Model Predicive Control An Introduction

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Model Predicive Control- An Introduction

• Johan Åkesson
• Department of Automatic Control
• Lund Institute of Technology

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Introduction

• Why MPC?
• On-line optimization
• Deals explicitly with common problems
• Centralized control
• Complex plants
• Multi-level
• Constraint handling
• Input saturation
• State constraints

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Short History of MPC

• Dynamic Matrix Control (Early 1970s)
• Industrial initiative
• Linear step response models
• IDCOM (Mid 1970s)
• Impulse response models
• Input and output constraints
• Many extensions during the following decades
• Several industrial products

(Cutler and Ramaker 1979)
(Richalet et al. 1976)
(Qin 2002)
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Short History of MPC contd

• Several streams of theory
• Process control
• Generalized predictive control
• Theoretical foundations
• Major research contributions during the last two
  decades

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Areas of application

• Process industry
• Petroleum
• Steel
• Foods
• (Pulp and paper)
• New areas
• Air plane control
• Supply chain management
• Waste water treatment

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Receding Horizon Control

• At time k, solve an open loop optimal control
  problem over a predefined horizon and apply the
  first input.
• At time k1 repeat the same procedure. (The
  previous optimal solution is discarded!)

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Chess as RHC

• Prediction of opponents moves
• Optimization of outcome a few moves ahead
• An unexpected move from the opponent change of
  strategy!
• Good players thinks several moves ahead long
  prediction horizon!

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Receding Horizon Control

• u constant after Hu samples
• Optimization over Hp

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MPC for Linear Systems

• Model assumptions
• Measured outputs
• Controlled outputs
• Constrained outputs

Notation from (Maciejowski 2003)
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An Optimal Control Problem

• The cost function
• Prediction horizon, Hp
• Control horizon, Hu
• First sample to be included, Hw

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Why Penalize ?

• Simple to handle
• No need to specify
• Still possible to penalize !

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Prediction Matrices

• Re-formulation of optimization problem

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Prediction Matrices contd

• Controlled variables

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Prediction Matrices contd

• The prediction Matrices should be calculated
  off-line

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The Optimization Problem

• The free response
• Cost function to minimize
• Control law (calculated off-line)

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Constraints

• Practically all control systems must handle
  constraints
• MPC does it explicitly
• Control increments
• Control variables
• Constrained variables

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Constraints contd

• Constraint matrices
• LICQP problem
• Convex problem
• Efficient algorithms

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Quadratic Programming Solvers

• Active set algorithms (QP)
• Given an active set of constraints, the solution
  may be written on closed form
• Modify the active set until solution found!

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Quadratic Programming Solverscontd

• Primal-dual interior point algorithms
• Solves the problem simultaneous in the primal and
  dual spaces
• Works well for very large systems
• Much to gain by exploring the structure of the
  MPC LICQP problem

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State Estimation

• Usually, the state vector is not available
• Observer necessary, e.g. a Kalman filter
• Optimization based on current state estimates

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Blocking Factors

• Increase horizons without increasing complexity
• Example Let every other be
  zero in the control horizon
• New cost function

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Linearity of the Controller(No active
constraints)

• The controller is linear!
• Only the first control is applied
• Pitfall constraints

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MPC as a 2-DOF Controller

• Linear feedback (if no constraints active)
• Enables standard analysis methods

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Theoretical issues

• Feasibility
• Is there a solution of the optimization problem?
• Stability
• Under what conditions is the closed loop system
  stable?
• Computational complexity
• Performance degradation
• due to delay?

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MPC Stability

• Non-linear controller!
• Use the cost function as a Lyapunov function
• Modifications needed to guarantee stability

(Mayne et. al. 2000)
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MPC Stability contd

• Terminal equality constraint
• Terminal cost function
• Dual mode control infinite horizon
• Terminal constraint set
• Increase feasibility region
• Terminal cost and constraint set

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An MPC Stability Theorem-Terminal equality
constraint

(Bemporad et. al. 1994)
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An MPC Stability Theorem contd

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An MPC Stability Theorem contd

• Optimality not needed for stability!

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An MPC Stability Theorem contd

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An MPC Stability Theorem contd

• What about asymptotic stability?

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Error-free Tracking

• Integral action necessary
• Disturbances
• Plant-model mismatch
• Include explicit integrators?
• How to penalize the integrator states?

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Integral Action

• Use a disturbance observer!
• Assume constant input disturbances
• If pgtm, introduce constant output disturbances on
  additional outputs

(Åkesson and Hagander, 2003)
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Implementational Aspects

• On-line optimization is computationally
  demanding!
• Execution times highly varying

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Explicit MPC

• Linear system model and inequality constraints
• Idea
• The MPC controller is piecewise linear!
• Calculate all controllers off-line
• Partitioning of state space
• At run-time find the right control law!
• On-line optimization eliminated!
• Exactly the same controller!
• Memory consuming

(Bemporad et al. 2002)
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Non-linear MPC

• Assume a non-linear system model
• General cost function

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Non-linear MPC contd

• Pros
• Increased area of applicability
• Stability proofs hold
• Optimality not needed for stability
• Cons
• Convexity lost
• Local minima
• Increased computation time
• Active area of research

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Hybrid MPC

• Mixed Logic Dynamics (MLD)
• Dynamics
• Logic
• State machines
• Hard optimization problems
• Mixed Integer Quadratic Programs
• Specialized solvers
• Successful implementations!

(Bemporad et al. 1999)
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MPC tools for Matlab

• Enable detailed analysis
• Controller behavior
• Optimization algorithm behavior
• Linear system models
• Hard constraints
• Kalman filter state estimation
• Error-free tracking

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MPC tools for Matlab

• MPC controller implementation
• Matlab/Simulink
• QP solver implementations
• Active set
• Interior point
• Supports Matlabs quadprog

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MPC tools for Matlab
MPCInit MPCSim MPCController MPCfrsp
getfeasible qp_as qp_ip

• Tools will be used in exercises

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Case StudyThe quadruple tank

• MIMO system
• Non-minimum phase dynamics

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Case study contd

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Case Study contd

• Control of the linearized model

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Case Study contd

• Control of the nonlinear plant

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Demo of MPC Tools