Integral-Type Event-Triggered Receding Horizon Control of Nonlinear Systems with Additive Disturbance

# Integral-Type Event-Triggered Receding Horizon Control of Nonlinear Systems with Additive Disturbance

Qi Sun, , Jicheng Chen, , Yang Shi,  The authors are with the Department of Mechanical Engineering, University of Victoria, Victoria, BC V8W 2Y2, Canada (e-mail: sunqi@uvic.ca; jichengc@uvic.ca; yshi@uvic.ca).
###### Abstract

This paper studies the event-triggered receding horizon control (RHC) of continuous-time nonlinear systems. An integral-type event-triggered mechanism is proposed to save the communicational resource, and a less conservative robustness constraint is introduced to the RHC scheme for compensating the additive disturbance. Based on these formulations, the designed event-triggered algorithm is shown to have better performance on avoiding unnecessary communication. Furthermore, the feasibility of the integral-type event-triggered RHC scheme and the stability of the closed-loop system are rigorously investigated. Several sufficient conditions to guarantee these properties are established, which indicate that a trade-off exists for designing parameters such as the prediction horizon, the disturbance bound, the triggering level, and the contraction rate for the robustness constraint. The effectiveness of the proposed algorithm is illustrated by a numerical example.

Nonlinear receding horizon control, integral-type event-triggered mechanism, continuous-time nonlinear system, robust control.

## I Introduction

Recent research interests and efforts have been directed towards reducing the computation or/and communication load in control systems by using the event-triggered scheme. This control paradigm employs the so-called event-triggered mechanism (ETM) to avoid unnecessary communication. Compared with the conventional periodic control, event-triggered control treats sampling instants as a design parameter while the periodic scheme samples the states at fixed time instants. The event-triggered paradigm is mainly to design a scheduling mechanism with the capability of determining when the states or sensor outputs should be sampled. The benefit of using this method is that it can provide a smaller communication rate and thus reduce the communication load, especially in cases where the computation and communication resources are scarce. Therefore the event-triggered control scheme has been widely studied under such resource-limited scenarios [1, 2, 3, 4, 5, 6, 7].

Several pioneering works have been devoted to build the fundamental blocks of the event-triggered control [1, 2, 3, 4]. The works [1, 2, 3] used a constant threshold-based event-triggered scheme, which enable the next sampling when the norm of state or estimation error exceeds a certain constant bound. Compared with the conventional periodic control, this method is shown to have great advantages in terms of reducing communication rate [1]. In another early work [4], the proposed event-triggered scheme adopted a so-called relative threshold policy for a class of nonlinear systems, where it allows transmissions when the norm of measurement error exceeds a weighted norm of the state. In addition, a lower bound for inter-execution time is proved to exist for avoiding the Zeno behavior. The study of state-based event-triggered control can be found in [3, 8, 5], and output-based event-triggered control has been reported in [9]. In [7], the authors proposed a periodic event-triggered scheme for reducing the communication rate, where the event-triggering condition is checked periodically with a fixed time interval. To further reduce the communication rate, the authors in [10] proposed a novel integral-based event-triggered scheme by incorporating the integral of estimated errors to the event-triggering condition.

In control research area, Receding Horizon Control (RHC) has been one of the most successful control methodologies. The basic idea of RHC framework is to solve optimization problems online at each sampling instant, and apply the corresponding control action to the plant. In particular, introducing event-triggered scheme to RHC is of great importance in term of alleviating communication load, thus receives many research studies [11, 12, 13, 14, 15, 16]. By using the event-triggered scheme, the event-triggered RHC can avoid unnecessary communication, and thus reduce the frequency of solving heavy computational optimization problems. Therefore, employing event-triggered scheme on RHC is preferable in term of saving computational cost. Some research has been concentrated on event-triggered RHC of linear systems [11, 12]. The authors in [12] studied the event-triggered RHC of linear systems with additive disturbance by using a tube-based approach. There have been also research interests focused on nonlinear systems [13, 14, 15]. In [13], an event-triggered scheme was proposed for nonlinear systems with additive disturbance by continuously measuring the error between the actual and the predicted trajectories. In order to acquire the benefit of avoiding Zeno behavior, the authors in [14] proposed an event-triggered mechanism design which can guarantee that the inter-execution time is lower bounded. Event-triggered RHC can also be found in decentralized and networked control systems [17, 15]. In [15], an event-triggered RHC framework was proposed for stabilizing the distributed nonholonomic systems. However, such RHC-based event-triggered schemes only use the instantaneous information of the actual and predicted state. The integral of the error between actual and predicted trajectories can be used in the event-triggered RHC framework for further reducing the communication rate compared with the existing results.

