Stability analysis of constrained MPC with CLF applied to discrete-time nonlinear system

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1 . RESEARCH PAPER. SCIENCE CHINA Information Sciences November 214, Vol : :9 doi: 1.17/s y Stability analysis of constrained MPC with CLF applied to discrete-time nonlinear system RUAN XiaoGang 1, HOU XuYang 1,2 & MA HangYing 1 1 Department of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 1124, China; 2 Beijing Aerospace Control Instrument Research Institute, Beijing 1142, China Received December 19, 213; accepted February 21, 214; published online September 5, 214 Abstract The model predictive control (MPC) strategy with a control Lyapunov function (CLF) as terminal cost is commonly used for its guaranteed stability. In most of the cases, CLF is locally designed, and the region of attraction is limited, especially when under control constraints. In this article, the stability and the region of attraction of constrained MPC that is applied to the discrete-time nonlinear system are explicitly analyzed. The region of feasibility is proposed to substitute the region of attraction, which greatly reduces the calculation burden of terminal constraints inequalities and guarantees the stability of the MPC algorithm. Also, the timevariant terminal weighted factor is proposed to improve the dynamic performance of the close-loop system. Explicit experiments verify the effectiveness and feasibility of the relative conclusions, which provide practically feasible ways to stabilize the unstable and/or fast-dynamic systems. Keywords constrained MPC, stability analysis, discrete-time system, control Lyapunov function, region of attraction Citation Ruan X G, Hou X Y, Ma H Y. Stability analysis of constrained MPC with CLF applied to discrete-time nonlinear system. Sci China Inf Sci, 214, 57: 11221(9), doi: 1.17/s y 1 Introduction Model predictive control (MPC) is a very popular control method, which has been widely applied to the chemical process control [1,2]. Its characters of the close-loop optimization and constraint optimization have many advantages, which are of great importance for practical engineering control. Many researchers have tried to apply MPC to more control objects, such as fast-dynamic and/or unstable system. These new control objects require the control methods owning the characters of instantaneity and/or stability, and many MPC algorithms are proposed to solve these problems. In the beginning, a terminal state constraint was imposed such that the terminal state lies in the origin [3]. Michalska and Mayne [4] relaxed the terminal state constraint to be a suitable neighborhood of the origin, and the control law is switched to a locally stabilizing linear controller. Chen and Allgower [5] defined the terminal state penalty matrix of the terminal cost as the solution of a Lyapunov equation, and gave out the prescribed terminal region that is determined by the linear controller around the origin. Magni and Sepulchre [6] guaranteed the stability of the receding horizon control using end Corresponding author ( hopor123@126.com) c Science China Press and Springer-Verlag Berlin Heidelberg 214 info.scichina.coink.springer.com

2 Ruan X G, et al. Sci China Inf Sci November 214 Vol :2 point penalty if a locally stabilizing linear control law is applied at the end of the time horizon. All of the aforementioned methods suppose that the region of attraction or origin is reachable, and the region of attraction is decided by the linear controller around the origin. Primbs et al. [7] proposed a receding horizon control algorithm using a global control Lyapunov function (CLF) to globally stabilize the nonlinear system. The point is that once a global CLF is obtained, stability of the receding horizon controller is guaranteed by including additional state constraints that require the derivative of CLF along the receding horizon trajectory to be negative. This approach is difficult to achieve for its state constraints and the for finding the difficulty of the global CLF. An alternative approach was proposed in [8,9]. A priori CLF as the terminal cost guarantees the stability of proposed approach only if the CLF is an upper bound on the cost-to-go. Jadbabaie et al. [1] gave out the upper bound onthe costtogo, andprovedthat ifthe terminalclfsatisfiesthebound constraint,the stabilityis