Dual-mode Nonlinear MPC via Terminal Control Laws With Free-parameters

Ⅰ. INTRODUCTION

Nonlinear model predictive control (NMPC) has received great attentions in both academic and industrial communities [1]-[4]. The attentions of NMPC are motivated by its advantages for control of multivariable nonlinear systems and for explicitly handling constraints on process variables. Due to using the receding horizon principle, in general the closed-loop stability of NMPC cannot be guaranteed automatically by itself. To address this stability issue, many schemes have been proposed to achieve the closed-loop stability of NMPC for the last two decades [1]-[4]. One of main schemes used to ensure the closed-loop stability of NMPC is the dual-mode strategy [5]-[10], in which the MPC law is parameterized with a fixed state feedback computed offline (often called terminal controllers) and then, additional corrections are computed online. By the additional corrections, the closed-loop states are steered into an invariant set (often called terminal domain) corresponding to the feedback law computed offline. Therefore, not only most of the heavy computations related to invariance and feasibility constraints are handled offline, but also the closed-loop stability of NMPC is ensured by exploiting the feedback law computed offline.

In dual-mode NMPC schemes, the size of the maximal feasible invariant set (denoted $X_N$) is dependent on the terminal region (denoted $X_0$), the terminal controller $u_l(x)$ and the prediction horizon $N$. In general, the terminal region $X_0$ with larger size increases the size of the feasible set $X_N$. However, it is difficult to obtain the quantitative interplay between each other. Hence, in the conventional dual-mode NMPC schemes [5]-[10] the closed-loop states will be driven into a predesigned $X_0$ within or less than $N$ steps, where a linear terminal control law $u_l(x)=Kx$ is used and is required to satisfy system constraints. For simplicity of designing controllers, the terminal law $u_l(x)=Kx$ in most dual-mode MPC schemes is usually selected as a linear quadratic optimal control law. Due to the invariance of the gain matrix $K$, the size of $X_0$ may be greatly small if prediction horizon $N$ is given. Note that the size of $X_0$ can be extended by increasing horizon $N$, but this approach increases the computational demand of solving the optimization problem of NMPC online. The region $X_0$ with a small size will decrease the size of the initial feasible set ${X_N}$ [1], [2], [4], which makes the performances of the dual-mode NMPC become more conservative.

In order to enlarge the size of the set $X_N$, a straightforward method is to lengthen horizon $N$. However, a long $N$ will increase the computational demand of MPC since it requires satisfying a bigger optimization problem than the one with a short $N$. Hence, the authors in [11]-[13] employed the notion of equilibrium surface and the scheduling method to propose the dual-mode NMPC schemes with enlarged terminal regions, where several terminal subsets are defined based on the equilibrium surface. Another method in [14] was proposed for linear systems, where the horizon $N$ in the MPC problem was divided into prediction horizon $N_p$ and control horizon $N_c$ and a new control mode was defined over time interval $[N_c, N_p]$. Then there exist three control modes, i.e., the terminal law $u_l(x)$ in $X_0$, MPC law $u_m(x)$ out of $X_0$ and new control law $u_s(x)$ within $[N_c, N_p]$ and out of $X_0$. This change enlarges the size of the terminal region and decreases the conservativeness of MPC. Now this strategy is named as the triple-mode MPC scheme and has been extended to analyze and design robust MPC of uncertain linear systems.

In this paper we consider continuous-time nonlinear systems subject to state and control constraints and propose a stabilizing dual-mode NMPC scheme using terminal control laws with free-parameters. The local control Lyapunov functions (CLFs) [15] for nonlinear systems are used to design the terminal controller with some free-parameters. Then a set of feasible parameters is selected to estimate the size of the terminal region $X_0$ as large as possible. When the state belongs to $X_0$, an optimal control problem is formulated to online compute the free-parameters, which leads to the optimality of $u_l(x)$ with respect to given cost functions. Conversely, a correction term is computed online when the state is out of $X_0$. By the stability results of the general dual-mode NMPC framework, the equilibrium point of the closed-loop system with NMPC controller proposed here is shown to be asymptotically stable. Finally the numerical example of a spring-cart system [16] is considered and the presented controller is compared with the dual-mode NMPC with quadratic optimal terminal control law in terms of performance and terminal regions.

