Multi-parametric Optimization and Control Where do …webdav.tuebingen.mpg.de/robust_mpc_legged_robots/oberdieck_slides.pdfWhat type of system? ... The variable space 𝒱becomes discontinuous

Multi-parametric Optimization and Control – Where do we stand?

Richard Oberdieck, Nikolaos A. Diangelakis, Ruth Misener, Efstratios N. Pistikopoulos

Group details and acknowledgment

http://parametric.tamu.edu http://paroc.tamu.edu

We gratefully acknowledge the financial support of EPSRC (EP/M027856/1) Texas A&M University Texas A&M Energy Institute

http://parametric.tamu.edu/

http://paroc.tamu.edu/

Inp

ut

Process‘High Fidelity’ Dynamic Modeling

Output Set-point

Output

Advanced Optimization andControl Policies

Process Modelling to Advanced Optimization and Control Techniques

Inp

ut


Output Set-point

Output

Advanced Optimization andControl Policies

No unified platform

No commercially available tool

No generally accepted procedure or ‘protocol’

via

Multi-parametric

Programming

PAROC

Process Modelling to Advanced Optimization and Control Techniques


‘High Fidelity’ model Dynamic model

Ordinary Differential Equations

Differential Algebraic Equations

Partial DAE

First Principles Models

High complexity

Often non-linear Custom Models

Advanced Model Libraries

Dynamic and steady-state

simulation

Advanced Optimization

Algorithms

Flowsheeting environment

Process Systems Enterprise, gPROMS, www.psenterprise.com/gproms, 1997-2015

PAROC – PARametric Optimization and ControlA unified framework and software platform

System Identification

Model Reduction Techniques

Approximate Model


‘High Fidelity’ model

Model Approximation

Linear state-space models

Model reduction techniques

Statistical methods

Linearization via gPROMS®

Exchange of I/O data via

gO:MATLAB

Execution of gPROMS® model

of arbitrary complexity within

MATLAB®

System Identification Toolbox




Approximate Model

Multi-Parametric Programming



Model Approximation

Multi-Parametric

Programming

Formulation of the optimization

and/or control as a multi-

parametric programming

problem

Explicit map of solutions

mp-LP, mp-QP, mp-MILP

mp-MIQP problems

POP – The Parametric Optimization Toolbox, Pistikopoulos Research Group

http://paroc.tamu.edu/Software




Approximate Model



Multiparametric recedinghorizon policies


Model Approximation

Multi-Parametric

Programming

Multi-Parametric Receding

Horizon Policies

mp-MPC – Control

mp-MHE – State estimation

mp-RHO – Scheduling




Approximate Model


Inp

ut


Output Set-point

Output

Multiparametric recedinghorizon policies

Actions within this area happen once and offline


Model Approximation

Multi-Parametric

Programming

Multi-Parametric Receding

Horizon Policies

Closed-Loop Validation

via gO:MATLAB within

MATLAB®

via C++ within gPROMS®


Focus of this talk

Multi-parametric Optimization and Control

Nominal

controller

Robust

controller

Continuous

systems

Hybrid systems

What type of system?• Discrete time

• Continuous and hybrid systems• Nominal and robust controllers

?


Nominal

controller

Robust

controller

Continuous

systems

Hybrid systems


• 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘

• Only continuous variables

Nominal

controller

Robust

controller

Continuous

systems ?

Hybrid systems

Model Predictive Control (MPC)

Essence: compute the optimal sequence of manipulated variables (inputs) that minimizes

Given: the predicted outputs or states of the system (using a mathematical model) 13

Objective Function = (tracking error, profit, energy etc.)

subject to constraints on inputs and outputs

Model Predictive Control – how to:

1. At time t, given the measurement y(t) (or state x(t))

2. Solve a Constrained Optimisation Problem to obtain:

a. Predicted future outputs (or states): y(t+1|t), y(t+2|t), … , y(t+P|t)b. Optimal sequence of m.v.: U*={u*(t), u*(t+1), u*(t+2), … , u*(t+m-1)}

3. Apply first input of the sequence u*(t) until time t+1

4. At time t+1 repeat14

Explicit/multi-parametric MPC

15

Treat all uncertainty (initial state, measured disturbance etc.) as parameter

Solve for a range and as a function thereof

Obtain explicit solution of the problem

(2) Critical Regions

(1) Optimal look-up function

mp-QP

Multi-parametric Programming – An overview

In multi-parametric programming, an optimization problem is solved for a range and as a function of certain parameters

Θ

𝑥 𝜃 = 𝐾𝜃 + 𝑟

The POP Toolbox – The mp-QP solver

1. Fix 𝜽 = 𝜽𝟎, and solve QPusing the KKT conditions

2. Get parametric solution viaBasic Sensitivity Theorem

3. Define the (critical) regionby optimality and feasibility

4. Cross the facet and findnew 𝜽𝟎

𝐶𝑅0𝐶𝑅1

𝐶𝑅2

𝐶𝑅3

Θ


Nominal

controller

Robust

controller

Continuous

systemsmp-QP

Hybrid systems


• 𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘

• Continuous and discrete variables

Nominal

controller

Robust

controller

Continuous

systemsmp-QP

Hybrid systems ?

Hybrid Systems – From Model to Optimization

Hybrid systems

Systems with continuous and discrete elements, e.g.

Hybrid systems

Hybrid Model Predictive Controller

The optimal control of hybrid systems results in a MIQP

Hybrid Model Predictive Controller

Multi-parametric MIQP

This problem can be solved explicitly as a mp-MIQP

mp-MIQP problems – Solution framework

Pre-Processing

Integer Handling

mp-QP solution

Comparison

Termination? STOPNO YES

mp-MIQP problems – The exact solution

Dua et al. (2002) Axehill et al. (2014)

Oberdieck et al. (2014) Oberdieck and Pistikopoulos (2015)

Comparison over entire CR

Linearization using McCormick

relaxation

The exact solution


Nominal

controller

Robust

controller

Continuous

systemsmp-QP

Hybrid systemsmp-MIQP


• 𝑥𝑘+1 = 𝐴(𝑘)𝑥𝑘 + 𝐵(𝑘)𝑢𝑘

• Continuous variables

Nominal

controller

Robust

controller

Continuous

systemsmp-QP ?


Robust MPC – Conceptual description

Nominal MPC: Robust MPC:

Robust MPC – Open-loop versus Closed-loop

Open-loop robust MPC: Closed-loop robust MPC:

• Find a single optimization sequence

that guarantees feasibility over the

entire horizon

• Ignores receding horizon nature of

the problem, i.e. the state is

measured at every stage

Conservative approach

• Identify the states for which

stage-wise feasibility can be

guaranteed

• Reachability analysis, i.e.

Recursive approach, i.e.

dynamic programming is applied

Nominal controller with reduced

feasible space

Robust MPC – The uncertainty set Ω

General polytope Box-constrained

Extreme points only via verticesExtreme points available in

halfspace representation

Robust Counterpart – Key concept

Robust counterpart

Reformulation of a robust optimization problem into anequivalent (regular) optimization problem

Robust counterpart

Key simplification

Instead of general polytopic Ω, we consider

Key simplification

The reformulation

This allows us to write the following:

Ben-Tal and Nemirovski (2000) Robust solutions of Linear Programming problems contaminated with uncertain data.

Mathematical Programming 88(3), 411 – 424.

Robust mp-MPC – Example problem


Nominal

controller

Robust

controller

Continuous

systemsmp-QP mp-LPs + mp-QP

Progress



Nominal

controller

Robust

controller

Continuous


Progress

Hybrid systemsmp-MIQP ?

• 𝑥𝑘+1 = 𝐴(𝑘)𝑥𝑘 + 𝐵(𝑘)𝑢𝑘

• Continuous and discrete variables

Robust hybrid mp-MPC – Conceptual developments

Apply the same principle: but what changed? The variable space 𝒱 becomes discontinuous (and thus non-

convex):𝒱 = ℝ𝑛 × {0,1}𝑝

Thus, the stage-wise problem becomes mp-MILP This means the reachability analysis becomes more complicated:

…

𝒳𝑁 = 𝒳𝒳𝑁−1 =

𝑖∈ℐ

𝒳𝑁−1𝑖


Nominal

controller

Robust

controller

Continuous


Progress

Hybrid systemsmp-MIQP mp-MILPs + mp-MIQPs

Progress

Multi-parametric Optimization and control – Conclusion

We presented1. An overview over the state-of-the-art in multi-parametric

optimization and control2. Recent results on the exact solution of mp-MIQP problems3. An intuitive way to solve closed-loop robust mp-MPC problems4. The extension to robust hybrid mp-MPC

In future1. Develop automated implementation of robust hybrid mp-MPC

code2. Validate in similar fashion to robust mp-MPC approach3. Tighten the suboptimality of the robust counterpart using novel

counterpart descriptions

Multi-parametric Optimization and Control – Where do we stand?

Richard Oberdieck, Nikolaos A. Diangelakis, Ruth Misener, Efstratios N. Pistikopoulos

Documents

Multi-parametric Optimization and Control Where do …webdav.tuebingen.mpg.de/robust_mpc_legged_robots/oberdieck_slides.pdfWhat type of system? ... The variable space 𝒱becomes discontinuous