Robust model predictive control for nonlinear parameter varying systems without computational delay

This article proposes a one‐step ahead robust model predictive control (MPC) for discrete‐time Lipschitz nonlinear parameter varying (NLPV) systems subject to disturbances. Within the proposed design framework, the optimization that generates the MPC policy to be implemented at next time instant is executed in advance during the current sampling period based on future state prediction. This new feature allows avoidance of the online computational delay existing in the traditional MPC settings and improves the control performance. The proposed MPC is proved to be recursively feasible with the guarantee for robust closed‐loop system stability and satisfaction of input and output constraints. A tractable linear matrix inequality (LMI) optimization problem is formulated to compute the control gains at each time instant. The computational complexity of the obtained LMI problem is also analyzed. The one‐step ahead robust MPC is further developed to cover discrete‐time Lipschitz NLPV systems with disturbance compensation. Efficacy and performance improvement of the design are demonstrated through a numerical example and an application to adaptive cooperative cruise control for automated vehicles under variable road geometry.

However, in many cases, the LPV system modeling is conservative and overapproximated. 11 This is because the nonlinear structure is reduced to a linear one without considering properly the dynamics of scheduling parameters, especially when these parameters are system state. To address this issue, this article considers nonlinear parameter varying (NLPV) systems, where a nonlinear dynamic map is incorporated with the classical LPV model to better capture the nonlinear structure.
There is an increasing research interest in NLPV systems. [11][12][13][14][15][16][17][18][19] However, when compared with LPV systems, NLPV systems have more challenging analysis and design problems due to the coexistence of parameter varying and nonlinearity. To address the challenges, several works have been published, which are grouped below according to the types of NLPV systems considered: (i) Lipschitz NLPV systems, where the nonlinear function is Lipschitz with respect to the state variables. Research on such systems include H ∞ robust estimation of the damper force of a vehicle suspension system 12 and -stable controller design. 13 Differential inclusion control is also proposed for one-sided Lipschitz NLPV systems. 14 (ii) Quadratic NLPV systems (which is also called QPV), whose state equation contains crossproducts between state variables. Existing QPV works consider control design with quadratic boundedness, 11 control design with given region of attraction, 15 state observer design, 16 locally asymptotic stabilization, 17 and H ∞ control. 18 (iii) Interval NLPV systems, whose nonlinear functions have upper and lower bounds. Fault diagnosis for a wind farm is designed using the interval NLPV parity equation approach. 19 However, there is no existing work considering optimal control for constrained NLPV systems, let alone robust optimal control for constrained NLPV systems. This motivates us to carry out the research presented in this article, because it is always desirable to operate engineering systems (including NLPV systems) optimally under constraints that represent the physical limits and performance requirements.
Our primary research goal is to address the optimal control for NLPV systems with constraints and disturbances by using model predictive control (MPC). MPC is currently one of the most popular control methods due to its remarkable ability of optimally controlling the systems under constraints. There are many works dedicated to designing MPC for LPV systems without [20][21][22] or with disturbance, [23][24][25] and for quasi-LPV systems, 26 of which more relevant works can be found in the most recent survey article. 27 However, robust MPC for NLPV systems has not been investigated.
Another research goal is to address the issue of time delay caused by online MPC computation. Computation-induced time delay exists in most of the existing MPC designs, except explicit MPC whose control laws are calculated fully offline. 28 The time delay exists because in practice the MPC computation and control policy implementation are executed sequentially. Consequently, the controlled plant needs to wait for the control policy during the period of computation, meaning that the control input has some time delays. Such an input delay is usually ignored in the literature by assuming that the control policy is applied instantaneously. However, the delay is nonnegligible for systems with relatively fast dynamics, for example, vehicles, and it can lead to control performance degradation 29 or even control failure. 30 To avoid the computational delay, existing approaches include advanced-step MPC 31 and one-step ahead MPC. 32 These two approaches follow a similar design principle by solving the future optimization problem in advance during the current sampling period, based on future state prediction under the current control action. However, the advanced-step MPC approach is tailored for Lipschitz nonlinear systems (not LPV) involving nonlinear programming problems. Although the one-step ahead robust MPC 32 is for LPV systems, its design is based on a common Lyapunov function which gives conservative control solutions. In a word, the problem of computation-induced time delay in the MPC for NLPV systems has not been solved.
Considering the background above, the main contributions of this article are summarized below: • A one-step ahead robust MPC is first proposed for discrete-time Lipschitz NLPV systems subject to disturbances and input and output constraints. Compared with the existing one-step ahead robust MPC, 32 the proposed MPC addresses the more challenging control problem for NLPV systems and has reduced design conservativeness by using an LPV form controller derived based on a parameter-dependent Lyapunov function. Rigorous proof of recursive feasibility is provided and a tractable linear matrix inequality (LMI) formulation is derived to compute the optimal control law at each time step. The computational complexity and loss of optimality due to the one-step ahead setting are analyzed in details, which are not seen in the existing one-step ahead robust MPC work. 32 • The proposed one-step ahead robust MPC is extended for Lipschitz NLPV systems with disturbance compensation, when partial disturbance is known and matched to the control input. The obtained MPC includes a feedforward action for actively compensating the known disturbance to achieve enhanced robust system stabilization under constraints. This has not been considered in any existing work for NLPV systems.
• The proposed MPC is applied to the control of vehicle platoons. It addresses the necessary integration of cooperative adaptive cruise control (CACC) and lane keeping (LK) for realizing robustly stable platooning under variable road geometry.
The rest of this article is organized as follows. Section 2 describes the design problem and provides some preliminaries. Section 3 presents the one-step ahead robust MPC design with/without disturbance compensation. Section 4 provides comparative simulations of a numerical example and a vehicle platooning application. Section 5 draws conclusions and provides future perspectives.
Notation: R a×b is a a × b real matrix. || ⋅ || is the 2-norm. ⋆ induces symmetry in a matrix. x ∈ [a, b] indicates that the variable x takes any value satisfying a ≤ x ≤ b. 0 m × n denotes a m × n zero matrix and I m denotes a m × m identity matrix. diag(⋅ , … , ⋅) is a block diagonal matrix. For a matrix P, P > (≥)0 means that it is positive definite (semidefinite), ||x|| 2 P = x ⊤ Px, P jj is the element at position (j, j), and max (P) denotes its largest eigenvalue. x(i|k) represents the predicted state at time k + i based on the state x(k) measured at time k, and u(i|k) represents the control input at time k + i, computed at time k.

