---
title: Approximations of MPC: Dynamic KKT & Learning based Approaches
tags:  #制御工学 #モデル予測制御 #機械学習 #最適化  
author: [Masaki Inoue](https://www.docswell.com/user/minoue227)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/GJ8DY3PGJD.jpg?width=480
description: セビリア大学での招待講義の資料です。
published: February 18, 26
canonical: https://www.docswell.com/s/minoue227/ZX6M2N-2026-02-18-135816
---
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Approximations of MPC:
Dynamic KKT & Learning
based Approaches
Masaki Inoue
Keio University
minoue@appi.keio.ac.jp
Seminar, University of Seville
May 30, 2022


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Various Control Applications
Air Traffic Management
Energy Management
fuel efficiency
safety
Light Guidance Control
reduced
discomfort
complex driver
behavior
2
stability
economic dispatch
Smart Agriculture
reduced
workload
preferred environment


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Model Predictive Control (MPC)
• Plant Model
• extended vector
• Control Law
ex. model-based state prediction:
• find optimal action
based on the measured state
• handle complex design specifications
• require real-time solution of optimization problem
3


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Approximation of Control Law
4
➢Control Law in MPC
➢Optimality Condition:
inequality constraints
are omitted in this talk
(can be extended)
• Karush–Kuhn–Tucker (KKT) condition
essence of high
computational burden
• control law is given by implicit function like
➢Aim of this talk:
Approximation of control law by explicit function like


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Two Approaches
MPC
i.
5
approx MPC
Approximation by dynamic KKT (instant MPC)
• original work: [K. Yoshida et al, IEEE LCSS, 2019]
• extensions: [M. Figura et al, ACC, 2020], [K. Sato et al, SICE AC, 2020]
ii. Approximation by learning technique
• [K. Hara et al, IFAC WC, 2020], [K. Hara et al, Automatica (conditionally accepted)]
• Others by many researchers
• explicit MPC: [Bemporad et al, Automatica, 2002] etc
• neural network based MPC: [T. Parisini & R. Zoppoli, Automatica, 1995] etc


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Outline
1. Preliminary: Dissipation-based System Analysis
• tool for closed-loop stability of approximated MPC
2. Dynamic KKT-based Approximation of MPC
3. Learning-based Approximation of MPC
6


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1. On System Description
➢ State equation
• basic description of dynamical systems
• express detailed behavior of IO dynamical system
• systems interconnection increases state space dimension
state equation is not always
tractable for large-scale system
analysis & design


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1. Differential Equation & Inequality (revisit)
➢ 1-dim differential equation
➢ 1-dim differential inequality
• is a description of a set of dynamical systems
• express rough behavior of dynamical system
➢ Model “Reduction” of state equation
finding energy function
reduced to 1-dim diff inequality


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1. Description by Dissipation Inequality
➢ Dissipation inequality (Willems)
• special case: QSR dissipativity (Hill & Moylan)
upper bound of energy increase
•
Im
Example 1: bounded L2 gain
Im
0
•
Example 2: passivity
Re
Re
0
Nyquist diagram


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1. Feedback Connection
ー
➢ Assumption: dissipation in subsystems
connected as
➢ Dissipation in feedback system
• evaluated by sum of energies


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1. Stability Theorem
ー
➢ Dissipation in feedback system
➢ Theorem:
Suppose (*) is negative (semi)definite. Then, the origin
of FB system is asymptotically (Lyapunov) stable
• small-gain theorem:
,
• passivity theorem:
,
• only few parameters are required for stability analysis
(NO detailed model like state equation)


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Outline
12
1. Preliminary: Dissipation-based System Analysis
• tool for closed-loop stability of approximated MPC
2. Dynamic KKT-based Approximation of MPC
• Main Reference: [K. Yoshida et al, LCSS, 2019]
3. Learning-based Approximation of MPC
MPC
approx MPC


