Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control

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1 International Journal of Automation and Computing 04(2), April 2007, DOI: /s Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control Xiao-Bing Hu 1 Wen-Hua Chen 2 1 Department of Informatics, University of Sussex, Falmer, Brighton, BN1 9QH, UK 2 Department of Aeronautical and Automotive Engineering, Loughborough University, LE11 3TU, UK Abstract: This paper proposes a new method for model predictive control (MPC) of nonlinear systems to calculate stability region and feasible initial control profile/sequence, which are important to the implementations of MPC. Different from many existing methods, this paper distinguishes stability region from conservative terminal region. With global linearization, linear differential inclusion (LDI) and linear matrix inequality (LMI) techniques, a nonlinear system is transformed into a convex set of linear systems, and then the vertices of the set are used off-line to design the controller, to estimate stability region, and also to determine a feasible initial control profile/sequence. The advantages of the proposed method are demonstrated by simulation study. Keywords: Model predictive control (MPC), stability region, terminal region, linear differential inclusion (LDI), linear matrix inequality (LMI). 1 Introduction Model predictive control (MPC) has been widely adopted in industry, and stability of MPC has attracted much attention of researchers in the past decades [1 4]. Terminal penalty is a most widely used technique to guarantee stability of MPC, which introduces a terminal weighting term in the performance index and (or) imposes extra constraints on terminal state during on-line optimization. Terminal penalty technique has achieved a great success in both linear systems, e.g., see [5] and [6], and nonlinear systems, e.g., see [7-10]. For constrained nonlinear systems, a terminal equality constraint was firstly used to establish stability under some assumptions in [7]. That means the terminal state is required to arrive at a specific point in the state space during on-line optimization. However, solving a nonlinear optimization problem with equality constraints is very time-consuming, and therefore is difficult to finish in a given time period. Furthermore, the stability region of the proposed MPC is very small. To avoid this, a dual mode control scheme was proposed in [8]. This method employs a local linear state feedback controller and a receding horizon controller, which replaces terminal equality constraints with terminal inequality constraints. The receding horizon controller is used to drive the terminal state into a terminal region determined by the terminal inequality, and then the local linear controller is employed to guarantee stability. Obviously, the advantage of MPC is lost when the local controller is activated. Recently, Reference [9] proposed a quasi-infinite MPC algorithm. Different from the dual mode control scheme in [8], the local linear state feedback controller is just used to calculate the infinite horizon cost of nonlinear system starting from terminal region. To guarantee stability, a terminal cost which covers this infinite horizon cost is added into the performance index of MPC. Manuscript received February 14, 2006; revised November 22, This work was supported by an Overseas Research Students Award to Xiao-Bing Hu. *Corresponding author. address: Xiaobing.Hu@sussex.ac.uk Therefore, the local controller is called a virtual linear stabilizing controller, and the advantage of MPC never loses until the system arrives at the equilibrium. Reference [10] even applied terminal penalty to more complicated nonlinear systems where computational delay and loss of optimality must be considered. The terminal region discussed in the above papers refers to a region where once the terminal state arrives under the control sequence yielded by solving online optimization problem, there exists a terminal control sequence, MPCbased or not, to steer the system state to the equilibrium. This is quite different from the definition of stability region, which is a set of initial states from which the state trajectory, under the control sequence yielded by solving online optimization problem, will