PROBLEM STATEMENT AND PRELIMINARIES
Consider a class of discrete-time NLPV systems where x(k) ∈ R n , u(k) ∈ R m , and y(k) ∈ R p are the system state, control input and output, respectively. f (x(k), (k), k) is the known nonlinear function and w(k) ∈ R q is the unknown disturbance. The system matrices A , B and D depend on the time-varying scheduling parameter (k) ∈ R g , which can be system state, inputs, outputs, or some exogenous signals. The set of scheduling parameters satisfy The LPV part of the system is assumed to have a polytopic representation with is replaced by f (x(k)) for simplicity. Assumption 1. The pairs (A l , B l ), l ∈ [1, ], are stabilizable. The state x(k) and scheduling parameters (k) are measurable.
where W is a known constant matrix of compatible dimensions, but not necessarily nonsingular.
This article aims to address the following problem: Problem 1. Under Assumptions 1 and 2 and initial state x(0), design an MPC controller for the system (1) to guarantee robust closed-loop stability under input and output constraints. In other words, at each time instant k, find the optimal control policy u(k) as the solution to the optimization problem below: where u s (i|k) is the sth input, y r (i|k) is the rth output, matrices Q, S > 0 are suitable weights and is some positive constant.
The above optimization problem has a H ∞ -type cost function (2), 25 which differs from the one in the existing robust MPC designs. 21,23,24,32 This cost is similar to the  2 gain used in receding horizon control. 33,34 By using such a cost function, the considered optimization problem is a two-person zero-sum dynamic game, 35 with the controller u(k) being the minimizer player and the disturbance w(k) being the maximizer player.
Hence, minimizing the cost can ensure the closed-loop system robust against any disturbance w(k) ∈ W. However, the cost function cannot be optimized directly as the normal MPC cost due to having infinite number of terms. Hence, this article will derive and minimize the upper bound of this cost. Note that the input and output constraints (4) and (5) are componentwise. This allows imposing different constraints on each individual input and output, which is more realistic than the norm constraints 20,21 in control engineering practice. The input constraint is imposed over the entire horizon, while the output constraint is imposed over the predicted future outputs but not at the current output y(k). This is because the current output cannot be influenced by the planned control action and hence it is meaningless to impose any constraint on it.
Before presenting the design in Section 3 to solve Problem 1, some useful definition and lemmas are recalled below.