retained without the use of terminal constraints for unconstrained system. The region of attraction of the constrained MPC was given out in [11 14], and the stability is guaranteed without terminal constraints. The region of attraction proposed in these references is conservatively analyzed, which means that the area out of the region of attraction may be stabilizable, and it is waiting for further research. What is more, relative experiments were not given, and the feasibility of the proposed methods was not verified. These works will be done in this article. The article is organized as follows. The problem setting is described in Section 2. In Section 3, the region of feasibility is inferred from the region of attraction based on the analysis of nonlinear discretetime MPC, the related factors that affect the region of feasibility are analyzed, and the methods to improve the dynamic performance are also discussed. The main results are verified using a discrete-time nonlinear system in Section 4. Finally, the conclusion is presented in Section 5. 2 Problem setting and optimal MPC Consider the nonlinear discrete-time dynamic system x(k +1) = f (x(k),u(k)),k N, (1) where x(k) R n is the system state at sampling time k, x(k+1)is the successor state, and u(k) R m is the control vector at sampling time k. The system is subject to constraints on state and control vector such that x(k) X,u(k) U,k N, (2) where X is a closed set and U is a compact set, both of them containing the origin. The control sequence applied to the system at sampling time k is denoted as u(k) = { u k/k,u k+1/k,...,u k+/k }, (3) where, u j/k,j = k,k+1,...,k+n denotes the predictive control vector at sampling time j based on the information at sampling time k. The corresponding state sequence is denoted as x(k) = { x k/k,x k+1/k,...,x k+n/k }. (4) The first element of x(k) is the state x k/k = x(k) at sampling time k, and the other elements of x(k) can be denoted as x j/k = φ(x j 1/k,u j 1/k ),j = k + 1,...,k + N, where x j/k is the predicted state at sampling time j. The control sequence u(k) is calculated by solving the finite-horizon optimal control problem(fhocp) P N (x(k),k), which is denoted as { } VN o (xo (k),u o (k),k) = min V N (x(k),u(k),k) = min l(x i+k/k,u i+k/k )+F(x N+k/k ), (5) u U u U

3 Ruan X G, et al. Sci China Inf Sci November 214 Vol :3 where, x j/k X,u j/k U,x N/k Ω, l(x,u) = x T Qx + u T Ru is the stage cost function, F (x) = x T Px is the terminal cost function, Ω is the region of attraction, and Q, R, P are coefficient matrixes respectively related to the stage state, stage input, and terminal state. The set of initial state in which the system can be steered into Ω by the FHOCP P N (x(k),k) is denoted as a set X N (Ω). The region of attraction Ω is defined in Assumption 1, and Lemma 1 gives out the stability ofthe MPC algorithm under Assumption 1 [15]. Assumption 1. Let F(x) be a control Lyapunov function (CLF) and the region of attraction Ω is given by Ω = {x R n : F(x) α}, where α > and satisfies that for all x Ω min{f(x,k +1) F(x,k)+l(x,u,k)}. (6) u U Lemma 1. If the system (1,2) is controlled by the optimal control sequence u o (k), which is calculated by solving FHOCP P N (x(k),k), and the terminal state set Ω of the close-loop system satisfies Assumption 1, then the optimal cost V o N (xo (k),u o (k),k) is a Lyapunov function and the close-loop system is asymptotically stable in the set X N (Ω). Remarks: It is easy to prove Lemma 1. According to inequality (6), we have that V o N (x o (k +1),u o (k +1),k +1) = N 2 l(x o i+k+1/k+1,uo i+k+1/k+1 )+F(xo N+k+1/k+1 ) l(x o i+k+1/k+1,uo i+k+1/k+1 )+F(xo N+k/k ) = V o N (x o (k),u o (k),k) l(x o k/k,uo k/k ) V o N (x o (k),u o (k),k). The optimal cost VN o (xo (k),u o (k),k) is monotonically decreasing as k. Considering the property of cost function V N (x(k),u(k),k), it is clear that VN o (xo (k),u o (k),k) is a Lyapunov function. Hence, it is easy to conclude that the close-loop system is asymptotically stable. We note that Lemma 1 dose not give out an explicit method to calculate the set X N (Ω). In Assumption 1, it is assumed that the stage cost function l( ) and the terminal cost function F ( ) are positively defined and satisfy quadratic bounds as follows: ( x 2 + u 2) l(x,u) M l ( x 2 + u 2), (7) m F x 2 F (x) x 2, (8) where,,m l,m F, are positive constants. These constraints are easily satisfied in most of the cases. 3 Analysis of the stable region First, we give out one of the explicit definitions of the set X N (Ω), and denote it as the region of feasibility X N (β). Assumption 2. Let β = α(1+n ml ) be a positive constant, where α is defined in Assumption 1 and N is the predictive horizon, and the region of feasibility X N (β) is given as X N (β) = {x(k) R n V o N (x o (k),u o (k),k) β},k N. (9) Lemma 2. If the system (1,2) is controlled by the optimal control sequence u o (k), which is calculated by solving FHOCP P N (x(k),k), and the initial state x() lies in the region of feasibility X N (β) defined