The contribution of the paper is to present a closed-loop dual-mode NMPC scheme where the terminal controller is updated using the current state at each sampling time and the size of the terminal region is decoupled from the local optimality of the terminal controller. The terminal controller and the terminal region have interplay in conventional dual-mode NMPC schemes. Hence, it is hoped that the proposed scheme provides a more convenient way than the conventional ones in terms of application. It is noted that the design of the terminal controller is motivated by the results in [17].

Ⅱ. PROBLEM SETUP AND PRELIMINARIES

A System Description

Consider an input-affine nonlinear system described by

where $x\in {\mathbb R}^n$ is the state variable, $u\in {\mathbb R}^m$ is the control variable, $f(x)$ and $g(x)$ are continuous functions in some neighborhood regions of the origin. The system is subject to the constraints on the state and control

where $X\in {\mathbb R}^n$ and $U\in {\mathbb R}^m$ are compact and convex sets containing the origin in their interior. It is assumed that the origin is the equilibrium point of the system and the solutions to the system (1) exist and are continuous for any initial state $x(0)$ $\in$ $X$ and piecewise, right-continuous input $u(\cdot)\in U$. We also assume that the states can be measured at each time instant.

Definition 1 [18]: Consider set $S\subseteq {\mathbb R}^n$ and controller $u:$ $S\rightarrow U$. The set $S$ is called a positively invariant set of the system (1) in closed-loop with $u(x)$ if its solutions $x(t;x(0),$ $u(x))$ $\in$ $S$ for any $t\geq 0$ and any $x(0)\in S$.

Definition 2 [15]: Consider the system (1) and a positive definite function $V(x)$. If $V(x)$ satisfies that

then $V(x)$ is said to be a control Lyapunov function (CLF) for the system, where the Lie derivatives $L_f V(x)=\frac{\partial V(x)}{\partial x}f(x)$, $L_g V(x)=\frac{\partial V(x)}{\partial x}[g_1(x), \ldots, g_m(x)]$ and $g_i(x)$, $i=1, \ldots, m$ is the $i$th column of matrix function $g(x)$. Moreover, if $V(x)$ $\rightarrow$ $\infty$ when $\|x\|\rightarrow \infty$, $V(x)$ is a global CLF of the system.

If we can construct a (global) CLF, $V(x)$, of system (1), we have an explicit control law that stabilizes the system to the equilibrium point.

Lemma 1 [17]: Consider the system (1) and a positive definite function $V(x)$. Let the set $D=[D_1, D_2]\times[D_3, D_4]$ with $D_i$ $>$ $0$, $i=1, \ldots, 4$, $\alpha(x)=L_fV(x)$ and $\beta(x)=L_gV(x)$. If $V(x)$ is a CLF of the system, then there exists a controller with free-parameters

where two free-parameters $(\mu_1, \mu_2)=\mu\in D$ and gain

such that the closed-loop system is asymptotically stabilized to the origin.

Note that the control law (4) gets to the standard Sontag's one if the parameter $\mu=(1, 1)$. In this case, the controller has no degrees of freedom with respect to the current state information and its performance heavily relies on the selected CLF of the controlled system. Here the control law (4) has two free-parameters that have an effect on the decaying velocity of the closed-loop system [17]. Hence, one can adjust these parameters to achieve a satisfied performance of the controller while ensuring stability of the closed-loop system.

The aim of the paper is to use the control law (4) to compute a terminal region with larger size and then is to obtain a new dual-mode NMPC scheme with guaranteed recursive feasibility and stability of the obtained NMPC.