Definition 1.
(Robustly positive invariant set 32) A set Ω is a robustly positive invariant (RPI) set for the system Let M and N be real constant matrices and P be a positive definite matrix, then for any scalars > 0 and ≥ max (P), the following inequality holds: 37 Remark 1. The stabilization assumption of all the pairs (A l , B l ), l ∈ [1, ], is standard in the context of polytopic system control. 20,21,32 Availability of full state x(k) and the scheduling parameter (k) for control design is also widely assumed in the existing MPC designs for LPV systems. [20][21][22]38 In the case when only partial state are measurable, dynamic output feedback robust MPC 25,32 or observer-based robust MPC 39 can be pursued. In Assumption 2, the Lipschitz condition is given with a matrix W. This kind of Lipschitz condition is also used in the literature 12,13 because it is less conservative than the traditional one with a positive scalar. 14 Further reduction of the conservativeness may be achieved by defining W as a matrix function of the scheduling parameters. 13 This article considers the unknown bounded disturbance w(k), which can be viewed as a lumped term that captures the total influences of disturbance, model uncertainty and process noise. Several methods for building such lumped disturbance models can be found in chapter 5 of the book. 40 In Problem 1, the proposed design uses a min-max optimization strategy to guarantee system robustness with the price of conservativeness. However, this is the common drawback of existing robust MPC designs. To reduce the design conservativeness, robust adaptive MPC 41 may be used, where the unknown parameters are estimated in real-time.

Robust MPC design
To solve Problem 1, the proposed robust MPC controller takes the following form over the optimization horizon i ≥ 0: where F(k) = ∑ l=1 l (i|k)F k,l and F k, l , l ∈ [1, ], are the controller gains to be determined. Different from the literature 32 where F k, 1 = · · · = F k, , the proposed controller is parameter-dependent and less conservative. The closed-loop system performance under this controller is analyzed in Lemma 3. An LMI optimization problem is also formulated to solve the gains F k, l , l ∈ [1, ]. (6) is robustly stable and satisfies the constraints |u j (k) |≤u s, max , s ∈ [1, m], and |y r (k) |≤y r, max , r ∈ [1, p], if the following optimization problem is feasible: [1, ] s.t.

Proof. (i) Closed-loop stability
Substituting the controller (6) into (1) gives the closed-loop system According to the quadratic boundedness condition, 32 stability can be guaranteed for the closed-loop system (12) under disturbance by ensuring the set Ω = {x(k)|x(k) ⊤ P(k)x(k) ≤ } as a RPI set for the system (1). Define the parameter-dependent Lyapunov function Hence, the closed-loop system (12) is stable, if the following condition holds: A sufficient condition of (13) is given as Since (14) is satisfied if there exists a scalar ∈ (0, 1] such that Let = w −2 max , then the above inequality is rearranged as By using (12), it can be derived that where According to Assumption 2 and Lemma 2, for any positive constants 1 and 2 and ≥ max (P(k + i + 1)), one has Applying (17) to (16) gives Combining (15) and (18) and reformulating it as a quadratic form of the vector [x(i|k) ⊤ −w(k + i) ⊤ ] ⊤ , then it gives where Applying Schur complement 36 to (19) yields A sufficient condition of (20) can be given as Define X l = P −1 l , X j = P −1 j , and = −1 . Substitute P l = X −1 l and P j = X −1 j into (21), and multiply both its sides with diag{ −1∕2 I, −1∕2 I, 1∕2 I, 1∕2 I, 1∕2 I, 1∕2 I, 1∕2 I}. Further using Schur complement, it then yields It is known that there exists a matrix G l satisfying 21 Multiplying both sides of (22) with diag{G ⊤ l , I, I, I, I, I, I, I} and its transpose, respectively, and using (23), then the inequality (8) is obtained by defining Y l = F k, l G l .
(ii) Optimality The proof above implies that under (8), the condition (15) is satisfied. Hence, the following inequality holds Summarizing both sides of (24) from i = 0 to i = ∞ gives This gives an upper bound of the cost function J(k). It can be derived that minimizing J(k) is equal to minimizing subject to the constraint (7). 20 (