arrive in the terminal region by the end of receding horizon. Terminal region can be used as an estimation of stability region, just as the above papers do, because it is included in the associated stability region, but usually it is conservative due to the gap between terminal region and stability region. As will be proved later in this paper, a method which distinguishes stability region from terminal region can make a better estimation of stability region. Another issue worthy of investigation is the importance of initial control profile/sequence to start online optimization. Simply speaking, a properly chosen initial control profile/sequence can help to make good estimation of stability region, and also to improve computational efficiency of MPC. For MPC of nonlinear systems, due to the heavy computational burden of solving online optimization problem, computational delay is normally too large to be ignored. Sometimes, a sampling time interval runs out even before any feasible solution has been found, let alone optimal ones. In such a case, a properly chosen feasible initial control profile/sequence is crucial to successful implementations of MPC to nonlinear systems. Actually, the idea of distinguishing stability region from terminal region and the importance of initial control pro-

2 196 International Journal of Automation and Computing 04(2), April 2007 file/sequence have already been studied in some papers on MPC for linear systems, e.g., see [11] and [12]. This paper aims to address the same issues of MPC for nonlinear systems, where both estimation of stability region and feasible initial control profile/sequence are practically more important than in the case of linear systems. In the proposed method, instead of using terminal region as estimation of stability region, an offline algorithm is introduced to estimate stability region, and at the same time, to find a series of state feedback control laws which are used to calculate feasible initial control profile/sequence. The remainder of this paper is organized as follows. Constrained MPC problem for nonlinear systems is formulated in Section 2. The new method is described in Section 3. Stability and feasibility are analyzed in Section 4. Section 5 reports some simulation results. The paper ends with some conclusions in Section 6. 2 Problem formulation Consider a nonlinear system ẋ(t) = f(x(t), u(t)), x(t 0) = x 0 (1) subject to control constraints u(t) U (2) where x R n and u R m are state and control vectors, respectively, and 0 U R m is a compact and convex set. In this paper, hatted variables are used in the receding horizon time frame, in order to distinguish the real variables. In general, a nonlinear MPC problem can be stated as: for any state x at time t, find a continuous function û(τ; x(t)) : [t, t + T ] U, in a receding horizon time frame T, such that the performance index J = g(ˆx(t + T )) + Z T 0 (ˆx(t + τ) T Qˆx(t + τ)+ û(t + τ; x(t)) T Rû(t + τ; x(t)))dτ (3) is minimized, where Q 0 and R > 0 are weighting matrices, and û( ; x(t)) is the control profile. û depends on the state measurement x(t) at time t. It is required that g(x) should be a continuous differentiable function of X, g(x) = 0 and g(x) > 0 for all 0 x R n. A typical choice of g(x) is given by g(x(t)) = x(t) T P x(t) (4) where P R n n is a positive definite matrix. The above MPC problem can be mathematically formulated as min J (5) û(τ,x(t)):(t,t+t ) subject to the system dynamics (1), input constraints (2) and terminal state constraint ˆx(t + T ) v, where ν is a terminal region. Let the optimal solution to the optimization problem (OP) (5) be denoted as û. Then the nonlinear MPC law is determined by u(t) = û (t, x(t)). (6) Similar to [10], the following assumptions on the system (1) are imposed: Assumption 1. f: R n R m R n is twice continuously differentiable and f(0, 0) = 0, 0 R n is an equilibrium of the system with u = 0. Assumption 2. System (1) has a unique solution for any initial condition x 0 R n and any piece-wise continuous and right-continuous u( ) : [0, ) U. Assumption 3. The nonzero state of system (1) is detectable in the cost. That is, Q 1/2 x 0 for all nonzero X such that f(x, 0) = 0 [13]. Assumption 4. All states, x(t), are available. Basically, to solve the OP (5), a feasible initial control profile