][18][19][20][21]. This method cannot guarantee that the derivative of Lyapunov function is negative, because the use of Young's inequality introduces the unknown term into the stability analysis. In this paper, the smooth function sg is utilized, and thus, the use of Young's inequality is avoided. With function sg, we can achieve thatV ≤ 0. The control algorithm for nonlinear system in [22] cannot be directly extended to MASs because of the uncertainties from agent's neighbor.

Remark 7
The consensus tracking error is defined in (12) only using the available information. Moreover, due to the existence of state hysteresis, the design of virtual controller α i, p at step p may require the states x i, p+1 and x j, p+1 . The rate-dependent termξ i, p introduces the state x i, p+1 into the virtual controller design. However, according to [46,47], x i, p+1 cannot be obtained in step p. Thus, r i, p is unavailable. According to Lemma 2, it is easy to prove that r i, p is bounded and nonnegative. Then, the upper and lower bound of the time-varying term r i, p are approximated byˆ i, p andη i, p , respectively. Moreover, Lemma 1 and NNs are introduced to handle the unaccessible state from its neighbor x j, p+1 . With these adjustments, we can continue the controller design.

Remark 8
The use of backstepping technique requires to calculate the derivatives of virtual control laws repeatedly, which impedes the implementation of our control scheme. To address this problem, the derivatives of virtual control laws are contained in the unknown function which is estimated by a NN. And then, the adaptive law is designed to approximate the ideal norm of the NN. Thus, the calculation of derivatives is avoided. 1 ] T , according to Lemma 1, the virtual control law is designed as: where γ i , i and c i,2 are positive constants to be selected.

Remark 9
The proof of M i,2 ≤ 0 needs thatˆ i,1 > 0. By designing the updated law˙ i,1 (43) as given in step 2,ˆ i,1 > 0 holds. Accordingly, the updated law˙ i, p will be designed in step p + 1.
Step p Similarly, we choose a Lyapunov function Calculating the time derivative of (12), we havė The control law and adaption laws are directly given as where c i, p is a positive design parameter. Then, we can derive the following equation in the same way as in step 2.
Step n In this case, we consider the MASs without input hysteresis. The plant can directly receive control signal, i.e., H i (v i ) = v i . The distributed consensus controller and the updated laws are given to achieve our control goal: where τ θ i,n i and ω θ i,n i are defined in (53) and (54). Define the Lyapunov function for this step.
In view of (57)-(61), we can get the derivative of V i,n i With the design of control signal and updated laws, the following theorem can be concluded.

Theorem 1
Considering the MASs (4) with states hysteresis, if the controller (57) and adaptation laws (59)-(61) are implemented, we can obtain that: • all variables of the MASs (4) are SGUUB.
• the consensus tracking error asymptotically converges to a predefined bound along with time, i.e., Proof Define the Lyapunov function Thus, z i, p ,θ i ,ˆ i, p−1 andη i, p are bounded. Moreover, through the definition of z i, p (12), we can get thatx i,1 is bounded because the output of leader y 0 is bounded. α i,1 is consisting of variablesθ i , i,1 , η i,1 ,x i,1 andx j,1 , such that α i,1 is also bounded. Subsequently, we can further get thatx i,2 is bounded from (12). In the same way, the boundedness of α i, p andx i, p can be proved. Then the control signal v i is also bounded. Noticing Eq. (9), the real system states x i, p are ensured to be bounded. Next it is proved that the consensus errors converge to a predefined neighborhood of zero. For convenience, we define Then one haṡ Integrate both sides of the inequality According to Barbalat's lemma, it is proved that lim t→∞ s i = 0, i.e., lim t→∞ |z i,1 | ≤ i,1 . Along with Lemma 3, y − Y ≤ 1 is achieved.
The L 2 -norm of the consensus error can be further proved Remark 10 It is proved that the consensus tracking errors can be designed in advance by using the Lyapunov function with smooth sign function. The L 2norm performance is obviously adjustable. By increasing the design parameters ζ i , i and γ i , the transient performance can be improved. The consensus tracking errors can be reduced by selecting a suitable i,1 .

Case 2 states hysteresis and input hysteresis
In the just described work, the control method for MASs with states hysteresis was developed. Now the case that actuators affected by hysteresis are further considered. The inverse compensation strategy is given as follows: where h id , l id and χ id are positive constants to be chosen. In addition, h id ≥ |χ id |. The error between actual control signal and desired control signal is given by Since ξ i and ξ id are bounded, i has an unknown upper bound such that | i (t)| ≤¯ i .
Only the actual control law in step n i needs to be modified. We havė The RBFNN is employed to approximate the unknown control gain. One has where Then, where β i is a parameter to be designed. iμ and iŴ are well-chosen positive definite matrices. Proj(·) represents the projection operator developed in [46].