4 Ruan X G, et al. Sci China Inf Sci November 214 Vol :4 by Assumption 2, then the terminal state x(k +N) will lie in the region of attraction Ω defined by Assumption 1. Proof. We suppose that x(k +N) / Ω and F (x(k +N)) > α. Then, we have that V o N (x o (k),u o (k),k) = = > l(x o i+k/k,uo i+k/k )+F(xo N+k/k ) l(x o i+k/k,uo i+k/k )+F(x(k +N)) l(x o i+k/k,uo i+k/k )+α ( xi+k/k 2 + ui+k/k 2 ) +α xi+k/k 2 +α F ( x i+k/k ) /MF +α > α(1+n / ). The above inequality is contradictory to the definition of X N (β). Hence, we conclude that for any initial state x(k) X N (β), the terminal state of the close-loop system (1,2) that is controlled by MPC algorithms with CLF satisfies F (x(k +N)) α and lies in the region of attraction Ω. Then, Lemma 2 is proved. Lemma 2 gives the conservative estimation of the feasible initial state set X N (β), which means that some initial states out of the set X N (β) may be stabilizable. In some special cases, we can get an explicit parameter β to make the region of feasibility X N (β) contain all of the feasible initial states. Lemma 3. Suppose that ( x 2 + u 2 ) l(x,u) M l ( x 2 + u 2 ), where = M l, and m F x 2 F (x) x 2, where m F =, the initial state lying in the region of feasibility X N (β) will be a necessary and sufficient condition to guarantee the terminal state lying in the region of attraction Ω, where β = α(1+n / ). Proof. Lemma 2 has proved the sufficient condition of guaranteeing the terminal state lying in the region of attraction Ω, and we only need to discuss the necessary condition of that. Since F (x(k +N)) α, we have that Thus, Lemma 3 is proved. V o N (xo (k),u o (k),k) = l(x o i+k/k,uo i+k/k )+F(xo N+k/k ) l(x o i+k/k,uo i+k/k )+α M l ( xi+k/k 2 ) +α M l F ( x i+k/k ) /mf +α α(1+n M l /m F ) α(1+n / ).

5 Ruan X G, et al. Sci China Inf Sci November 214 Vol :5 The supposed conditions = M l and m F = are not so hard to satisfy, and they can be satisfied by choosing proper coefficient matrixes of the stage cost and terminal cost. For example, if we set the coefficient matrixes as Q = I n n, R = I m m and P = m F I n n, then it is easy to get that = M l = and m F = = m F, where and m F are some fixed positive values, and I is the unit matrix. Lemma 4. If the system (1,2) is controlled by the optimal control sequence u o (k), which is calculated by solving FHOCP P N (x(k),k), and the initial state x() lies in the region of feasibility X N (β) defined by Assumption 2, then the optimal cost of FHOCP V o N (xo,u o,k) is a Lyapunov function and the closeloop system is asymptotically stable in the region of feasibility X N (β) X N (Ω). Remarks: Since Lemma 1 has proved that if the terminal state lies in the region of attraction Ω, the close-loop system will be asymptotically stable. Based on Lemma 2, we know that any initial state lying in the region of feasibility X N (β) can converge to the region of attraction Ω at the terminal instant of the predictive horizon N. Then, we can conclude that the close-loop system with initial state lying in the region of feasibility X N (β) is asymptotically stable. Thus, Lemma 4 is proved. The benefit of Lemma 4 is that the terminal constraint is removed once the initial state of system belongsto theregionoffeasibilityx N (β), which greatlyreducesthe calculationburden ineachoptimization calculation circle. Also, we discuss the explicit way to calculate the region of feasibility X N (β). The nonlinear model (1,2) can be simply described as We define that x(k +1) = x(k)+f(x,k)+g(x,k)u(k),x(k) X,k N. (1) W (x,u,k) = F(x,k +1) F(x,k)+l(x,u,k) = x T (k +1)Px(k +1) x T (k)px(k)+x T (k)qx(k)+u T (k)ru(k), (11) where, P is the coefficient matrix of terminal cost, Q is the state coefficient matrix of stage cost, and R is the