iii) Constraint satisfaction
It is straightforward to prove that the input constraint (4) is satisfied if (9) holds. 20,21 Since V(x(i|k)) ≤ , ∀w ∈ W, the output constraint |y r (i + 1|k) |≤y r, max , ∀i ≥ 0, is guaranteed if the following inequality holds Note that |y r (i + 1|k) where C r is the rth row of C. Hence, similar to (18), it can be derived that where 3 and 4 are any given positive constants and r = max (C ⊤ r C r ). Combining (26) and (27) and reformulating it as a quadratic form of the vector [ Applying Schur complement to (28) yields A sufficient condition of (29) can be given as Substituting P l = X −1 l to (30) and using Schur complement results in Multiplying both sides of (31) with diag{G ⊤ l , I, I, I} and its transpose, respectively, and using (23), then it gives Further substituting Y l = F k, l G l to (32) gives the inequality (10).
To complete the proof, the following requirements on the decision matrices and variables are given in (11). Since ≥ max (P(k + i + 1)), the matrix P l is set to satisfy P l ≤ I, which means that X −1 l ≤ −1 I and equivalently X l ≥ I, l ∈ [1, ]. In addition, the matrix U is symmetric and the decision variables and are positive. ▪ In Lemma 3, the LMI condition (7) ensures the optimality, (8) guarantees closed-loop system stability, while (9) and (10) ensure satisfaction of the input constraints (4) and output constraints (5), respectively. Moreover, it is straightforward  (1) is It is not unrealistic to assume that the MPC optimization can be solved within one sampling period (see Section 3.4.2), that is, = t s . Then the system (33) can be rewritten in the discrete-time form as This is equivalent to Hence, at sampling instant k ≥ 0, the control input applied to the plant is u(k − 1) instead of u(k), where u(− 1) = 0. This means that there is always a one-step time delay in the control input, which will cause consequent loss of closed-loop system performance or even stability. To avoid the computational delay, a one-step ahead robust MPC is proposed in the following section.

One-step ahead robust MPC design
In this section, a one-step ahead robust MPC is proposed to solve Problem 1. Its key idea is shown in Figure 1 through comparison with the robust MPC in Section 3.1. In this article, the optimization of computing the MPC policy is assumed to be accomplished within one sampling period. In the one-step ahead robust MPC framework, the optimization task for generating the controller u(k + 1) is shifted to the previous sampling period k. In such case, the optimization is executed in the background parallel to implementation of the current control action. Therefore, it avoids the waiting time caused by the optimization. Different from (6), the one-step ahead robust MPC controller takes the following form over the optimization horizon i ≥ 0: where k ≥ 1 and F(k) = ∑ l=1 l (i|k)F k,l with the design gains F k, l . As shown in Figure 1, at time k, the controller u(k) with the gains F k − 1, l , l ∈ [1, ], obtained at time k − 1 is applied to the system (1). Closed-loop system performance under the controller (34) is analyzed in Theorem 1, where an LMI optimization problem is also formulated to solve the optimal control gains. (34) is robustly stable and satisfies the constraints |u j (k) |≤u s, max , s ∈ [1, m], and |y r (k) |≤y r, max , r ∈ [1, p], if the following optimization problem is feasible:

Theorem 1. Under Assumptions 1 and 2, the system (1) with the controller
where r = max (C ⊤ r C r ), r ∈ [1, p], while , ∈ (0, 1] and i , i ∈ [1,6], are given constants. The controller gains are obtained as F k,l = Y l G −1 l , l ∈ [1, ]. Proof. The one-step ahead robust MPC always considers the system invariance at the next sampling time k + 1, while the invariance at current time k is already guaranteed at time k − 1. Hence, value of the variable i in Problem 1 starts from i = 1. Despite of this change, the proof of Lemma 3 can be directly used to show that the system (1) under the controller (34) is stable with satisfaction of the input and output constraints, given the conditions (36), (37), (38), and (39). However, by replacing i ≥ 0 with i ≥ 1, the optimality condition (7) in Lemma 3 no longer works. The new optimality condition is derived below. When i ≥ 1, the inequality (25) becomes This result in J(k) ≤ V(x(1|k)) ≤ . Hence, minimization of J(k) is equal to minimization of subject to the constraint Since ||w(k) ||≤w max , (41) holds if there is a scalar ∈ (0, 1] such that According to (18), for any positive constants 5 and 6 , it holds that Applying (43) to (42), and formulating it in a quadratic form of the vector [1 −w(k) ⊤ ] ⊤ , then it gives Applying Schur complement to (44) and using = −1 , then it gives By defining X j = P −1 j , then the inequality (35)

One-step ahead robust MPC with disturbance compensation
This section describes design of a one-step ahead robust MPC for discrete-time NLPV systems with partially known disturbance. Such a setting is not impractical. For example, the upstream wind speeds can be measured by LIDAR and used in MPC wind turbine control for compensating their influence. 42 Another example is the vehicle platooning considered in our simulation example 2, where the acceleration of the preceding vehicle can be transmitted via vehicle-to-vehicle (V2V) wireless communication. It has been shown that in general the combination of feedback and feedforward actions will have performance gain in MPC. 43 The considered NLPV systems are presented by where d(k) ∈ R t is the known disturbance. The distribution matrix E is assumed to satisfy the matching condition rank([B E ]) = rank(B ). This enables compensation of d(k) via the control actionū(k). The other variables and matrices are defined the same as in the system (1).
For the system (45), the following one-step ahead robust MPC controller combining feedback and feedforward actions is proposed:ū , substituting the controllerū(k) (46) into the system (45) gives This system is in the same form with the system (1). Therefore, design of the one-step ahead robust MPC controller u(k) follows directly from the results in Section 3.2 considering the cost function where only the feedback control effort u(i|k) is concerned. In this case, the input constraint (4) is shrunk as |u s (k) |≤u s, max , where u s,max =ū s,max − û s,max with |û s (k)| ≤ û s,max . Design details the controller u(k) are not repeated here.
Remark 2. An effective alternative approach to enhance MPC robustness in the presence of system disturbance is the hierarchical control, 44 which combines a high-level MPC controller and a low-level unconstrained controller. The low-level controller works at a faster rate to make the low-level closed-loop system behaviors like the nominal system without disturbance, which facilitates the high-level MPC design. The low-level controller can be a PID controller or an advanced controller such as sliding mode controller (SMC). 45 PID is well-known to have inherent robustness to disturbance because of its integral action. The SMC owns its robustness to disturbance by using a feedforward action, as in this article, to actively compensate the disturbance. [46][47][48] It should be noted that by using either the approach proposed in this article or the hierarchical control, the magnitude of the control signal û(k) in this article (or the low-level controller in the hierarchical control) is always needed to tighten the input constraint of MPC. Compared with PID, one advantage of using active disturbance compensation is the possibility of saving the effort for tuning good PID parameters for NLPV systems. However, to have the measured disturbance d(k) compensated by u(k) using the approach in this article or the existing SMC, a necessary assumption is that d(k) is matched, that is, rank . This means that d(k) acts at the system through the control channels. In the cases when this rank condition is unsatisfied, it is always possible to partition the disturbance into matched and unmatched parts. 45 Then the matched part can be compensated using feedforward action, while the unmatched part can be lumped together with the unmeasured disturbance w(k) and handled through robust design.