needs to be determined online, such that terminal constraint will be satisfied and stability can then be guaranteed. Let δ denote a sampling time interval and δ < T. Usually, the sub-profile û initial (τ; x(t+δ)) : [t+δ, t+t ] U of an initial control profile can simply inherit the subprofile û (τ; x(t)) : [t + δ, t + T ] U of the last optimal solution to the OP (5), and only the sub-profile û initial (τ; x(t + δ)) : [t + T, t + T + δ] U, i.e., the initial terminal control, needs to be determined online according to stability requirements. Starting from this initial control profile, the result of online optimization can usually steer the state trajectory into the terminal region ν by the end of receding horizon. In most literature, terminal region is directly used as an estimation of stability region, but little information is given about which state out of terminal region can be chosen as initial state. Clearly, it could be very conservative to use terminal region, which is related to terminal state, to estimate stability region, which is defined as a set of initial states. The basic idea of the new method proposed in this paper is to distinguish estimated stability region from terminal region, and then calculate a feasible initial control profile û initial (τ; x(t))[t, t + T ] U which will lead to a feasible solution to the OP (5) to steer any state from estimated stability region into terminal region by the end of receding horizon. However, it is not an easy task to online calculate feasible initial control profile, particularly for nonlinear systems. To avoid this, global linearization and linear differential inclusion (LDI) techniques are adopted to transform the nonlinear MPC problem given by (1) (6), to make it possible to offline determine feasible initial control laws. Firstly, a LDI is defined as h x i ẋ Θ, u» x(0) u(0) =» x0 where Θ R n (n+m). Consider system (1). Suppose for each [x(t), u(t)] and t, there is a matrix G(x(t), u(t)) Θ such that» x(t) f(x(t), u(t)) = G(x(t), u(t)). (8) u(t) Then every trajectory of the nonlinear systems (1) is also a trajectory of the LDI defined by Θ. If we can prove that every trajectory of the LDI defined by Θ has a certain property (e.g., reduces into the terminal region), then every trajectory of the nonlinear system (1) has this property. Conditions that guarantee the existence of such a matrix G u 0 (7)

3 X. B. Hu et al./ Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control 197 are f(0, 0) = 0 and» f f Θ for all x(t), u(t) and t. (9) x u By the Relaxation Theorem [14], one may also assume Θ is a convex set for each x(t), u(t) and t. The LDI given by h x i»» x(0) x0 ẋ C 0Θ, = (10) u u(0) is called the relaxed version of LDI (7). Since C 0Θ Θ, every trajectory of the nonlinear system (1) is also a trajectory of relaxed LDI (10). Actually, we will not need the Relaxation Theorem, or rather, we will get it for free in this paper. The reason is that if a quadratic Lyapunov function, e.g., a quadratic performance index as used by MPC in this paper, is adopted to establish some properties for the LDI (7), then the same Lyapunov function establishes the same properties for the relaxed LDI (10) [14]. The properties of every point in C 0Θ, a convex set, can be revealed by studying the properties of the vertices. If for all vertex systems, there exists an initial control profile which can steer any state from estimated stability region into terminal region, then it is feasible to any system within C 0Θ. For more details about the problem formulation, readers are suggested to refer to [10], [14] and [15]. 3 Stability region and new MPC Definition 1. Terminal region ν is defined as a region where once the state ˆx(t + T ) arrives, under the control û (τ; x(t)) : [t, t + T ] U yielded by solution to the OP (5), there exists a control û : [t+t, ] U which can steer the state to the origin. Definition 2. Stability region M refers to a set of initial states from which the optimal state trajectory ˆx( ) : [t, t + T ], under the optimal open-loop control profile û (τ; x(t)) : [t, t + T ] U yielded by solving the OP (5), will arrive in the terminal region ν by terminal time t + T. As mentioned above, many existing MPC methods simply use terminal region as an estimation of stability region. The new MPC proposed in this paper will distinguish estimated stability region from terminal region, particularly for nonlinear systems. The new method is composed of two parts: offline algorithm and online algorithm. The offline algorithm, which is the core of the new method, aims to make an estimation of stability region as large as possible, and at the same time, to find a series of feasible initial control laws which can improve the computational efficiency of online optimization. The online algorithm is used to calculate optimal control profile over receding horizon. 