In this paper, we investigate the robust integral-type event-triggered RHC problem for the continuous-time nonlinear systems with additive disturbance, aiming at alleviating the computational load while ensuring the feasibility of proposed RHC problem and the stability of the closed-loop system. The main contributions of this work are three-fold:

• An integral-type event-triggered RHC algorithm has been designed for the continuous-time nonlinear system in the presence of additive disturbance. This ETM is proposed by using the integral of the error between the actual and predicted states. By using the ETM, the optimization problem will be only solved when the accumulated error reaches the designed triggering level. The triggering level is also designed to avoid the Zeno behavior;

• A novel robustness constraint is proposed to compensate the additive disturbance for the closed-loop system. The nominal state in the optimization problem is required to satisfy a time-varying constraint, which is decreasing proportionally to time. Moreover, this constraint will shrink into an ellipsoidal terminal region after a prediction horizon. By using this unique configuration, the optimization problem admits a less conservative initial feasible region.

• The feasibility of the integral event-triggered RHC and the stability of the closed-loop systems are thoroughly studied. Sufficient conditions for ensuring the feasibility and stability are provided, respectively. It is also shown that the feasibility and stability conditions are subject to the prediction horizon, the bound of disturbance, the triggering level, and the contraction rate for the robustness constraint. Moreover, there exists a design trade-off for the parameters when the control performance and the computational load are considered.

The paper is organized as follows: Section II describes the problem formulation of the proposed event-triggered RHC algorithm. Section III states the feasibility and stability results of the proposed algorithm. Section IV gives a simulation example to verify the effectiveness and computational efficiency of the proposed algorithm.

Notations: The real space is denoted by and the symbol represents the set of all positive integers. For a given matrix , we use and to denote its transpose and inverse. For a symmetric matrix , we write or if is positive definite (PD) or positive semidefinite (PSD). The largest and smallest eigenvalues of are denoted by and , respectively. Given a column vector , represents its Euclidean norm, and is the -weighted norm. We also use the notation . Given a continuously differentiable vector-valued function on , we use to represent its Jacobian matrix.

## Ii Problem Formulation

### Ii-a System Dynamics and Optimization Problem

We consider a continuous-time nonlinear system with additive disturbance as follows

 ˙x(t)=f(x(t),u(t))+ω(t), (1)

where is the state variable, is the control input, and is the additive disturbance. The system satisfies and has a Lipschitz constant . The control input , where is a compact set containing the origin. Moreover, the disturbance is also in a compact set containing the origin, which is bounded by . By linearizing the nonlinear system (1) at the equilibrium , we can obtain the linearized state-space model:

 ˙x(t)=Ax(t)+Bu(t)+ω(t), (2)

where and .

In the following, we introduce a conventional assumption for the linearized model (2), which is necessary for analyzing the closed-loop performance of the nonlinear system (1).

###### Assumption 1.

There exists a feedback control law such that the closed-loop system matrix is Hurwitz.

In addition, we also make use of a conventional result of the control invariant property of the nonlinear system (1).