Parameter tuning
In Theorem 1, the positive constants i , i ∈ [1,6] are given to ease the solving of the obtained optimization problem. Currently there is no known method to select their values optimally. However, simulation experiments have shown that using different values for them has no significant influence on the control performance. Hence, they are usually chosen as small positive constants in real implementation, as in the simulations presented in Section 4. Similarly, the constants , ∈ (0, 1] are introduced to reduce the design conservativeness and have negligible influence on the control performance. Hence, they can be chosen as any value within (0, 1].

Computational complexity
It is worth to analyze and compare the computational complexity of the robust MPC in Section 3.1 and the one-step ahead robust MPC in Section 3.2. This is realized by analyzing and comparing the computational complexity of the LMI optimization problems in Lemma 3 and Theorem 1. The computational complexity (or number of flops) of solving an LMI can be approximated by Gahinet et al: 49  =  3 , where  is the LMI row size and  is the total number of scalar decision variables. For the LMI optimization problem in Lemma 3, it can be derived that For the LMI optimization problem in Theorem 1, it can be derived that  2 = (10n + m + 2q + 1) 2 + (2n + m + 2np + pq + p 2 ) + 2m + 2, It can be seen that  1 <  2 and  1 =  2 . Hence, the one-step ahead robust MPC has increased computational complexity when compared with the robust MPC. More specifically, the increase is Δ =  2  3 2 −  1  3 1 = [(4n + q + 1) 2 + (m − 1) ] 3 1 . Since the state dimension n is generally not less than the input dimension m, disturbance dimension q, and output dimension p, it is concluded that the computational complexity of the proposed one-step ahead robust MPC  2 depends mainly on the system dimension n and the total scheduling parameters . More specifically,  2 is a function of n 7 5 . To have a more intuitive view, we consider a system with dimensions n = 10, = 2, m = 2, q = 1, and p = 2, then  2 is about 25 gigaFLOPS. For an Intel i9 8-core CPU, the computation performance can be 576 gigaFLOPS per second. Hence, the LMI optimization problem in Theorem 1 can be solved within 0.044 s. The computation time will be much less if solving the LMI optimization problem using a GPU, whose computation performance can be up to 1150 gigaFLOPS. Therefore, the proposed algorithm can be effectively implemented with online computation of the LMI optimization problem.

Loss of optimality
The proposed one-step ahead robust MPC executes the optimization in advance by leveraging the predicted future system state under the current control action. Hence, there is expected to be a loss of optimality when compared with the ideal robust MPC that can be implemented without any computational delay. According to (25) and (40), at time k the costs of the ideal robust MPC and one-step ahead robust MPC satisfy J(k) ≤ V(x(k)) and J(k) ≤ V(x(1|k)), respectively. Hence, the loss of optimality J (k) is bounded as 0 ≤ J (k) ≤ V(x(1|k)) − V(x(k)), which clearly results from the use of x(k) to predict x(k + 1) for computing u(k + 1) in advance. • Existing one-step ahead MPC 32 . It takes the form of (34) but using a constant gain F(k) = F k instead of the parameter-dependent gain as in the Proposed one-step ahead MPC. The gain F k is designed based on the common Lyapunov function method, which is formulated as a special case of the optimization problem in Theorem 1 with F I G U R E 2 Trajectory of displacement x 1 (k) and velocity X 1 = · · · = X , G 1 = · · · = G , and Y 1 = · · · = Y . The optimization problem is solved using the same parameters Q, S, , , and i , i ∈ [1, 6] as the Proposed one-step ahead MPC, and the initial gain F 0 = [− 0.0430 −0.0861].
• Ideal MPC. This is referred to the robust MPC controller with the same gains as the Delayed MPC, but being applied to the system (48) instantaneously without delay. This controller serves as the baseline case to see how much optimality the One-step ahead MPC loses.
The closed-loop systems under four MPC controllers are simulated with the same initial state x(0) = [4 0] ⊤ . It is seen from the trajectories in Figure 2 that all controllers can regulate the displacement x 1 (k) and velocity x 2 (k) of the cart-spring system to the equilibrium (0, 0). Convergence of the Proposed one-step ahead MPC is quicker than the Delayed MPC, and very close to the Ideal MPC. Due to the use of the common Lyapunov method, the Existing one-step ahead MPC is more conservative than the Proposed one-step ahead MPC, and thus has lower converging speed. As observed from Figure 3, all the controllers satisfy the input constraint u(k) ∈ [− 1, 1], but as expected there is a one-step delay in the Delayed MPC. The above results demonstrate efficacy of the Proposed one-step ahead MPC and its advantages over the Existing one-step ahead MPC and Delayed MPC. Moreover, the loss of optimality is negligible compared with the Ideal MPC.