3.1 Offline algorithm 1) Make a global linearization of system (1): f(x(t), u(t)) = [ f» f x(t) x u ]. (11) u(t) 2) Choose a relaxed LDI defined by C 0Θ = [ f x u 0 f u ] (12) such that Condition (9) is satisfied. 3) Suppose C 0Θ has l vertices, and they are V (r) = [A(r) B(r)], r = 1,..., l. Then, we can construct l linear vertex systems ẋ(t) = A(r)x(t) + B(r)u(t), u(t) U; r = 1,..., l. (13) Given a sampling time interval δ, discretize the above continuous-time vertex systems. Suppose the corresponding discrete-time vertex systems are x(k + 1) = Ā(r)x(k) + B(r)u(k), u(k) 2 ū 2, r = 1,..., l (14) with discrete-time performance index J(k) = x(k + N k) T P x(k + N k)+ N 1 X (x(k + i k) T Qx(k + i k) + u(k + i k) T Ru(k + i k)). i=0 (15) 4) Determine a terminal region ν according to any existing MPC method. In this paper, the method reported in [6] is used to determine the terminal region ν based on those vertex systems given in (14): ν = {x R n : x T P x 1} (16) where the matrix P is optimized such that the terminal region ν is as large as possible. For the sake of identification, hereafter, TMPC is used to denote the MPC method adopted to determine terminal region, and SMPC denotes the new MPC proposed in this paper. Based on ν, an estimation of stability region and a feasible initial control profile will be determined in the following steps. 5) Solve the following optimization problem min log(det(s 1 )) (17) S, S subject to " S (Ā(r)N S + Γ (r) S) # T Ā(r) N S + Γ (r) S 0, r = 1,..., l W " # (18) Y S i 0, Y (S i) T jj ū 2 S j, i = 0,..., N 1; j = 1,..., m (19) where W = P 1 (20) S = [S0 T... SN 1] T T (21) Γ (i) = [Ā(r)N 1 B(r)... Ā(r) 0 B(r)] (22) and 0 < S R n n. 6) Estimate the stability region M as M = {x R n : x T Zx 1}, where Z = S 1. (23) 7) Calculate the discrete-time initial control sequence as K(k + i) = S iz, u(k + i k) = K(k + i)x(k), i = 0,..., N 1; k 0 (24) and the associated continuous-time initial control profile is û(τ) = u(k + i k), τ [t + iδ, t + iδ + δ), i = 0,..., N 1. (25)

4 198 International Journal of Automation and Computing 04(2), April Online algorithm 1) Measure the state x(t). Let x(k) = x(t). If x(k) M but x(k) / ν, then calculate the initial control profile according to (24) and (25), in order to steer the system state from M into ν; otherwise, determine the initial control profile according to the TMPC. 2) If x(k) M but x(k) / ν, set the performance index as (15); otherwise, set it according to the TMPC. Then solve the OP (5). Basically, the OP (5) is a quadratic programming problem, which can be solved by some standard algorithms such as active set methods and interior point methods [2]. 3) Let t = t + δ and go to 1). 4 Stability and feasibility Stability is guaranteed by the TMPC which is used to determine the terminal region ν. Based on this ν, the SMPC is implemented to estimate the stability region M as large as possible, and also to guarantee that the system state can be driven from M into ν. The following theorem establishes the feasibility of the SMPC. Theorem 1. For those discrete-time vertex systems given in (14), suppose there exist S and s such that conditions (18) and (19) hold. Then the SMPC proposed in Section 3 is feasible to steer any initial state from M into ν. Proof. Set the initial control sequence according to (24), i.e., u(k+i k) = K(k+i)x(k), K(k+i) = S iz, i = 1,..., N 1. (26) Then one has x(k +N k) = (Ā(r)N + Γ (r)k N (k))x(k), r = 1,..., l (27) where If K N (k) = K(k). K(k + N 1) (28) Z (Ā(r)N + Γ (r)k N (k)) T P (Ā(r)N + Γ(r)K N (k)) (29) then because x(k) T Zx(k) 1, one has 1 x(k) T (Ā(r)N + Γ (r)k N (k)) T P (Ā(r)N + Γ (r)k N (k))x(k) = x(k + N k) T P x(k + N k). This means: if Condition (29) is satisfied, there is at last one control sequence to steer