B Review of Dual-mode MPC Principle

Considering the constrained system (1) and (2), we recall the general dual-mode feedback MPC controller

where $X_0$ is the terminal region of (1), $\pi(x)$ is the local optimal controller (also called terminal controller) defined in $X_0$ and $v$ is the additional correction term determined by solving the finite horizon optimal control problem

where $x(t)$ is the state at current time $t\geq 0$, $N>0$ is the prediction horizon (for simplicity, let the prediction horizon $N_p$ equal the control horizon $N_c$, i.e., $N_p=N_c=N$) and $J(x)$ is the cost function

where the stage cost $L(x, u)$ is a positive definite function on $x$ and is a semi-positive definite function on $u$. Therefore, a dual-mode MPC algorithm can be summarized as follows:

Algorithm 1:

Step 1: Set the prediction horizon $N>\delta$, stage cost $L(x, u)$ and a sufficiently small sampling time $\delta>0$, and let $t=0$.

Step 2: Measure the state $x(t)$ at time $t$. Then solve the optimization problem (6). If feasible, apply $u^{DM}(x)$ to system (1), go to Step 3; Otherwise, let $N=N+\delta$, and repeat Step 2.

Step 3: Let $t=t+\delta$ and $N=N-\delta$, go back to Step 2.

Lemma 2 [5], [7], [12]: Consider the system (1) and optimization problem (5)-(7). If the optimization problem is feasible at the initial time, the system (1) in closed-loop with (5) is then asymptotically stable in the face of constraints (2).

Remark 1: From the optimization problem (5)-(7), it is seen that the dual-mode MPC controller is related to the triplet of the terminal region $X_0$, terminal controller $\pi(x)$ and correction term $v$. In traditional dual-mode NMPC, the terminal controller $\pi(x)$ is typically computed offline by solving linear quadratic optimal control problems. In this setting, $\pi(x)$ is predefined and $X_0$ might be considerably small. As a result, the performance (e.g., the admissible set of initial states) of the whole MPC is conservative. In what follows, we give an alternative design of the triplet using Lemma 1, which yields a new dual-mode NMPC scheme of the constrained nonlinear system (1) and (2).

Ⅲ. NEW DUAL-MODE NMPC SCHEME

In this section, we re-design the terminal region $X_0$ and terminal controller $\pi(x)$ by the assumption of the known CLF of system (1) and then propose a new dual-mode NMPC scheme for the system.

A Terminal Regions

Let the function $V(x)$ be a known, local CLF of system (1). We define a level set of $V(x)$

where the radius $c>0$. Clearly, the size of region $X_0$ is determined by the value of $c$. To derive an $X_0$ as large as possible, we define a set $X_h$ via the terminal controller (4)

Then we increase $c$ from 0 until the region $X_0(c)$ approaches the set $X\bigcap X_h$ to the maximal extent. In this case, let $c=\hat{c}$. Hence, there exists at least a $\mu$ such that

Since $X_0(\hat{c})$ is the level set of $V(x)$, it is a positively invariant set of the system (1) in closed-loop with the controller (4) subject to the constraints (2). In the following, we present an algorithm to calculate $X_0(\hat{c})$ offline.

Algorithm 2:

Step 1: Partition $X$ near the origin into $l$ discrete points and $D$ into $r$ discrete points with sufficiently large integers $l>0$ and $r>0$.

Step 2: Initialize $c=\infty$ and let $i=1$.

Step 3: Compute $h(x^i, \mu^j)$ for $j=1$ to $r$, where $\mu^j=$ $(\mu_1^j,$ $\mu_2^j)$. If $h(x^i, \mu^j)\notin U$ for all $j=1, \ldots, r$, then go to Step 4; Else, go to Step 5.

Step 4: If $V(x^i)\leq c$, then let $c=V(x^i)$ and go to Step 5.