4.2
Example 2: Vehicle platooning under variable road geometry CACC for automated vehicle platooning has gained its popularity due to the great potential in improving traffic throughput, safety and fuel economy. 50 The goal of CACC is controlling the followers in the platoon to track the leader velocity, whilst maintaining a safe intervehicle distance, by using wireless V2V and/or vehicle-to-infrastructure communication networks. Most of the existing CACC designs investigate vehicles on flat roads where the longitudinal control is efficient to realize platooning. This article considers CACC on roads with bank and curvature. In such a context, the vehicle lateral and longitudinal stability are affected by the road conditions and the nonlinear coupling between them. 51 Hence, it is necessary to integrate LK with CACC to achieve robustly stable vehicle platooning. For simplicity and clarity, a two-vehicle platooning system model is built for robust MPC design. This model is directly applicable for multiple vehicles by regarding each two consecutive vehicles as a two-vehicle platooning system where the predecessor acts as the leader. Define p l , v l x , and a l as the position, velocity, and acceleration of the leader, respectively, and p, v x , and as the position, velocity and desired torque of the follower, respectively. The objective of CACC is to regulate the vehicle following errors Δp = p l − p − d safe and Δv = v l x − v x to zero, where d safe is a given safe intervehicle distance. To perform the CACC design, the following error system is used 52 : wherėe y anḋe are the lateral velocity and yaw rate of the follower with respect to the road, respectively. C d is the drag coefficient, air is the air density, A f is the frontal area, and v wind is the longitudinal component of the wind speed. m is the vehicle mass, g is the gravitational acceleration, and c is the rolling resistance coefficient. The signal a l is sent from the leader to the follower via V2V communication network, and thus the disturbance d is known and can be compensated using feedforward action.
To design the LK controller, the lateral tracking error system of the follower is given as 51 where e y and e are the lateral displacement and yaw angle error with respect to the road, respectively. is the front wheel steering angle. c f and c r are the cornering stiffness of the front and rear tires, respectively. l f and l r are the distances from the center of gravity of the vehicle to the front and rear tires, respectively. I z is the yaw moment of inertia. =̇d es is the desired yaw rate calculated bẏd es = v x ∕R with the road radius R. is the road bank angle.
where the integration terms ∫ Δpdt and ∫ e y dt are included to remove steady-state errors of Δp and e y . The overall error system composed of (49) and (50) is given aṡ with , v x ]. Define = 1∕v x , then it holds that ∈ [ , ] with = 1∕v x and = 1∕v x . Under the sampling time t s , the system (51) can be converted into a discrete-time NLPV system where Note that x 6 =̇e =̇−̇d es anḋd es = v x ∕R. Hence, |x 6 |≤r max + v max /R min , where r max is the maximum yaw rate, v max is the maximum longitudinal velocity and R min is the minimum road radius. According to Assumption 2, the Lipschitz constant matrix W can be chosen as This choice of W considers the worst scenario and imposes design conservativeness. To reduce the conservativeness, the nonlinear function f (x(k)) can be formulated in a quadratic form where v x (k) is defined as a scheduling parameter. In such case, the results of NLPV systems with quadratic nonlinearity 15,53 can be adapted for control design. However, it is expected that the design complexity increases.  [10,40] m/s, the maximum yaw rate is r max = 0.03 rad/s and the minimum road radius is R min = 100 m. The input constraints are max = 10 • and max = 6.5 kN, and the output constraint is Δp max = 5 m. Since the combined CACC and LK error system (52) is in the same form as (45), design of the controller u(k) follows the method in Section 3. Comparative simulations are performed for the following three MPC designs: • Delayed MPC. It is referred to the robust MPC controllerū(k) = ∑ l=1 (F k,l (k)x(k) + L k,l d(k)) applied to each follower with one-step delay. The feedback gains F k, 1 and F k, 2 are solved from the optimization problem in Lemma 3 using Q = 100I 8 , S = 0.01I 2 , = 0.01, 1 = 0.01, and i = 0.1, i ∈ [2, 4].
• Ideal MPC. This is referred to the robust MPC controller with the same gains as the Delayed MPC, but applied to each follower instantaneously without delay.
In the simulations, the profiles of leader velocity v l x , road curvature 1/R, and bank angle depicted in Figure 4 are used. The wind speed v wind takes random values within [− 1, 1] m/s. The leader has the initial state (p l (0), v l (0)) = (20, 10). Follower 1 has the initial state (p(0), v(0)) = (9.5, 10) and (e y (0),̇e y (0), e (0),̇e (0)) = (0.1, 0, 0, 0). Follower 2 has the initial state (p(0), v(0)) = (0, 10) and (e y (0),̇e y (0), e (0),̇e (0)) = (0.1, 0, 0, 0). It is seen from Figure 5 that under the One-step ahead MPC and Ideal MPC, the intervehicle distances of Leader & Follower 1 and Followers 1&2, are controlled to be the desired value d safe = 10 m. Moreover, the output Δp(k) is within the given range [− 5, 5] m. However, the intervehicle distance under the Delayed MPC fluctuates around the safe distance. From Figure 6, it is observed that the One-step ahead MPC and Ideal MPC achieves LK with smaller lateral displacement than the Delayed MPC. It can be seen from Figure 7 that the desired torques under the Delayed MPC are larger than those of the other controllers and even violate the input constraint [− 6.5, 6.5] kN at some time instants. Moreover, as shown in Figure 8, there are oscillations in the steering angles of both followers when using the Ideal MPC.
The above simulation results demonstrate that the One-step ahead MPC achieves robustly stable CACC and LK with satisfaction of given input and output constraints, despite with the time-varying leader velocity, road bank and curvature. Moreover, performance of the One-step ahead MPC is almost as good as that of Ideal MPC. However, under the Delayed MPC, the platoon is not stable and there are fluctuations in the steering angles, deteriorating the driving comfort.

