Robust Model Predictive Control of Networked Control Systems under Input Constraints and Packet Dropouts

and Applied Analysis 3 data that will be sent to the state observer. If the data is not updated, the control input will keep unchanged. More important is that taking the packet dropouts into account is consistent with the real situation design in industry. Therefore, signals transmitted through the communication network to the estimated state observer can be expressed as u (k) = δ (k) u c (k) , (9) where u c (k) is the control input that is computed by using the feedback gain K(k) which is determined at each instant k. u(k) is the actual measurement signal of u c (k) and transmitted to the plant (1) and state observer (4).We assume that the stochastic variable δ(k) satisfies Prob{δ(k) = 1} = E{δ(k)} = δ and Prob{δ(k) = 0} = 1 − δ. Let δ(k) = δ(k) − δ. Then, E{δ(k)} = 0 and E{δ2(k)} = δ(1 − δ) ≜ φ. In a similar situation, we consider stochastic variables α(k) and β(k) to describe successive packet dropouts in a random way: u (k) = β (k) u c (k) , y c (k) = α (k) y (k) , (10) where u c (k) is the control input, u(k) is the actual measurement signal of u c (k)which is transmitted to the plant (1), and y c (k) is the actual signal of y(k) which is transmitted to the dynamic output controller, respectively. We assume that the stochastic variables α(k) and β(k) satisfy Prob{α(k) = 1} = E{α(k)} = α and Prob{α(k) = 0} = 1 − α and Prob{β(k) = 1} = E{β(k)} = β and Prob{β(k) = 0} = 1 − β, respectively. Let ?̂?(k) = α(k) − α and β(k) = β(k) − β, respectively. Then, E{?̂?(k)} = E{β(k)} = 0, E{?̂?2(k)} = α(1 − α) ≜ l, and E{β 2 (k)} = β(1 − β) ≜ m , respectively. The following Lemma is used in the process of proof. Lemma 1. Let F and G and H and E be the real matrices of appropriate dimensions; then F + GHE + ETHTGT < 0 if and only if there exists a positive scalar ε > 0 such thatF+εGG+ εE T E < 0 or, equivalently,


Introduction
Model predictive control (MPC) [1,2] is an important method to handle control problems with systems having input, state, and output constraints [3][4][5][6][7].Its current control action is obtained by solving an online, at each time instant, open-loop constrained infinite horizon optimization problem.The current state of the system is treated as the initial state of optimal control problems [8], and only the first optimal control sequence is implemented [9].In this line of the research, a model is used to predict the future behavior of the real system and obtain an optimal control sequence which satisfies the input, state, and output constraints.So the model quality is very vital in the robust MPC.
From such a viewpoint, it is a significant problem to develop the MPC algorithms, which are robust against model uncertainties, and guarantee a certain control performance objective [10][11][12][13][14][15][16][17].The type of MPC has been studied for many years [18][19][20][21].Recently, MPC is an efficient method in automatic control theory and industrial processes.In [22], a robust MPC is presented for a class of uncertain systems and then applied to angular positioning system.Besides, the robust MPC developed based on explicit model uncertainty descriptions has been proposed [23].As one of such descriptions, polytopic system model is considered to be an effective one for the uncertainty modeling of linear timevarying (LTV) systems.
With the increasing requirement of reliability, environmental sustainability, and profitability, we begin to apply the communication networks to the practical industrial process [24][25][26][27][28][29][30][31].However, in the real systems, the measurement data may be transferred through multiple sensors.The sensors are connected to the controller via a network, which is shared by other networked control systems.Due to sensor aging and sensor temporal failure, the measured data may be transferred through many sensors in the real process via networks so that successive packet dropouts are unavoidable in the real industry process [32][33][34].Moreover, another significant factor, input constraints, which stems from the restrictions of engineering equipment, cannot be ignored in industrial processes.Therefore, the robust MPC controller should be developed by considering packet dropouts and constraints.
In this paper, we describe an industrial process system as the linear time-varying system with packet dropouts from the MPC controller and dynamic output controller to the plant.Furthermore, a Bernoulli random binary distribution with known probabilities is used to describe the packet dropouts Notation.The notation used in the paper is standard.The superscript "" represents the matrix transposition and R   shows the   -dimensional Euclidean space.And  and 0 denote the identity matrix and zero matrix, respectively.The notations  > 0 and  > 0 mean that  and  are real symmetric and positive definite; the notation ‖‖ refers to the norm of matrix  that is defined by ‖‖ = √ tr(  ), where "tr" stands for the trace operator.‖ ⋅ ‖ 2 refers to the usual Euclidean vector norm.The symbol " * " denotes the elements below the main diagonal of a symmetric block matrix.Prob{⋅} means the occurrence probability of the event "⋅".{} and { | } represent the expectation of event  and the expectation of  conditional on , respectively.