Remark 11
Unlike [25], the model of input hysteresis is unknown, such that the perfect inverse compensation in [25] cannot be applied in this paper. In this article, we design the updated law Eq. (80) to estimate the model's parameters instead of parameter identifications, which is easier to be implemented in practice.
Compared with [28][29][30], the control input of MASs is not only affected by the unknown hysteresis in actuator, but also by the unknown control gain caused by states hysteresis, as shown in Eq. (73). The control coefficient is time-varying such that we cannot directly design an inverse compensation for input hysteresis. A NN is employed to approximate the unknown function. • all variables of the MASs (4) are SGUUB.
• the consensus tracking error asymptotically converges to a predefined bound along with time, i.e., Proof Consider the following Lyapunov function and defineV = N i=1V i,n i . From (82), one haṡ Moreover, through the definition of z i, p (12), we can get thatx i,1 is bounded because the output of leader y 0 is bounded. α i,1 is consisting of variablesθ i , i,1 , η i,1 ,x i,1 andx j,1 , such that α i,1 is also bounded. Subsequently, we can further get thatx i,2 is bounded from (12). In the same way, the boundedness of α i, p and x i, p can be proved. Then the control signal v i is also bounded. Noticing Eq. (9), the real system states x i, p are ensured to be bounded. The property that the consensus errors converge to a predefined neighborhood of zero is also given.
According to Barbalat's lemma, it is proved that lim t→∞ s i = 0, i.e., lim t→∞ |z i,1 | ≤ i,1 . Along with Lemma 3, y − Y ≤ 1 is achieved. Since the measured errors are always bounded, the boundedness of the actual system states can be obtained by the boundedness of the measured value.
The L 2 -norm of the consensus errors can be further proved Remark 12 If we directly design the updated laws , the term λ i which includesW i andμ i cannot be ensured to be bounded. Accordingly, the compensation of λ i cannot be achieved and the system stability cannot be guaranteed. With the projection operator, we can make sure thatW With this property, we can continue our work and design the distributed controllers.
Remark 13 In Eq. (76), we cannot confirm the sign of item z i,n i − i,n i f i,n i sg i,n iμ TŴ T i R i in advance, and therefore, we cannot estimate the norm instead of the weight vector to reduce the computational burden. However, the number of estimated parameters depends on the number of NN nodes, so that we can reduce the number of NN node to avoid the computational problem. It does not affect the system stability or increase tracking errors. As shown in Eq. (85), only the transient performance of the system is affected.

Corollary 1
In both two cases, the MASs 4 still remain stable even if there is bounded disturbances in the system dynamics. Moreover, the conclusion in Theorems 1 and 2 still can be achieved.
Proof Let ϒ i, p represent the disturbance. When the disturbances exist in system dynamics, according to where r i, p and ϒ i, p are bounded. Rewriting Eq. (16), one has whereῩ i, p = r i, p ϒ i, p . Then, the unknown parameter need to be estimated is still θ i . Thus, according to the same process of controller design and stability analysis, Theorems 1 and 2 can be obtained.

Illustrative examples
In this section, two examples are provided to illustrate the effectiveness of the aforementioned distributed consensus control method. Figure 2 denotes the communication graph of the MAS which is consisting of a leader and 5 followers.

Numerical example
The dynamics of follower agents are given aṡ wherex i,2 is the exact system state vector which cannot be used in our controller. The measuring sensors are subject to the hysteresis which is model bŷ It is more challenging when the input and states are both subjected to hysteresis. The control scheme in Case 2 is implemented for the example. The hysteresis in actuator is modeled by The leader's dynamics are given as follows: We can conclude that adaptive parameters are bounded as shown in Fig. 6. The consensus errors converge to a small domain of zero rapidly as shown in

Remark 14
As we mentioned in the proof of Theorem 1 and 2, the proposed control method can ensure that the tracking error lim t→∞ |z i,1 | ≤ i,1 . If we choose i,1 = 0, the perfect tracking performance can be achieved. However, in this case, the function sg i,k will be discontinuous which may cause the chattering phenomenon of actuator. Therefore, in practical application we should choose a suitable tracking accuracy according to the actuator. In this simulation example, the tracking accuracy i,1 is selected as 0.75, and there- shows that the exact system state x 1,2 is very different from the measured onex 1,2 , which is the greatest difficulty of the controller design in this paper.

Application example
Consider the spring-mass-damper MAS that has the same topology as Fig. 2. Each follower is modeled aṡ The leader's dynamics is given as: The system parameters are selected as: m i = 1. k i = 1 + 0.2i and c i = 1.5. The design parameters are selected as follows: c i,1 = 6, c i,2 = 1 and i,1 i,2 = 0.8. The initial MAS's states and the leader's output are the same as the numerical example. Some initial values of the adaptive parameters have been changed as below:η i,1 (0) =η i,2 (0)) = 1,λ i (0) = 0.5 andμ i (0) = [0.2, 0.6] T . To illustrate the effectiveness of the proposed method, the control scheme in [23] is considered to make a comparison. We select the same design parameters. Figures 9, 10 and 11 shows the simulation results. It can be obtained from Fig. 11 that the proposed scheme can provide a better system performance. Note that the system performance is adjustable by selecting different design parameters. However, the

Conclusion
The consensus control strategy for MASs with unknown states hysteresis and input hysteresis is researched. Based on backstepping technique, we employ 2 adaptive laws to approximate the upper and lower bounds of the unknown term introduced by state hysteresis. To handle the input hysteresis, NNs are utilized to approximate the unknown control gain which is coupled by input hysteresis and states hysteresis. The proposed scheme guarantees the boundedness of all signal and the consensus errors are ensured to converge to a predefined neighborhood of zero asymptotically. In addition, the L 2 -norm of the consensus error can be further ensured. Two simulation examples are provided to illustrate the effectiveness of the proposed control approach.
There are still several questions worth exploring in the future:   • As mentioned in Remark 14, chattering phenomenon may occur if we choose 0 tracking error. How to achieve perfect tracking performance without chattering is an interesting problem. • The inverse compensation for input hysteresis employs NN, which will bring lots of learning parameters. It is an important work to reduce the computed burden. • Besides, the control for MAS with quantized states and sensor faults is under our consideration.