control coefficient matrix of stage cost. Then, we have that W(x,u,k) = (x(k)+f(x,k)+g(x,k)u(k)) T P(x(k)+f(x,k)+g(x,k)u(k)) x T (k)px(k)+x T (k)qx(k)+u T (k)ru(k) = 2x T (k)p(f(x,k)+g(x,k)u(k))+(f(x,k)+g(x,k)u(k)) T P(f(x,k)+g(x,k)u(k)) +x T (k)qx(k)+u T (k)ru(k) = u T (k) ( g T (x,k)pg(x,k)+r ) u(k)+2(x T (k)+f T (x,k))pg(x,k)u(k) +2x T (k)pf(x,k)+f T (x,k)pf(x,k)+x T (k)qx(k). The problem min u(k) W can be dealt as a simple QP problem, and the optimal solution is such that u opt (k) = (g T (x,k)pg(x,k)+r) 1 g T (x,k)p (x(k)+f(x,k)). (12) In some special cases, Eq. (12) can be simplified. Suppose that there are no input constraints, and the input coefficient matrix R is assumed to be a zero matrix, the optimal control vector u opt (k) turns out to be u opt (k) = g 1 (x,k)(x(k)+f(x,k)). (13) The necessary condition of the above formula is that g 1 (x,k) exists. Then, we have that W opt = 2x T Px+x T Px+x T (k)qx(k). If the condition P Q is satisfied, the formula W opt will be true for any x R n. Thus, we have the following lemma. Lemma 5. For a nonlinear system (1), the region of attraction Ω = R n is infinite and so will be the region of feasibility X N (β) = R n, if the following conditions are satisfied: (1) control coefficient matrix of

6 Ruan X G, et al. Sci China Inf Sci November 214 Vol :6 stage cost R = ; (2) the matrix function g(x,k) is invertible; and (3) the coefficient matrix of terminal cost P is no less than the state coefficient matrix of stage cost Q. However, the matrix function g(x,k) is not invertible in most of the cases, and the region of attraction Ω should be analyzed in specific cases. Remarks: Though the three conditions of Lemma 5 are critical for most of the real systems, Lemma 5 gives out the limits of region of attraction and region of feasibility. The conditions in Lemma 5 reveal that if large areas of the region of feasibility are got, the input coefficient matrix R should be small and the coefficient matrix of terminal cost P should be larger than the state coefficient matrix of stage cost Q. This helps to design large regions of feasibility for common systems. Proposition 1. Suppose a weighted terminal cost is given by F λ (x) = λ F (x),λ 1, if F (x) satisfies Assumption 1, the set Ω λ = {x R n : F λ (x) λ α} will be a region of attraction, and the set X N (β λ ) is a region of feasibility, where β λ is denoted as β λ = λα(1+n ml ). Proof. min {F λ(f (x,u)) F λ (x)+l(x,u)} = min{λ (F (f (x,u)) F (x))+l(x,u)} u U u U min{λ ( l(x,u))+l(x,u)} (14) u U. According to the inequality (14), we conclude that Ω λ satisfies Assumption 1 and is a region of attraction. Also, it is easy to conclude that X N (β λ ) is a region of feasibility according to Lemma 2. Remarks: As the weight factor of terminal cost λ increases, the region of attraction Ω λ dose not change, but the region of feasibility X N (β λ ) expands correspondingly. It is clear that increasing λ or N can enlarge the set X N (β λ ). As λ + or N +, the region of feasibility X N (β λ ) R n. Hence, we can conclude that there exists constant λ or N ensuring that X X N (β λ), which means that any initial state in X can be steered in to Ω λ by MPC without the terminal constraint, and the close-loop system is asymptotically stable in X. However, large λ or N will bring in extra problems. For example, if weighted fact λ is too large, the dynamic performance of the close-loop system will be poor, and if the predictive horizon N is too large, the calculation burden will sharply increase. To improve the performance of optimization, the weight factor λ with initial value λ can be reduced at each sampling time such that λ(k +1) = max ( λ(k) ) l(x k/k,u k/k ) α(1+n / ),1. (15) Proposition 2. If the weight factor λ(k) with initial value λ() = λ is chosen as (15), and the set X N (β λ) is the region of feasibility for λ() = λ, then the close-loop system is asymptotically stable in the set X N (β λ), and the set X N (β λ) is also a region of feasibility for close-loop system with time-variant weight factor λ(k) such as (15). Proof. To prove that the set X N (β λ) is the region of feasibility for close-loop system with time-variant weight factor, we have to prove that ( ) VN o (xo (k),u o (k),k) β λ(k) = λ(k)α 1+N. (16) For λ() = λ, Eq. (16) is established. For k, VN o (xo (k +1),u o (k +1),k +1) VN o (xo (k),u o (k),k) l(x(k),u(k)) ( ) λ(k)α 1+N l(x(k),u(k))