Step 5: Let $i=i+1$. If $i\leq l$, then go back to Step 3; Else, go to Step 6.

Step 6: Let $\hat{c}=c$ and save $\hat{c}$.

Note that the value $\hat{c}$ computed by Algorithm 2 may be conservative and the computational burden is expensive due to enumeration calculation. Nevertheless, this computation process does not increase the computational burden of online solving NMPC optimization problems since it is an algorithm operated offline.

Lemma 3: Let $V(x)$ be a known, local CLF of system (1). Then the system (1) in closed-loop with the controller (4) satisfies the constraints (2) while asymptotically stabilizing the origin in the invariant set $X_0(\hat{c})$.

Proof: Since the constraint sets $X$ and $U$ contain the origin as their interior, the level set of $V(x)$ exists and is non-empty. Then applying Lemma 1 directly yields Lemma 3 and hence, the proof is omitted.

Remark 2: It should be pointed out that there are no systematic methods to find a global CLF for the nonlinear systems, such as $\dot{x}=f(x, u)$. For the system (1), however, the local CLF can be computed by some existing methods [15], [18]-[20], e.g., the linearization at the equilibrium points, the feedback linearization, the backsteping control, etc. In this paper, we focus on the presentation of a new dual-mode MPC scheme for system (1). Hence, it is always assumed here that a local CLF of (1) has been constructed a prior.

Remark 3: Considering the nonlinear systems with an equilibrium surface, we use the proposed approach to calculate a terminal region for every equilibrium point on the surface. Then by the gain scheduling technique, we obtain an enlarged terminal region of the whole nonlinear system. For details, see e.g. [11]-[13].

B Terminal Controllers

Unlike the conventional dual-mode NMPC, where the terminal controllers are typically selected as the linear quadratic regulator (LQR) controllers with the cost functions of NMPC, here we use the controller (4) as the terminal control law of the new dual-mode NMPC scheme. Note that the controller (4) has free-parameters $\mu_1$ and $\mu_2$. From [17], it is known that the parameters $(\mu_1, \mu_2)$ are the measurement of decaying velocity of the CLF of (1). In this way, the parameters have an important effect on the control performance of the closed-loop system if the CLF is viewed as a virtual energy function of the system. Consequently, the parameters can be determined online by some additional requirements.

Here we consider the cost function (7) and define an optimal control problem

where the function $h(x, \mu)$ is given by (4) and the set $X_0(\hat{c})$ is computed by (8)-(10). From the determination of $\hat{c}$, we know that there is at least a $\mu$ satisfying the constraints in problem (11), i.e., the problem is always feasible for any current state $x(t)$ $\in$ $X_0(\hat{c})$.

Let $\mu^*(t)$ be an optimal solution to (11) at time $t$. Then the terminal control law defined in the terminal region $X_0(\hat{c})$ is defined as

Note that in order to decrease the computational demand of problem (11), the parameters $\mu(t)$ are assumed to be constant within the prediction horizon window $[t, t+N]$. However, from the receding horizon principle we know that $\mu(t)$ at each time $t$ is computed based on the state $x(t)$ at the current time $t$. Therefore, the obtained controller (12) is a closed-loop feedback control law since the parameters are dependent upon the current state $x(t)$.

C Dual-mode Controllers

Consider the nonlinear system (1) with constraints (2), the terminal region $X_0(\hat{c})$ and terminal controller (12). We define a new dual-mode controller

where $\hat{\mu}\in D$ is an offline identified number such that $h(x, \hat{\mu})$ satisfies constraints (2), $\mu^*(t)$ is the optimal solution to (11) at time $t$ and the correction term $v(t)$ is determined by solving the finite horizon optimal control problem

where the cost function $J(x)$ is given by (7), function $h(x, \mu)$ is given by (4) and set $X_0(\hat{c})$ is computed by (8)-(10). Then an overall dual-mode NMPC algorithm is presented below:

Algorithm 3:

Step 1: Set $N>0$, $L(x, u)$ and $0 < \delta < N$, and let $t=0$.