any initial state within M into ν. Using the transforms (20), (23) and (26), one can see that (29) is equivalent to condition (18). Similar to [6], it is easy to prove that condition (19) guarantees that input constraints are satisfied. According to the LDI theory, when the vertex system has a certain property, then any system within the C 0Θ has the same property. Remark 1. Theorem 1 gives an estimation of stability region for initial state instead of terminal region ν for terminal state. It also gives a feasible initial control sequence to steer any state from M into ν. Remark 2. Theorem 1 can not guarantee that starting from the initial control sequence, the solution to the OP (5) will always drive the system state from M to ν. There are two ways to improve Theorem 1. One way is to impose a terminal constraint x(k + N k) ν on the OP (5). The other is to modify the offline algorithm of SMPC such that the optimal control sequence yielded by solving the OP (5) automatically steers terminal state into ν. In general, a larger stability region is achieved by the former, but at the cost of heavier online computational burden. Theorem 2 gives the set by which the system state automatically arrives in ν under the control sequence yielded by solving the OP (5). Theorem 2. Suppose there exist matrices S > 0 and S such that (30) and (31) are obtained. " # Y S i (S i) T 0, Y jj ū 2 j, i = 0,..., N 1; j = 1,..., m S (31) hold where Φ(r) Q N = (QN )1/2 Φ(r)N, Γ (r) Q N = (QN )1/2 Γ (r)n (32) Q N = diag {Q,..., Q}, {z } N 2 Γ (r) N = 6 4 R N = diag {R,..., R} {z } N (33) B(r) Ā(r) B(r) B(r) Ā(r) N 2 B(r) Ā(r) N 3 B(r) B(r) (34) I Φ(r) N = (35) Ā(r) N 1 where S and Γ (r) are given in (21) and (22) respectively. Then the optimal control sequence of SMPC is feasible to automatically steer any initial state from M into ν S (Ā(r)N S + Γ (r)s) T ( Φ(r) Q N S + Γ (r) Q S) T N ST Ā(r) N S + Γ (r)s W 0 0 Φ(r) Q N S + Γ (r) Q S N 0 I 0 S 0 0 (R N ) ; r = 1,..., l, (30) 5

5 X. B. Hu et al./ Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control 199 Proof. According to (26) (28), one can re-write the performance index (15) as If J(k) = x(k) T ((Ā(r)N + Γ (r)k N (k)) T P ((Ā(r)N + Γ (r)k N (k)) + ( Φ(r) N + Γ (r) N K N (k)) T Q N ( Φ(r) N + Γ (r) N K N (k)) + K N (k) T R N K N (k))x(k) (36) Z (Ā(r)N + Γ (r)k N (k)) T P ((Ā(r)N + Γ (r)k N (k))+ ( Φ(r) N + Γ (r) N + Γ (r) N K N (k)) T Q N ( Φ(r) N + Γ (r) N K N (k)) + K N (k) T R N K N (k) (37) then because x(k) T Zx(k) 1, one has 1 J(k) J(k) x (k + N k) T P x (k + N k), where J(k) and x ( ) represent the optimal performance index and the associated state respectively. One can see that if condition (37) is satisfied, the solution to the OP (5) can automatically steer any terminal state from M into ν. By using the transforms (20), (23) and (26), one can see that condition (37) is equivalent to condition (30). Condition (31) guarantees that input constraints are satisfied. According to the LDI theory, when the vertex system has a property, then any system within the C 0Θ has the same property. Remark 3. Theorem 2 also gives an estimation of stability region M, but it is more conservative than Theorem 1. The advantage of Theorem 2 is that it gives a feasible initial control sequence, by which the SMPC can automatically steer the system state trajectory from M to ν. Remark 4. According to Theorem 2, Step 5 in the offline algorithm needs to be modified, i.e., when the optimization problem (17) is to be solved, conditions (30) and (31) instead of (18) and (19) must be satisfied. 5 Simulation results Consider the following system ( ẋ 1 = x 2 + u(µ + (1 µ)x 1) ẋ 2 = x 1 + u(µ 4(1 µ)x 2) or ẋ = f(x, u) (38) which is borrowed from [7] and is unstable for any µ (0, 1). Assume µ = 0.5 in this simulation. The performance index is chosen as (3) with g(x) defined by (4). Weighting matrices are " # Q =, R = 1.0. (39) Input constraint is given as U = {u R 1 u 1}. (40) A sampling time interval δ = 0.1 time-units. The length of receding horizon is 1.5 time-units. In other words, the receding horizon is 15 steps long. In the following simulation study, the MPC in [6] is used to determine terminal region and the new MPC aims to estimate stability region as large as possible. For the sake of identification, the new MPC based on Theorem 1 is denoted