###### Lemma 1.

[18] If is twice continuously differentiable, , is piece-wise right-continuous and suppose that Assumption 1 holds, then given a stabilizable , and two symmetric positive-definite matrices and , there exists a constant such that: (1) The Lyapunov equation admits a unique solution , where and satisfies ; (2) and is control invariant by the feedback control law for the nonlinear system (1).

The nominal system of (1) can be defined as

 ˙^x(s)=f(^x(s),^u(s)), (3)

where and are the predicted states and control sequence, respectively. Note that the nominal system dynamics (3) will be used for constructing the equality constraint of the following optimization problem. In order to avoid ambiguity, we take explicit notations and for the predicted state and control trajectory at the event-triggered instant . It should be noticed that the event-triggered instants are generated by using the integral-type ETM, which will be elaborated after introducing the optimization control problem.

Then the nonlinear optimization problem can be formulated as

 ^u∗(s;tk)= argmin^u∈UJ(^x(s;tk),^u(s;tk)) (4) s.t. ˙^x(s;tk)=f(^x(s;tk),^u(s;tk)), (5) ^u(s;tk)∈U,s∈[tk,tk+T], (6) ∥^x(s;tk)∥P≤(tk+T−s)M+s−tkTαϵ. (7)

The cost function is defined as follows

 J(^x(s;tk),^u(s;tk))\lx@stackrel△=∫tk+Ttk∥^x(s;tk)∥2Q+∥^u(s;tk)∥2Rds+∥^x(tk+T,tk)∥2P, (8)

where , , is the identity matrix, is defined by using the method from Lemma 1, is the prediction horizon, is the designed parameter for defining the terminal set, is the scaling ratio, and is the contraction rate for the robustness constraint (7). It should be noted that the terminal constraint is .

By using the aforementioned system dynamics and optimization problem, we construct the following closed-loop system as

 ˙x(t)=f(x(t),^u∗(t−tk;tk))+ω(t),k∈{0,1,2,…}, (9)

where is the event-triggered instant. In this control framework, we use the optimal control sequence as the control input generated by solving the nonlinear optimization problem (4). In order to reduce the communication between the controller and the actuator, we take an integral-type ETM for determining when the next optimization should be conducted, i.e. solving the optimization problem and transmitting the optimal control sequence to the actuator.

###### Remark 1.

In the optimization problem (4), the robustness constraint (7) is used for compensating the additive disturbance when applying the optimal control sequence to the closed-loop system (9). The shrinking constraint-based method for robust RHC has been firstly used in [19]. In their configuration, the robustness constraint shrinks very fast to the terminal constraint as time evolves. In our approach, a less conservative constraint is proposed, where the shrinking rate is a constant. Intuitively, our proposed robustness constraint will provide a larger feasible set for solving the optimization problem.

### Ii-B Integral-type Event-triggered Mechanism

For the nonlinear system (1), we assume that the event-triggered instants set is with , where is determined by the proposed event-triggered scheme (10). We also make the assumption that there is no time delay and inaccuracy when transmitting the sensor measurements to the digital controller. By using this configuration, an integral-type ETM is introduced for scheduling and implementing the sampling tasks, i.e. determining the event-triggered instants . As shown in Fig. 1, the ETM produces an ON/OFF signal to the sensor by measuring the error between the actual state and the predicted optimal state. Based on this configuration, we can acquire the benefit of reducing both the computational and communicational load. The integral-type event-triggering condition is designed as

 ¯tk+1=infh>tk{h:∫htk∥x(s;tk)−x∗(s;tk)∥Pds=δ},tk+1=min{¯tk+1,tk+T}, (10)

where is a minimum time instant satisfying the triggering condition (10). This ETM can take account of accumulated error between the measured state and optimal predicted state generated by the RHC algorithm (4) over the current period , which is different from the event-triggered setting in [14]. The following theorem shows some important properties of the proposed integral-type ETM.

###### Theorem 1.

For the nonlinear system (1), if the event-triggered time instants are implemented as (10), then the following result holds: The upper bound for inter-execution time is ; the lower bound can be guaranteed by properly designing the triggering level as

 δ=ρ¯¯¯λ(√P)[eLβT(βTL−1L2)+1L2], (11)

where is a scaling parameter.