CONCLUSION AND FUTURE PERSPECTIVE
This article proposes a one-step ahead robust MPC for discrete-time Lipschitz NLPV systems under disturbances and constraints in inputs and outputs. The one-step ahead manner shifts optimization of generating the next control action in advance to the current time instant, which avoids the computational delay and improves control performance. The proposed design is proved to be recursively feasible and the control gains are obtained via solving an LMI optimization problem. The one-step ahead robust MPC is further extended to having disturbance compensation capability when partial disturbance is known. The computational complexity and loss of optimality by using the one-step ahead setting is also analyzed. Efficacy of the proposed MPC is illustrated by simulations of a cart-spring system and a platoon of three vehicles under time-varying leader velocity and road geometry. The results confirm that the one-step ahead robust MPC achieves better control performance than the robust MPC. The results also show that the proposed MPC has small loss of optimality compared with the ideal robust MPC without computational delay. Future research will focus on solving several open questions: (i) Reducing the design conservativeness by replacing the Lipschitz nonlinearity with less conservative ones such as quadratic nonlinearity. (ii) Developing one-step ahead robust MPC for NLPV systems by considering system uncertainties and disturbances separately. (iii) Developing an observer-based one-step ahead robust MPC for NLPV systems whose state are partially measured and corrupted by noise.