Dear Majid,

you raise an interesting question.

Let's consider a discrete-time Optimal Control Problem (OCP) which implicitly defines an MPC feedback:

V_N(\hat x_i) := min \sum_{k=0}^{N-1} l(x_k, u_k) + V_f(x_N)

subject to

x_{k+1} = f(x_k, u_k), x_0 = \hat x_i

x_k \in \mathbb{X}, u_k \in \mathbb{U} \forall k \in {0, ..., N}

x_N \in \mathbb{X}_f

here l(x_k, u_k) is the cost function or stage cost, while V_f(x_N) is the terminal cost. Moreover, V_N(\hat x_i) is denoted as optimal value function of the OCP.

For most of the MPC designs (set stabilization, tracking etc) one proves stability choosing V_N(\hat x_i) as a Lyapunov function candidate, see e.g. Mayne et al. 2000 or Chen & Allgöwer 1998.

In the proof one typically requires that the terminal cost V_f(x_N) satisfies a local Lyapunov function-like decrease property on the terminal set \mathbb{X}_f to show asymptotic stability. In other words, it is required that

\forall x \in \mathbb{X}_f \exists u \in \mathbb{U} s.t.

V_f(f(x,u))- V_f(x) \leq -\l(x, u)

To come back to your question: the typcial Lyapunov function candidate in MPC is the optimal value function V_N. However, to show that V_N is a Lyapunov function of the closed loop, many works require that the terminal cost satisfies a Lyapunov function like descrease property on terminal set.

Other works directly consider V_f to be a control Lyapunov function, see Jadbabaie et al. 2001. Also note that in economic MPC one often considers shifted value functions as Lyapunov function candidates.

I hope this helps to understand what kind of role Lyapunov functions play in stability of MPC.

References:

Mayne, David Q., et al. "Constrained model predictive control: Stability and optimality." Automatica 36.6 (2000): 789-814.

Chen, H., and F. Allgower. "A Quasi-Infinite Horizon Nonlinear Model Predictive Control Scheme with Guaranteed Stability." Automatica 34.10 (1998): 1205-1217.

Jadbabaie, Ali, Jie Yu, and John Hauser. "Unconstrained receding-horizon control of nonlinear systems." IEEE Transactions on Automatic Control 46.5 (2001): 776-783.

Dear Majid,

It should be noted that the MPC control algorithm in general does not require the terminal cost to be a local control Lyapunov function. However, the use of the Lyapunov function in the MPC control is to look for more stability, more robustness, compliance with constraints or remove uncertainties for non-linear systems...

For more more details and information about this subject please see links and attached files in topic.

-Stability and robustness - Centro Piaggio - Unipi

www.centropiaggio.unipi.it/.../3_MPCcourse_stability-robustness.p...

-Economic MPC of Nonlinear Systems with Non-Monotonic Lyapunov ...

https://arxiv.org/pdf/1707.05044

-Stabilizing Model Predictive Control - TU Ilmenau

https://www.tu-ilmenau.de/fileadmin/.../Slides_Part_1_and_2.pdf

-On Input-to-State Stability of Min-max Nonlinear Model Predictive ...

www.mate.tue.nl/mate/pdfs/8071.pdf

Best regards

Dear Timm,

Dear Mohamed,

According to your explanations, I realized the diversity of Lyapunov functions in stability of mpc, and a lot of ambiguities in this regards became clear to me.

Thanks for your valuable guidance