###### Proof.

This proof can be done by two steps.

Step 1: The upper bound of inter-execution intervals is . From the design of the integral-type ETM, it can be directly deduced that all the intervals is less than or equal to the prediction horizon .

Step 2: The lower bound for inter-execution intervals can be designed as . To prove this result, we firstly consider the upper bound for at . We assume here that the sensor measurements of the states are accurate, thus it follows that . By using the triangle inequality, we have Then by applying the integral form of Gronwall-Bellman inequality, it can be obtained that Substituting the previous inequality to (10), we can deduce that Since is strictly larger than zero for , we can choose as (11) by setting . Therefore, it can be guaranteed that the lower bound of the triggered time interval is . The proof is completed. ∎

###### Remark 2.

Note that the designed ETM is based on the integral of the error between actual states and predicted states. The main difference of the integral-type ETM is that the accumulated error between two consecutive event-triggered instants is taken into account. It is worthwhile to point out that this event-triggered scheme is only valid for the system in the presence of disturbance.

## Iii Main Results

### Iii-a Feasibility Analysis

Following a conventional setup for RHC framework, we construct a classical feasible control sequence for the optimization problem (4). This same control policy has been widely exploited by [20, 21], and it can be given as

 ~u(s;tk)={^u∗(s;tk−1),if s∈[tk,tk−1+T]Kx(s;tk),if s∈[tk−1+T,tk+T] (12)

Then the sub-optimal feasible control and state trajectory evolves as

 ˙~x(s;tk)=f(~x(s;tk),~u(s;tk)). (13)

Before presenting the result of this section, we would like to introduce a lemma which will be used in the following analysis.

###### Lemma 2.

Given a continuously differentiable vector-valued function on , then the following inequality holds

 supt∈[a,b]∥g(t)∥≤12∫ba∥g′(t)∥%dt+12∥g(a)+g(b)∥. (14)
###### Proof.

For every , we have two results: Subtracting the aforementioned two equations, it yields By employing the triangle inequality, we can deduce from the above equality that Since every defined on the closed interval is equal or less than the right side of the above inequality, thus the result (14) holds. ∎

To simplify the analysis of integral-type event-triggered configuration, we make use of a term for , which can be found continuously differentiable. Note that is defined as the candidate state trajectory generated by the nominal system (13), and is the solution of the optimization problem (4). In addition, we let the sub-optimal feasible state by sampling the state at the triggered time instant . For convenience, we denote , where is the square root of the . Then we can propose the following result.

###### Corollary 1.

Given and defined on , the following inequality holds

 sups∈[tk,tk−1+T]∥e(s;tk)∥P≤L2βTLβT−1eL(1−β)Tδ. (15)
###### Proof.

By Lemma 2, we can obtain that

 sups∈[tk,tk−1+T]∥√Pe(s;tk)∥≤12L∫tk−1+Ttk∥√Pe(s;tk)∥ds+12∥√Pe(tk;tk)+√Pe(tk−1+T;tk)∥. (16)

Note that , then it can be deduced from the above inequality that

 sups∈[tk,tk−1+T]∥e(s;tk)∥P≤12L∫tk−1+Ttk∥e(s;tk)∥Pds+12∥2e(tk;tk)+∫tk−1+Ttk˙e(s;tk)ds∥P≤L∫tk−1+Ttk∥e(s;tk)∥Pds+∥e(tk;tk)∥P. (17)

Next we show that the upper bound of is related to the triggering level . By using the result from Theorem 1, the upper bound of is for . Thus, it can be obtained that for . Note that we also have for , where the triggering level is designed as the integral of from to . By following simple calculation, we can obtain that