Problem Formulation
First, we consider the linear time-varying system as follows: where () ∈ R   is the plant state vector; () ∈ R   is the output of the plant; () ∈ R   is the control input; (), (), and  are system parameters.Ω is a given set.For the polytopic systems, the set Ω is the polytope: where Co represents the convex hull.Besides, if [() ()] ∈ Ω, then for   ≥ 0,  = 1, 2, . . ., , and . When  = 1, it means that the system in (2) is becoming a linear time-invariant system.
In this paper, the system matrices [() ()] are known clearly.However, the future matrices [( + ) ( + )],  ≥ 1, are indefinite but well known which change in a given polytope set Ω. We assume that the linear discrete-time system (1) has input constraints [35], which satisfies at each instant  ≥ 0, as follows: where  max refers to the peak bound of the input .
In this paper, we consider a state observer to estimate the state of the plant (1) as follows: x ( + 1) =  () x () +  [ () − x ()] +  ()  () , where x() ∈ R   refers to the estimated state of () and  refers to the state observer gain.It is assumed that the initial estimate state x0 = x(0) and the initial output  0 = (0) are known exactly.Note that system matrices [() ()], output (), and the estimated state x(), at each instant , are well known so that x( + 1) can be calculated by using (4).However, the estimated state x( +  + 1),  ≥ 1, is uncertain.The MPC controller is to be determined as follows: where () is the feedback control gain at time instant . In where   ,   , and   are the unknown parameters of dynamic output controller.
Combining (1), (6), and (10), the augmented system is given as follows: where In practical industrial processes, we have to not only consider the uncertainties in (2), but also consider a more important factor that is data loss.Data losses are inevitable in communication networks, because the measurement data may be transferred through many sensors in the real process.
Here, we consider a stochastic variable () to describe successive packet dropouts in a random way.The characteristic of the successive packet dropouts is the latest received Abstract and Applied Analysis 3 data that will be sent to the state observer.If the data is not updated, the control input will keep unchanged.More important is that taking the packet dropouts into account is consistent with the real situation design in industry.Therefore, signals transmitted through the communication network to the estimated state observer can be expressed as where   () is the control input that is computed by using the feedback gain () which is determined at each instant .() is the actual measurement signal of   () and transmitted to the plant (1) and state observer (4).We assume that the stochastic variable () satisfies Prob{( In a similar situation, we consider stochastic variables () and () to describe successive packet dropouts in a random way: where   () is the control input, () is the actual measurement signal of   () which is transmitted to the plant (1), and   () is the actual signal of () which is transmitted to the dynamic output controller, respectively.We assume that the stochastic variables () and ( The following Lemma is used in the process of proof.
Lemma 1.Let  and  and  and  be the real matrices of appropriate dimensions; then  +  +       < 0 if and only if there exists a positive scalar  > 0 such that + −1   +    < 0 or, equivalently,

Infinite Horizon Robust Performance Objective Analysis
In this paper, the aim of the robust MPC is to determine the model predictive control law {( | ), ( + 1 | ), . . ., ( +  | )},  ≥ 1, which regulates the initial state of system reaching the origin, making the robust performance objective minimum in the worst case and ensuring that the closedloop system is asymptotically stable.In order to determine the control input () under input constraints, we minimize the performance objective  ∞ () through minimizing its upper bound.Consider the performance objective by minimizing the infinite horizon objective function  ∞ () as follows: min where in which  1 and  2 are the positive-definite matrices that should be given firstly.It is worth noting that the choices of  1 and  2 are unclear, which are usually adjusted according to the practical situation.
Proof.Firstly, we construct a Lyapunov function ( x()) = x () x(), where  > 0 is a symmetric matrix.We assume that {( x(∞ | ))} = 0.By summing (14) from  = 0 to  = ∞, we can obtain the following inequality: Let  be the upper bound of  ∞ () and we have the following inequality: In a similar situation, we consider transferring the minimization of  ∞ () into minimizing its upper bound.Therefore, we assume that the following inequality holds for all ( | ), ( | ),  ≥ 0, satisfying the input constraints

State Observer Design
In this section, we consider the design of state observer.The form of state observer is (4).Therefore, the state estimation error is () = () − x().Combining (1) with (4), we can derive the error dynamics as follows: Next we use LMI to determine the state observer gain .First, we should construct a Lyapunov function Ṽ(()) as follows: Ṽ ( ()) =   () P () , P > 0. ( We assume that the Lyapunov function Ṽ(()) satisfies the following inequality: where  (0 <  < 1) is a decay rate and L ≥ 0 is a weighting matrix which are determined by the designer.According to the assumption (20), the state observer gain  can be obtained by the following theorem.
Theorem 2. If there exist P > 0 and Q = P, one can derive the following inequality: where the state observer gain is derived by  = P−1 Q.Therefore, the estimated state x can track the original system states  and x() → () when  → ∞.
Proof.Inequality ( 20) is equivalent to where Π() = () − .However, inequality (22) can be converted into the following form: By using Schur complement, we can obtain the following LMI constraint: Let Q = P; inequality ( 24) is equivalent to It is established for all [() ()] ∈ Ω.And the state observer gain  is obtained by  = P−1 Q.