Step 2: Offline compute $X_0(\hat{c})$ by (8)-(10) and pick a feasible solution $\hat{\mu}\in D$.

Step 3: Measure the state $x(t)$ at time $t$. If $x(t)\in X_0(\hat{c})$, then solve (11) and go to Step 5. Else, solve (14). If (14) is feasible, then $v(t)=v^*(t)$ and go to Step 4. Else, let $N=N$ $+$ $\delta$ and go to Step 3.

Step 4: Apply $u^{DM}(t)=h(x(t), \hat{\mu})+v^*(t)$ to system (1) and let $t=t+\delta$ and $N=N-\delta$; go back to Step 3.

Step 5: Apply $u^{DM}(t)=h(x(t), \mu^*(t))$ to system (1) and let $t=t+\delta$; go back to Step 3.

Note that the optimal control problems (11) and (14) in Algorithm 3 are nonlinear programming problems and in general, are non-convex due to the nonlinear equality constraints. Several optimization methods, such as successive quadratic programming (SQP) and genetic algorithm (GA), can be used to solve these optimization problems. Some reviews on optimization methods for solving constrained model predictive control can be consulted elsewhere, see e.g. [21], [22].

It should be pointed out that the decision variables of the optimal control problem (14) can be selected as $\mu$ and $v$, which increases the degrees of freedom of (14) to be optimized. However, this selection increases the computational demand of solving (14) although it may enlarge the feasible set of initial states of the controller. In addition, the effects dictated by fixing $\mu$ can be compensated to some degree by the online optimization of the correction term $v$.

In the sequel, we consider the optimization feasibility and stability of the dual-mode NMPC controller (13) obtained from Algorithm 3.

Definition 3: Consider a state $\xi\in X$ of system (1). If there is a solution $v(s)$ satisfying the problem (14) with $x(s)|_{s=t}$ $=$ $\xi$, then $\xi$ is said to be a feasible state. All feasible states constitute a feasible set $X_N$ of (13).

In general, we have $X_0\subseteq X_N\subseteq X$ and $X_N\subseteq X_{N+1}$ for nominal system (1).

Lemma 4: Consider two positive integers $N_1$ and $N_2$ with $N_1$ $<$ $N_2$. Then the feasible sets of (1) satisfy $X_{N_1}\subseteq X_{N_2}$.

Proof: Consider any state $x(t)\in X_{N_1}$ at time $t$. Solving the problem (14) with $N=N_1$, we have an optimal solution $v^*(s)$ and its associated controller $u(s)=h(x(s), \hat{\mu})+v^*(s)$ for $s\in [t, t+N_1]$. Meanwhile, the terminal state $x(t+N_1)\in X_0(\hat{c})$.

Let $N=N_2$ and consider a candidate solution to (14)

which leads to an associated controller

From Lemma 3 and the construction of the dual-mode controller (13), there exists a $\hat{\mu}\in D$ such that $X_0(\hat{\mu})$ is a positively invariant set of the constrained system (1) and (2) in closed-loop with the controller (4). Substituting (15) into the problem (14) with the initial state $x(t)$ and the horizon $N_2$, it is obtained that the constraints in (14) are fulfilled. This implies that the candidate (15) is a feasible solution to (14) at time $t$. From the Definition 3, $x(t)\in X_{N_2}$ and it leads to $X_{N_1} \subseteq X_{N_2}$ due to arbitrariness of $x(t)$.

Lemma 4 implies that one can use a larger prediction horizon $N$ to enlarge the feasible set $X_N$, i.e., to enhance the initial feasibility. Nevertheless, the larger $N$ generally increases the computational demand of solving the optimal control problem (14).