 ρ¯¯¯λ(√P)[eLh(h−1L)+1L]=Lδ, (18)

and consequently it follows that

 ∥e(h;tk)∥P≤ρ¯¯¯λ(√P)eLhh≤Lδhh−1L, (19)

where . Since the function gets its maximum at , the above inequality becomes

 ∥e(tk;tk)∥P≤∥e(h;tk)∥P≤L2βTLβT−1δ. (20)

According to Gronwall-Bellman inequality, one can obtain (15) by substituting (20) to (17). The proof is thus completed. ∎

Now we can analyze the iterative feasibility of the RHC problem (4), implying that if the RHC problem admits a solution at current time instant then a feasible solution exists for the next time instant. To prove this result, we use a conventional feasible control sequence candidate at time instant defined in (12), where for and for . In the following theorem, we will show that the designed control sequence candidate can steer the feasible state into if some conditions can be satisfied. In addition, it is also necessary to show that the candidate state will remain in the designed state constraint (7).

###### Assumption 2.

The optimization problem (4) admits a feasible solution for the initial time .

###### Theorem 2.

Suppose that the Assumptions 1 and 2 hold. The RHC problem (4) is iteratively feasible under the following conditions:

 L2βTeL(1−β)TLβT−1ρ¯¯¯λ(√P)[eLβT(βTL−1L2)+1L2]≤(1−α)ϵ, (21) T≥−2¯¯¯λ(P)λ––(Q∗)βlnα, (22) M≥max{L2βTeL(1−β)TLβT−1δαϵ+1,1−1β+1αβ}. (23)

Moreover, the maximum allowable disturbance can be given as

 ρ≤(1−α)ϵL2βTeL(1−β)TLβT−1¯¯¯λ(√P)[eLβT(βTL−1L2)+1L2]. (24)
###### Proof.

First, we show that the designed control sequence for drives into , i.e. . Let us construct an error norm for . By using Corollary 1, we can obtain that

 sups∈[tk,tk−1+T]∥~x(s;tk)−^x∗(s;tk−1)∥P≤L2βTLβT−1eL(1−β)Tδ. (25)

Then it follows that . By using the Triangle inequality, we have

 ∥~x(tk−1+T;tk)∥P≤∥^x∗(tk−1+T;tk−1)∥P+L2βTeL(1−β)TLβT−1δ, (26)

which implies that .

Note from Theorem 1 that the designed triggering level . In order to steer the candidate state trajectory into , one can simply deduce that the following inequality must holds

 L2βTeL(1−β)TLβT−1ρ¯¯¯λ(√P)[eLβT(βTL−1L2)+1L2]≤(1−α)ϵ. (27)

From (27), it can be also obtained that the maximum bound for disturbance satisfies .

Second, we consider the candidate trajectory for . By using Lemma 1, we can verify that is an invariant set for the closed-loop system . Consequently, we can deduce that . By the virtue of comparison principle for , it follows that

 V(~x(s;tk))≤ϵ2e−λ––(Q∗)¯λ(P)(s−tk−1−T), (28)

which indicates that . By using Theorem 1, we can have . To obtain , it is equivalent to show that . With some calculation, one can obtain to guarantee the previous inequality holds. Similar argument can be found in [14].

Third, we show that will satisfy the state constraint (7). For , one can get

 ∥~x(s;tk)∥P≤∥^x∗(s;tk−1)∥P+L2βTeL(1−β)TLβT−1δ, (29)

which can be easily derived from (15). Then we need to prove

 (tk+T−s)M+s−tkTαϵ≤(tk−1+T−s)M+s−tk−1Tαϵ+L2βTeL(1−β)TLβT−1δ. (30)

By some calculation, it can be obtain that . For , it can be derived from (28) that

 ∥~x(s;tk)∥P≤ϵe−λ––(Q∗)¯λ(P)(s−tk−1−T)/2. (31)

In order to prove , it is equivalent to show

 (tk+T−s)M+s−tkTαϵ≥ϵe−λ––(Q∗)¯λ(P)(s−tk−1−T)/2. (32)