MPC Controller Design
Now our goal is to design a robust MPC to generate an optimal control law so that the performance objective can be achieved.In the following, Theorem 3 is given to guarantee that the desired performance objective can be implemented by computing () in ( 5) for linear discrete-time systems (1) under constraints (3) and packet dropouts (9).We can obtain the following theorem.
Theorem 3. Assume that the uncertainty Ω is a prescribed polytope and the probabilities  take values from 0 to 1.The control input (5) under a series of constrained input (3) that minimizes  ∞ () is obtained by where  > 0 is symmetric and the upper bound  of  ∞ () can be solved by the following minimization problem: subject to where  =  and  > 0 is a positive scalar and Γ =    +     .
At each future time instant, there exists the optimal control law to minimize the performance objective.Finally, the closed-loop system will be asymptotically stable.
Proof.In order to achieve the robust MPC performance objective under input constraint and packet dropouts, there exist three LMIs that need to be feasible.The first LMI is obtained by (16); by minimizing the upper bound of , we can get the second LMI; the third LMI is obtained by solving the input constraint (3).In the following, we give the proof.

Dynamic Output Controller
Theorem 4. Assume that the uncertainty Ω is a prescribed polytope and the probabilities  and  take values from 0 to 1.
The control input (5) under a series of constrained inputs (3) that minimizes  ∞ () is obtained by where  2 > 0 is symmetric and the upper bound  of  ∞ () can be solved by the following minimization problem: subject to where Proof.Let  =  −1 > 0 and, using Schur complement, we can obtain inequality (41).Combining ( 10) and ( 7) with ( 18), we have by premultiplying and postmultiplying (46) by   and , respectively.By further transformation, we can achieve where  = [ where , and  > 0 is a positive scalar.

Simulation Results
Consider the following uncertain polytope system: where the polytope formed by the two local discrete models is as follows: In Theorem 2, the   decay rate  and weighting matrix L are given as Then, the infinite horizon robust performance objective  ∞ () has the following weighting matrices:  1 = 0.01,  2 = 0.00001.Another weighting matrix Λ is given as where the initial states of the real system (1) and the state observer are  0 = [0.020]  and x0 = [0.02−0.01]  , respectively.
In Figure 1, the two curves are the state of the original closed-loop system.The solid line represents the real state   () ( = 1, 2) of the original closed-loop system and the dashed line denotes its estimated state x () ( = 1,2), respectively, in Figures 2 and 3.In Figure 4, the curve denotes the control input () under the robust performance objective with the constrained input.The curve shows the output of closed-loop system, which converges to zero when  → ∞ in Figure 5.   Consider the following uncertain polytope system: where the polytope formed by the two local discrete models is as follows: In the simulation, let [() ()] = [ 1 ].The input constraint is as follows: Then, the infinite horizon robust performance objective  ∞ () has the following weighting matrices: where the initial states of the real system    In Figure 6, the two curves are the state of the original open-loop system.In Figure 7, the two curves are the state of the original closed-loop system.In Figure 8, the two curves are the state of the dynamic output controller.The curve denotes the control input () under the robust performance objective with the constrained input in Figure 9.The curve represents the control output   () of dynamic output controller in Figure 10.The curve shows the output of closed-loop system, which converges to zero when  → ∞ in Figure 11.

Conclusion
In this paper, we have designed an output-feedback MPC to solve the problem of the robust MPC with input constraints and successive packet dropouts.The method makes use of infinite horizon min-max algorithm with LMI constraints.First, we have constructed a state observer.Then, the optimization problem can be solved by dealing with some LMI constraints.We can obtain the control input sequence by dealing with the infinite robust MPC and input constraint based on the estimated state of observer.From the simulation, the design method of robust MPC with input constraint has been verified feasiblely.As a future work, we will develop the output-feedback MPC algorithm combined with nonlinear MPC and its optimization.

Figure 1 :
Figure 1: The state response of closed-loop system.

Figure 6 :
Figure 6: The state response of open-loop system.
State response of closed-loop system

Figure 7 :
Figure 7: The state response of closed-loop system.

Figure 8 :Figure 9 :
Figure 8: The state response of dynamic output controller.