as SMPC1, the one on Theorem 2 as SMPC2, while the MPC in [6] as TMPC. Four methods are used to set initial control profiles for TMPC in order to investigate the feasibility of TMPC under computational time limit, and they are denoted as TMPC1, TMPC2, TMPC3 and TMPC4 respectively. Table 1 explains these methods as well as the methods for SMPC1 and SMPC2 to set up initial control profile. In Table 1, K term is the terminal gain determined offline by TMPC, û ( ; x(t 0)) is the optimal control profile at time t = t 0, K stab1 ( ) and K stab2 ( ) are feasible control gain sequences determined offline by SMPC1 and SMPC2 respectively, and Inheriting means to inherit the optimal solution of last run of online optimization except its first element. 5.1 Estimation of stability region Firstly, global linearization technique is used to determine a C 0Θ:» " # f f 0.5u x 1 C 0Θ = =. (41) x u 1 2u 0.5 2x 2 For the sake of simplification, we assume x i [ 1 1], i = 1, 2. (42) Therefore, C 0Θ has 8 vertices. Terminal region, estimated stability region and feasible initial control laws are then calculated based on the vertices of C 0Θ. Fig. 1 gives the terminal regions. Each dashed terminal region is related to a certain vertex system, while the solid region is calculated for all 8 vertex systems. All terminal regions are calculated by TMPC. The stability regions estimated by SMPC1 are illustrated in Fig. 2, and the stability regions estimated by SMPC2 in Fig. 3. From Figs. 1 3, one can make the following observations: 1) All estimated stability regions are larger than the associated terminal regions. 2) The terminal region or estimated stability region for a certain vertex system is much larger than that for all 8 vertex systems. Actually, the latter is just a subset of the intersection of all formers. 3) The stability regions estimated by SMPC1 are larger than those by SMPC2. 4) The stability regions estimated by SMPC 2 are similar to the associated terminal regions. This implies that the conditions in Theorem 2 are restrictive. To further enlarge estimated stability region, we introduce an iteration process which repeatedly applies SMPC1 or SMPC2. This means when a new estimation of stability region is made, it is then used as a terminal region to calculate another larger new estimation of stability region. In other words, increasing the length of receding horizon can effectively enlarge estimated stability region. However, for TMPC in [6], the length of receding horizon has no influence on terminal region. This implies TMPC is somehow conservative and unreasonable. Fig. 4 gives the result of repeating SMPC Online performance The following simulation study is conducted with two different initial states: Case 1, x 0 = (0.2, 0.2); Case 2, x 0 = (0.5, 0.6). The control performances are given in

6 200 International Journal of Automation and Computing 04(2), April 2007 Figs. 5 and 6, where dashed lines are related to Case 1, while solid lines to Case 2. Suppose a sampling time interval (0.1 time-units) is long enough for solving the online optimization problem, i.e., feasible global-optimal solution can always be found within a sampling time interval. We then have Fig. 5 and Table 2. Fig. 5 gives the online control performances of TMPC and SMPC, which are almost the same. This is understandable: if a sampling time interval is long enough, because the same performance index is applied to all controllers, they should find the identical optimal control profile, and then achieve the same control performance. Actually, the main difference between these controllers is that they adopt different initial control profiles to start online optimization. Table 2 gives the computational burdens under the assumption that a sampling time interval is long enough. For SMPC, the online optimization starts from a series of fixed initial control laws, which are determined offline in advance, while for TMPC, it starts by following some experiential guidelines, as listed in Table 1. From Table 2, one can reach the following conclusions: 1) TPMC1, TPMC2 and TMPC4 take relatively less computational time than TMPC3, SMPC1 and SMPC2; the possible reason is because TPMC1, TPMC2 and TMPC4 inherit the optimal solution of last run of online optimization. 