For brevity, we denote , and it turns out that . By evaluating the derivative of , it can be verified that is non-positive for , which indicates . Finally, the designing parameter should be set as for guaranteeing the satisfaction of the proposed robustness constraint. The proof is completed. ∎

###### Remark 3.

Note from Theorem 2 that the feasibility can be affected by the prediction horizon , the Lipschitz constant , and the disturbance bound . In order to achieve the recursive feasibility, the prediction horizon should be lower bounded, and the design parameter in (7) should be lower bounded as well. Specifically, a lager leads to a larger initial feasible region. It should be also noted that the maximum allowable disturbance can be decided as (27), which shows that the allowable disturbance and the prediction horizon are correlated to each other. Thus the trade-off between these two parameters must be taken into account when designing the algorithm.

### Iii-B Stability Analysis

In this part, we investigate the closed-loop stability of the proposed integral-type event-triggered RHC. In the following theorem, we mainly analyze the non-increasing properties of the cost function in (8). Due to the existence of the disturbance, it is worthwhile to point out that the closed-loop stability can be achieved to converge to an invariant set. Since we use the RHC configuration, the analysis for stability can be divided into two steps: One is to ensure that the optimal trajectory will enter the terminal set in finite time; the other is to prove that the closed-loop system is stable after the state enters the terminal set .

###### Theorem 3.

Suppose that the assumptions 1 and 2 hold, and the conditions in Theorem 2 are satisfied, then the closed-loop system (9) enters the designed set in finite time and converges to if the following condition holds:

 ¯¯¯λ(Q)λ––(P)L2βT(1−β)TLβT−1[L2βTLβT−1δ2+2[(1−β)M+β]αϵδ]≤λ––(Q)n¯¯¯λ(P)(n+1)βT(αϵ−δ)2,∃n∈N. (33)
###### Proof.

This theorem will be proved by two steps.

Step 1: For all initial state , we aim to show the state trajectory enters in finite time. In this situation, we construct an error term of two Lyapunov functions as follows . Expanding this term yields

 Δ~J(x(s;tk),u(s;tk))=∫tk+Ttk∥~x(s;tk)∥2Q+∥~u(s;tk)∥2Rds+∥~x(tk+T;tk)∥2P−∫tk−1+Ttk−1∥^x∗(s;tk−1)∥2Q+∥^u∗(s;tk−1)∥2Rds−∥^x∗(tk−1+T;tk−1)∥2P. (34)

Substituting  (12) to the above equation, we can obtain that

 Δ~J(x(s;tk),u(s;tk))=∫tk+Ttk−1+T∥~x(s;tk)∥2Q∗% ds+∫tk−1+Ttk∥~x(s;tk)∥2Q−∥^x∗(s;tk−1)∥2Q+∥~u(s;tk)∥2R−∥^u∗(s;tk−1)∥2Rds−∫tktk−1∥^x∗(s;tk−1)∥2Q+∥^u∗(s;tk−1)∥2Rds+∥~x(tk+T;tk)∥2P−∥^x∗(tk−1+T;tk−1)∥2P. (35)

Note from Lemma 1 that . Taking integral from to of the above inequality yields

 ∫tk+Ttk−1+T˙V(~x(s;tk))ds=∥~x(tk+T;tk)∥2P−∥~x(tk−1+T;tk)∥2P≤−∫tk+Ttk−1+T∥~x(s;tk)∥2Q∗ds. (36)

Applying this fact to , it can be shown that

 Δ~J(x(s;tk),u(s;tk))≤∫tk−1+Ttk∥~x(s;tk)∥2Q−∥^x∗(s;tk−1)∥2Qds−∫tktk−1∥^x∗(s;tk−1)∥2Q+∥^u∗(s;tk−1)∥2Rds. (37)

To analyze the above inequality, we firstly consider the term

 A=∫tk−1+Ttk∥~x(s;tk)∥2Q−∥^x∗