Theorem 1: If the optimization problem (14) is feasible at initial time $t=0$, then the constrained closed-loop system (1), (2) and (13) by Algorithm 3 is asymptotically stable in invariant feasible set $X_N$. Then $X_N$ is a domain of attraction of the closed-loop system.

Proof: First from Lemma 3 and Algorithm 3, it is known that the overall dual-mode NMPC controller (13) is always feasible provided that problem (14) is initially feasible in feasible set $X_N$. Then by Definitions 1 and 3 and Algorithm 3, we have that $X_N$ is an invariant set. Moreover, from Lemmas 2 and 3, it is obtained that the closed-loop system (1) and (13) by Algorithm 3 are asymptotically stable in $X_N$. Therefore, all trajectories of the closed-loop system starting in $X_N$ converge to the origin. Then the invariant set $X_N$ is a domain of attraction of the closed-loop system.

Remark 4: In the dual-mode NMPC scheme presented here, some free parameters are introduced into the terminal controller $\pi(x)=h(x, \mu)$. An advantage of this is to decouple the local optimality of the terminal control law from the size of the terminal region. So, using a feasible set of parameters $\mu$ leads to a terminal region as large as possible while minimizing cost functions of NMPC with respect to the parameters to regain the local optimality of the terminal control law to some extent. Note that the terminal control law and the terminal region have interplay in conventional dual-mode NMPC schemes. Moreover, the terminal controller in the proposed scheme is updated based on the current state at each sampling time. In addition, the overall dual-mode NMPC controller is a state feedback control law, i.e., a closed-loop controller. This is helpful in control of uncertain systems with small disturbances.

Ⅳ. NUMERICAL EXAMPLE

In this section, a spring-cart is used to illustrate the performance of the results presented here, compared to the dual-mode NMPC via the LQR method. All optimization problems in this example had been solved by the fmincon function with the SQP algorithm in MATLAB 7.1 on the laptop with MS Windows XP and an Intel (R) Core^TM 2 CPU with 2.0 GHz and 1 GB RAM. The Euler's first-order approximation had been used to obtain all derivatives in this simulation.

Consider the continuous time model of the cart-spring system [16]:

where $x_1$ is the displacement of the carriage from the equilibrium, $x_2$ is the carriage velocity, $u$ is the external force, $M$ is the mass, $k$ is spring elasticity and $h$ is a damping factor. The parameters of the model are $M$ = 1 kg, $k$ = 0.33 and $h$ = 1.6. Let the constraints on the state and control $X =[-2,2]$ $\times$ $[-2,2]$ and $U=[-1,1]$, respectively.

Define a candidate CLF of system (17)

It can be verified that $V(x)$ is a CLF of (17). Then we can construct the function $h(x, \mu)$ in (4) with

Compute offline a terminal region $X_0(\hat{c})$ with $\hat{c} =0.572$, which is pictured as an ellipsoid denoted by solid ellipse in Fig. 1. To compare with the performance of the conventional dual-mode NMPC, we use an LQR controller based on linearized model of system (17) on the origin to compute the terminal region.

Figure  1.  The terminal regions $X_0$ obtained by the proposed method (solid line) and the LQR method with $J_1(x)$ (dashed line), $J_2(x)$ (dash-dotted line) and $J_3(x)$ (dotted line).

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For the linearized system of (17)

we consider the following three quadratic stage costs

Then solving the Riccati equation of system (17) with respect to $L_1(x, u)$, $L_2(x, u)$ and $L_3(x, u)$, respectively, we have three corresponding LQR controllers $u_i(x)=K_ix$ for $i$ = 1, 2 and 3 with

Furthermore, we compute the terminal region $X_0(c_i)$ associated with $L_i(x)$ for $i$ = 1, 2 and 3

with $c_1 = 0.32$, $c_2 = 0.58$ and $c_3 = 1.10$. These terminal regions are shown in Fig. 1, where dashed ellipse denotes $X_0(c_1)$, dash-dotted ellipse $X_0(c_2)$ and dotted ellipse $X_0(c_3)$.