2) Generally, SMPC2 completes online optimization faster than SMPC1, due to the reason discussed in Remark 2 in Section 4. 3) The average computational time associated with a long simulation time is less than that with a short one, because at the beginning of simulation, the system state is far away from the origin and consequently it is more difficult to find out the optimal solution. A more general situation for nonlinear systems is that the online optimization can probably not be completed in a sampling time interval. For example, if a time-unit is 10 seconds, then a sampling time interval is just 1 second, less than those maximum computational times given in Table 2. If the online optimization can not be completed in time, because no feasible initial control profile is available, TMPC just simply adopts the latest solution to determine the actual control signal, no matter whether it is feasible or not. SMPC1 uses the latest solution if and only if this solution is feasible, i.e., it can drive the state trajectory into the terminal region at the end of receding horizon. Otherwise, the associated initial control laws, which have been determined offline, are used to calculate the actual control signal. For SMPC2, starting from the offline-determined feasible initial control laws, any half-done solution to the OP (5) is feasible to stabilize the system. As shown in Fig. 6, TMPC can not stabilize the system when the sampling time interval is 1 second and the initial state is out of the terminal region, while both SMPC1 and SMPC2 still have good online control performances. Fig. 6 demonstrates that a feasible initial control profile is very important to the implementations of MPC for nonlinear systems. A practicable MPC scheme for general nonlinear systems is proposed in this paper. Simulation study illustrates that the main advantages of the new method include: 1) different from many existing MPC methods, the new method distinguishes estimated stability region from conservative terminal region, and stability region can be estimated offline as large as possible; 2) a feasible initial control profile can be determined off-line to guarantee stability when computational time limit presents; 3) global optimum to online nonlinear optimization is not necessary to establish stability. Fig. 1 Terminal regions Fig. 2 Stability regions under SMPC 1 6 Conclusions Fig. 3 Stability regions under SMPC 2

7 X. B. Hu et al./ Model Predictive Control of Nonlinear Systems: Stability Region and Feasible Initial Control 201 Table 1 Methods to set initial control profiles Beginning Initial control profile û initial ( ; x(t)) time TMPC 1 TMPC 2 TMPC 3 TMPC 4 SMPC 1 SMPC 2 t = t 0 [0,..., 0] [0,..., 0] [0,..., 0] û ( ; x(t 0)) K stab1 ( ) x(t 0) K stab2 ( ) x(t 0) t > t 0 [Inheriting, K termx(t)] [Inheriting, 0] [0,, 0] [Inheriting, K termx(t)] K stab1 ( ) x(t) K stab2 ( ) x(t) Table 2 Comparison of computational time (the sampling time is long enough) Simulation time (time-unite) Maximum Case 1: x 0 = (0.2, 0.2); computational time(s) Case 2: x 0 = (0.5, 0.6). Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Case 1 Case 2 Average TMPC computational time of TMPC a run of online OP TMPC solver(s) TMPC SMPC SMPC Fig. 4 Stability regions under iteration of SMPC 1 Fig. 6 Control/state profiles under different MPC schemes with a sampling time of 1 second References Fig. 5 Control/state profiles under different MPC schemes with a sampling time long enough [1] D. W. Clarke. Advances in Model-based Predictive Control, Oxford University Press, Oxford, UK, [2] J. M. Maciejowski. Predictive Control with Constraints, Pearson Education, UK, [3] D. Q. Mayne, J. B. Rawlings, C. V. Rao, P. O. M. Scokaert. Constrained Model Predictive Control: Stability and Optimality. Automatica, vol. 36, no. 6, pp , [4] C. E. Garcis, D. M. Prett, M. Morar. Model Predictive Control: Theory and Practice A Survey, Automatica, vol. 25, no. 3, pp , [5] R. R. Bitmead, M. Gevers, V. Wertz. Adaptive Optimal Control: The Thinking Man s GPC, Prentice-Hall, New York, [6] J. W. Lee, W. H. Kwon, J. Choi. On Stability of Constrained Receding Horizon Control with Finite Terminal Weighting Matrix. Automatica, vol. 34, no. 12, pp , [7] D. Q. Mayne, H. Michalska. Receding Horizon Control of Nonlinear Systems. IEEE Transactions on Automatic Control, vol. 35, no. 7, pp , 1990.