7 Ruan X G, et al. Sci China Inf Sci November 214 Vol : Q=.1*I, R= Q=.1*I, R=.1 Q=I, R= Q=I, R= x 1 Figure 1 The regions of attraction of nonlinear system (17) with different coefficients Q and R Q=.1*I, R= Q=.1*I, R=.1 Q=I, R= Q=I, R= x 1 Figure 2 The regions of feasibility of nonlinear system (17) with different coefficients Q and R. ( )( α 1+N λ(k) l(x(k),u(k)) ) α(1+n / ) { ( α(1+n ) max λ(k) l(x(k),u(k)) )} α(1+n / ),1 ( ) = λ(k +1)α 1+N. Eq. (16) is proved to be right by the above inequality. According to Proposition 1, the set X N ( βλ(k) ) is the region of feasibility at each sampling time k. Hence, the close-loop system is asymptotically stable in the control progress. The Proposition 2 gives a feasible way to improve the dynamic performance of the MPC algorithm, and does not reduce the area of the region of feasibility at the same time. 4 Experiments To discuss the region of attraction Ω and the related propositions, the discrete-time model of a nonlinear system (16) is considered [5,16], which is unstable and its linearized system is stabilizable (but not controllable). { x1 (k +1) = x 1 (k)+t s ( (k)+(.5+.5 x 1 (k))u(k)), u U = [ 2,2]. (17) (k +1) = (k)+t s (x 1 (k)+(.5 2 (k))u(k)), The terminal coefficient P = [ ] is calculated by solving the Lyapunov function of the linearized system. With different values of state coefficient Q and control coefficient R, the region of attraction Ω of nonlinear system (17) is depicted in Figure 1. It is obvious that the larger the coefficients, including state coefficient Q and input coefficient R, the smaller the area of the region of attraction Ω. According to Lemma 2, the region of feasibility X N (β) can be calculated, and it is depicted in Figure 2. It shows that the larger the coefficients, including state coefficient Q and input coefficient R, the smaller the region of feasibility X N (β). The other control experiments have shown that if the initial states ofthe nonlinearsystem (17)lie in the regionoffeasibility X N (β), the MPC algorithmalways keeps the stability of the system. In Figure 3, we give out the response trajectories of system (17) with initial states lying on the boundary of the region of feasibility X N (β), and the parameters are chosen as Q =.1 I 2 2,R =. The result shows that all of the trajectories converge to the origin, which verifies that the proposed region of feasibility is effective. If the terminal cost is given by F λ (x) = λ F (x),λ 1, as the coefficient λ changes, the region of feasibility X N (β λ ) changes as well, which are depicted in Figure 4. The coefficients are chosen as Q =

8 Ruan X G, et al. Sci China Inf Sci November 214 Vol : Boundary Trajectories λ=.1 λ=1 λ=5 λ= x x 1 Figure 3 The response trajectories of nonlinear system (17) with different initial states. Figure 4 The regions of feasibility with different terminal cost coefficients λ (a) λ=1 Time-variant λ.5.5 (b) λ=5 Time-variant λ x 1 x 1 Figure 5 The response trajectories with different terminal matrix coefficients λ. (a) The response trajectories with initial terminal matrix coefficient λ = 1; (b) the response trajectories with initial terminal matrix coefficient λ = 5. Table 1 The cumulative errors of response trajectories in 5 s with different terminal matrix coefficients λ Initial λ 1 5 Initial state (1.2, 3.5) (.8, 2.5) Cumulative errors(constant λ) Cumulative errors(time-variant λ) I 2 2 and R =. It can be seen that the larger the coefficient λ, the larger will be the region of feasibility X N (β λ ). However, it does not mean that we can use a terminal cost coefficient λ as large as possible to get a large enough area of region of feasibility. The cost will be the drop of the dynamic performance. Intuitively speaking, stability and optimality are contrary to each other to some extent. The influence of the terminal cost coefficient λ on the dynamic performance is depicted in Figure 5 and Table 1. Two sets of experiments were performed with different terminal cost coefficients λ and different initial states. In the first set of experiment, as shown in Figure 5(a), the Initial terminal cost coefficients is λ = 1, and the initial state is x = (1.2, 3.5), which lies on the boundary of the region of feasibility when λ = 1. The solid line in the figure denotes the control trajectory of MPC algorithm with constant terminal cost coefficient λ = 1, and the dotted line denotes the control trajectory of MPC algorithm with time-variant terminal cost coefficient λ, which dynamically changes according to (15). The cumulative errors of the two control trajectories are depicted in Table 1, and it shows that the decreasing terminal cost coefficient λ reduces the cumulative error of the control trajectory. The Figure 5(b) shows the result of the other set of experiment. In this experiment, the Initial terminal cost coefficients is λ = 5, and