It is observed from Fig. 1 that the terminal region $X_0$ in the proposed dual-mode NMPC scheme is independent of the cost function used while $X_0$ in LQR-based dual-mode NMPC scheme couples with the cost function by the way of the terminal LQR controllers. Thus, in order to derive a maximal terminal region in the LQR-based dual-mode NMPC scheme, we should elaborately select the cost function (e.g., $L_3(x)$). However, the cost function of a controlled system is usually pre-given according to some control tasks of the system. In this setting, we may have a very small terminal region (e.g., $L_1(x)$ and $X_0(c_1)$) and a corresponding small feasible state set $X_N$ for the dual-mode NMPC controller.

In order to more clearly show the differences of $X_N$ associated with $X_0(c_i)$, Table Ⅰ tabulates the shortest prediction horizon $N_{\rm min}$ that is used to guarantee the feasibility of the optimization problem (14) at initial time $t_0 = 0$ with two initial states $A$ $(-1.0, 1.7)$ and $B$ $(1.5, -1.2)$. It can be seen that for the different stage cost functions (20)-(22), the $N_{\min}$ associated with $X_0(c_1)$ is longest, the $N_{\min}$ with $X_0(c_2)$ is the second longest and the $N_{\min}$ with $X_0(c_3)$ is shortest. The reason causing this difference is due to the different size of $X_0(c_i)$. This fact is also in accord with the accepted conclusion of NMPC, which inversely implies that the feasible state set $X_N$ with $X_0(c_3)$ has a larger size than the other's if the prediction horizon of each controller is selected to be equal.

Table  Ⅰ.  THE SHORTEST PREDICTION HORIZON $N_{\min}$ USED TO ENSURE INITIAL FEASIBILITY OF THE OPTIMIZATION PROBLEM

Initial states	$X_0(\hat{c})$	$X_0(c_1)$	$X_0(c_2)$	$X_0(c_3)$
$A$ $(-1.0, 1.7)$	3$\delta$	12$\delta$	3$\delta$	$\delta$
$B$ $(1.5, -1.2)$	3$\delta$	10$\delta$	5$\delta$	3$\delta$

 | Show Table

DownLoad: CSV

Denote by CLF-MPC and LQR-MPC the proposed dual-mode NMPC scheme and the LQR-based dual-mode NMPC scheme, respectively. We consider the stage costs $L_i(x, u)$ for $i=$ 1, 2 and 3 and the prediction horizon $N=15\delta$ with a sampling time $\delta = 0.25$ s, and let the sector of $\mu$ be $[0.1, 5]\times [0.1, 10]$. Figs. 2-4 show the state, control input and parameters $(\mu_1, \mu_2)$ time profiles of the separate closed-loop systems starting from the initial state $A$, where Fig. 2 is obtained by applying $L_1(x, u)$, Fig. 3 is obtained by $L_2(x, u)$, and Fig. 4 is obtained by $L_3(x, u)$. In these figures, the solid lines are the results of CLF-MPC, the dotted lines are the results of LQR-MPC, the solid lines in right-lower subfigure are the computed parameter $\mu_1$ and the dash-dotted lines in right-lower subfigure are the computed parameter $\mu_2$. It is observed from Figs. 2-4 that:

Figure  2.  The state and control time profiles of the closed-loop system under CLF-MPC (solid) and LQR-MPC (dotted) with the stage cost $L_1(x, u)$, and the time profiles of parameters $\mu_1$ (solid) and $\mu_2$ (dash-dotted).

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Figure  3.  The state and control time profiles of the closed-loop system under CLF-MPC (solid) and LQR-MPC (dotted) with the stage cost $L_2(x, u)$, and the time profiles of parameters $\mu_1$ (solid) and $\mu_2$ (dash-dotted).

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Figure  4.  The state and control time profiles of the closed-loop system under CLF-MPC (solid) and LQR-MPC (dotted) with the stage cost $L_3(x, u)$, and the time profiles of parameters $\mu_1$ (solid) and $\mu_2$ (dash-dotted).

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1) For any stage cost $L_i(x, u)$ for $i = 1, 2, 3$, the origin of the system (15) in closed-loop with CLF-MPC is asymptotically stable while satisfying the state and control constraints.

2) Although the state trajectories obtained by CLF-MPC are similar to those by LQR-MPC, there exists an obvious difference of the control inputs obtained separately by CLF-MPC and LQR-MPC. For instance, we can see that the profiles obtained by CLF-MPC are smoother than those by LQR-MPC, which benefits the running of the controlled systems.

Table Ⅱ presents the sum of stage costs $L_i(x, u)$ over the simulation time. From Tables Ⅰ and Ⅱ, it is observed that:

Table  Ⅱ.  THE SUM OF STAGE COSTS $L_i(x, u)$ OVER THE SIMULATION TIME

Controllers	$L_1(x, u)$	$L_2(x, u)$	$L_3(x, u)$
CLF-MPC	14.8126	6.0822	1.6797
LQR-MPC	14.8084	6.0842	1.6818

 | Show Table

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1) Although the size of $X_N$ associated with $X_0(\hat{c})$ is less than that with $X_0(c_3)$, the performance value of $L_3(x, u)$ of CLF-MPC evaluated over the simulation time is better than that of LQR-MPC.

2) The size of $X_N$ associated with $X_0(\hat{c})$ is more than that with $X_0(c_2)$ and the performance value of $L_2(x, u)$ of CLF-MPC is better than that of LQR-MPC.

3) The size of $X_N$ associated with $X_0(\hat{c})$ is much more than that with $X_0(c_1)$, although the performance value of $L_1(x, u)$ of CLF-MPC is worse than that of LQR-MPC.

These observations imply that the CLF-MPC is better than the LQR-MPC in the terms of the feasible state set and the sum of stage costs. Note that one should elaborately select the stage cost $L(x, u)$ in order to achieve some good performances of the feasible state set and the sum of stage costs. Hence, the proposed dual-mode NMPC presents much more flexibility and convenience than the traditional dual-mode NMPC in applications.

Ⅴ. CONCLUSION

The paper presented a new dual-mode nonlinear model predictive control scheme for constrained continuous-time nonlinear systems. The main idea is to use control Lyapunov functions of the nonlinear system to design a terminal controller with some free-parameters and to select the parameters such that the terminal region was as large as possible. Optimality of the terminal controller was regained by optimizing the free-parameters with respect to some given cost functions. Then a varying time-horizon dual-mode NMPC scheme was proposed, whose recursive feasibility and stability were established. The example of the spring-cart system was used to demonstrate the advantages of the proposed scheme here, by comparing with the LQR-based dual-mode NMPC scheme.

APPENDIX A PROOF OF LEMMA 1

Proof: Consider the system (1) and assume $V(x)$ is a known CLF of (1). For any given parameters $\mu\in D$, we have the derivative of $V(x)$ along the trajectories of the system in closed-loop with the control law (4)

From the property of CLFs, setting $u\equiv 0$ leads to $\dot{V}(x)=$ $\alpha(x)$ $\leq$ $0$ for the state $x$ satisfying $\beta(x)=0$. Integrating the state $x$ satisfying $\beta(x)\ne 0$, we have $\dot{V}(x)\leq 0$ for all states.

Based on the Lyapunov's arguments, the above inequality implies that the origin of the closed-loop system is stable. Furthermore, the function $V(x)$ satisfies that

for all states. If the closed-loop system has only trivial solution satisfying $\alpha(x)=0$ and $\beta(x)=0$, then the origin is asymptotically stable according to the LaSalle's theorem [19].