8 202 International Journal of Automation and Computing 04(2), April 2007 [8] H. Michalska, D. Q. Mayne. Robust Receding Horizon Control of Constrained Nonlinear Systems. IEEE Transactions on Automatic Control, vol. 38, no. 11, pp , [9] H. Chen, F. Allgower. A Quasi-infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability. Automatica, vol. 34, no. 10, pp , [10] W. H. Chen, D. J. Balance, J. O Reilly. Model Predictive Control of Nonlinear Systems: Computational Burden and Stability. IEE Proceedings-Control Theory and Applications, vol. 147, no. 4, pp , [11] W. H. Chen. Maximisation of Feasibility/Stability Regions of Model Predictive Control for Constrained Linear Systems. IEE Proceedings-Control Theory and Applications, vol. 149, no. 3, pp , [12] W. H. Chen, X. B. Hu. Model Predictive Control Algorithm with Nonlinear Terminal Control. International Journal of Robust and Nonlinear Control, vol. 14, no. 4, pp , [13] R. Sepulchre, M. Jankovic, P. V. Kokotovic. Constructive Nonlinear Control, Springer-Verlag, Berlin, [14] S. Boyd, L. E. Ghaoui, E. Feron, V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. The Society for Industry and Applied Mathematics, Philadelphia, [15] W. H. Chen, D. J. Balance, J. O Reilly. On Attraction Domain of Model Predictive Control of Nonlinear Systems with Input/State Constraints. Technical Report: CSC99009, [Online], Available: Publications/Reports/csc99009.ps, Nov [16] L. Magni, L. Sepulchre. Stability Margins of Nonlinear Receding-horizon Control via Inverse Optimality. Systems & Control Letters, vol. 32, no. 4, pp , [17] S. J. Qin, T. A. Badgwell. An Overview of Nonlinear Model Predictive Control Applications. In Proceedings of International Symposium on Nonlinear Model Predictive Control: Assessment and Future Directions, Ascona, Switzerland, pp , [18] J. A. Rossirer, J. R. Gossner, B. Kouvaritakis. Infinite Horizon Stable Predictive Control. IEEE Transactions on Automatic Control, vol. 41, no. 10, pp , [19] E. Mosca, J. Zheng. Stable Receding of Predictive Control. Automatica, vol. 28, no. 6, pp , [20] H. Demircioglu, D. W. Clarke. CGPC with Guaranteed Stability Properties. IEE Proceedings-Control Theory and Applications, vol. 139, no. 4, pp , [21] Y. I. Lee, B. Kouvaritakis. Linear Matrix Inequalities and Polyhedral Invariant Sets in Constrained Robust Predictive Control. International Journal of Robust and Nonlinear Control, vol. 10, no. 13, pp , [22] P. O. M. Scokaert, J. B. Rawlings, E. S. Meadows. Discretetime Stability with Perturbations: Application to Model Predictive Control. Automatica, vol. 33, no. 3, pp , Xiao-Bing Hu received his B. Sc. degree in aviation electronic engineering at Civil Aviation Institute of China, Tianjin, China, in 1998, the M. Sc. degree in automatic control engineering at Nanjing University of Aeronautics & Astronautics, Nanjing, China, in 2001, and the Ph.D. degree in aeronautical and automotive engineering at Loughborough University, UK, in He is currently a research fellow in Department of Informatics at Sussex University, UK. His research interests include predictive control, artificial intelligence, air traffic management, and flight control. Wen-Hua Chen received his M. Sc and Ph. D. degrees from Department of Automatic Control at Northeast University, China, in 1989 and 1991, respectively. From 1991 to 1996, he was a lecturer in Department of Automatic Control at Nanjing University of Aeronautics & Astronautics, China. He held a research position and then a lectureship in control engineering in Center for Systems and Control at University of Glasgow, UK, from 1997 to He holds a senior lectureship in flight control systems in Department of Aeronautical and Automotive Engineering at Loughborough University, UK. He has published one book and more than 80 papers on journals and conferences. His research interests include the development of advanced control strategies and their applications in aerospace engineering.