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2. Dynamic KKT
➢ KKT condition (recall)
optimization problem
➢ dynamic KKT condition
• reduced to KKT condition at steady state (
• primal-dual gradient algorithm:
)
[K. J. Arrow et al, Studies in Linear and Non-Linear Programming, 1958]
• extension to handle inequality constraints:
[A. Cherukuri et al, SCL, 2016], [A. Adegbege, IFAC WC, 2020]


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2. Instant MPC
➢ Instant MPC:
dynamic controller realized by optimization algorithm
associated with MPC
approx MPC
• less approx. error for slow dynamics
• No error at DC gain
• explicit control law like
• sketch for discrete-time algorithm case
MPC
iMPC
objective function


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2. Demonstration of instant MPC
➢ Attitude control of aircraft [K. M. Sobel et al, JGCD, 1985]
➢ Simulation results
• suboptimal performance
• constraint violation for some algorithm
• avoided by other PDG algorithm [A. Adegbege, IFAC WC, 2020]
• significantly reduce computational burden
MPC（cplex）
18.13 ms
iMPC
0.2070 ms


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2. Stability Analysis of instant MPC
➢ Assumption: dissipativity of plant system
iMPC
➢ Extension of instant MPC
• add tunable parameters that do not shift the steady state
Note:
at steady state
➢ Theorem: existence of parameters s.t.
• dissipativity of iMPC:
• close-loop stability:


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2. Short Summary on instant MPC
MPC
iMPC
➢ Instant MPC:
suboptimal control realized by solution algorithm
Main Reference: [K. Yoshida et al, LCSS, 2019]
• guarantee optimality at steady state
• related works: [A. Jokic et al, TAC, 2009], [L. Lawrence, TAC, 2021]
• guarantee closed-loop stability based on dissipation theory


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Outline
18
1. Preliminary: Dissipation-based System Analysis
• tool for closed-loop stability of approximated MPC
2. Dynamic KKT based Approximation of MPC
• main reference: [K. Yoshida et al, LCSS, 2019]
3. Learning-based Approximation of MPC
• main reference: [K. Hara et al, IFAC WC, 2020],
[K. Hara et al, Automatica (conditionally accepted)]
MPC
approx MPC


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3. Learning MPC
19
➢Idea of data-driven approximation
• collect input-output data on optimal control law offline
• apply data-driven modeling to find explicit control law
MPC
data collection
approx MPC
modeling
➢Related (classic) works
• neural network-based approximation of optimal control:
[D. Psaltis et al, CSM, 1988], [T. Parisini el al, Automatica, 1995]
➢This work: stability-preserving data-driven approximation


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3. Controller Structure
20
➢Approximation of observer-based MPC
• find suitable lifted state
•
observer
MPC
e.g. radial basis function
approximation
• state equation for controller
• focus trajectory of lifted state
• nonlinear in state
• linear in system matrices
, not of state
& lifted state
➢Preliminary for data-driven approximation
• perform offline simulation of MPC to generate
• transform to lifted state
& input


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3. Dissipativity-Constrained Learning
➢Preliminary 2:
• impose dissipation specification
approx MPC
➢Formulation: dissipativity-constrained learning
control
performance
closed-loop
stability
• nonlinear matrix inequality is reduced to LMI [Y. Abe et al, ECC, 2016]
• idea for reduced conservativeness [K. Hara et al, IFAC WC, 2020] etc
21


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3. Demonstration
22
➢Nonlinear plant:
• passive [H. Zekari & P. Antsaklis, IJC, 2019]
➢Suboptimal controller design:
Simulation result:
• MPC & observer design
• IO data collection
• passivity-constrained learning
ーpassivity-constrained
ーno-constrained
closed-loop stability


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Summary: Approximation of MPC
1. Dynamic KKT-based Approximation
• instant MPC is realized by optimization algorithm associated with MPC
• guarantee optimality at steady-state
• stability analysis by dissipativity theory
2. Learning-based Approximation
• guarantee stability by
dissipativity-constrained learning
• suboptimality analysis by Koopman operator theory (omitted)
• present LMI-based solution method (omitted)
MPC
approx MPC
23