9 Ruan X G, et al. Sci China Inf Sci November 214 Vol :9 the initial state is x = (.8, 2.5). The cumulative errors of the two control trajectories in Figure 5(b) are shown in Table 1. The two sets of experiments show that the method proposed in Proposition 2 is feasible, and it indeed improves the dynamic performance of the MPC algorithm. The point is that the region of feasibility is not affected at the same time. 5 Conclusion The region of attraction guarantees the stability of the MPC algorithm with CLF, and eliminates the terminal constraints in each optimization calculation circle, which benefits the application of MPC algorithm to real-time systems. However, it is just a conservative estimation of the region where the MPC algorithm is feasible. In this article, we propose the region of feasibility to expand the region of attraction, and analyze the related factors that affect the region of feasibility. At the same time, we note that as the terminal cost coefficient λ increases, which can expand the region of feasibility, the optimal performance will decrease. A time-variant terminal cost coefficient λ is designed to improve the optimal performance. The effectiveness of these methods are verified by a series of experiments using a classic discrete-time model. The results also show that the proposed methods are feasible for engineering practice. These works develop the MPC theory, and improve the MPC algorithm suitable for more objects, including unstable and/or fast-dynamic system. Acknowledgements We acknowledge support from National Natural Science Foundation of China (Grant Nos , ), Key Project of S&T Plan of Beijing Municipal Commission of Education (Grant No. KZ212151), and National Basic Research Program of China (973) (Grant No. 212CB72). References 1 Mayne D Q, Rawlings J B, Rao C V, et al. Constrained model predictive control: stability and optimality. Automatica, 2, 36: Magni L, Scattolini R. An Overview of Nonlinear Model Predictive Control. In: Automotive Model Predictive Control, London: Springer, Keerthi S S, Gilbert E G. Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations. J Optim Theory Appl, 1988, 57: Michalska H, Mayne D Q. Robust receding horizon control of constrained nonlinear systems. IEEE Trans Automat Contr, 1993, 38: Chen H, Allgower F. A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 1998, 34: Magni L, Sepulchre R. Stability margins of nonlinear receding horizon control via inverse optimality. Syst Control Lett, 1997, 32: Primbs J A, Nevistic V, Doyle J C. A receding horizon generalization of point-wise min-norm controllers. IEEE Trans Automat Contr, 2, 45: Jadbabaie A, Yu J, Hauser J. Receding horizon control of the Caltech ducted fan: a control Lyapunov function approach. In: Proceedings of IEEE Conference on Control Applications, Kohala Coast, Jadbabaie A, Yu J, Hauser J. Stabilizing receding horizon control of nonlinear systems: a control Lyapunov function approach. In: Proceedings of the American Control Conference, San Diego, Jadbabaie A, Yu J, Hauser J. Unconstrained receding-horizon control of nonlinear system. IEEE Trans Automat Contr, 21, 46: Limon D, Alamo T, Salas F, et al. On the stability of constrained MPC without terminal constraint. IEEE Trans Automat Contr, 26, 51: Graichen K, Kugi A. Stability and incremental improvement of suboptimal MPC without terminal constraints. IEEE Trans Automat Contr, 21, 55: Chen W, Cao Y. Stability analysis of constrained nonlinear model predictive control with terminal weighting. Asian J Control, 212, 14: Chen W H. Stability analysis of classic finite horizon model predictive control. Int J Control Autom, 21, 8: Mayne D Q. Control of constrained dynamic systems. Eur J Control, 21, 7: Pin G, Raimondo D M, Magni L, et al. Robust model predictive control of nonlinear systems with bounded and state-dependent uncertainties. IEEE Trans Automat Contr, 29, 54: