David Clarkeand Model Predictive Control by tzv97744

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									David Clarke and Model Predictive Control

     In celebration of David Clarke’s
           contribution to MPC
      St Edmunds Hall, Oxford University, January 9, 2009


                         David Mayne
                    Imperial College London


                                                            IC – p.1/30
           David

Congratulations on your many
       achievements!




                          IC – p.2/30
CONTENTS


 •   SOME OF DAVID’S ACHIEVEMENTS

 •   WHERE IS MPC NOW?

 •   A CURRENT ISSUE: ROBUST MPC

 •   FUTURE CHALLENGES

 •   CONCLUSIONS




                                    IC – p.3/30
SOME OF DAVID’S ACHIEVEMENTS

 •   Identification
 •   Adaptive and Self-tuning Control
 •   GPC
 •   Smart, self-validating sensors and actuators
 •   Director of Invensys: University Technology Centre for
     Advanced Instrumentation




                                                     IC – p.4/30
Identification


  •   Ph.D. Topic (1967)
  •   Generalised least squares for system identification
  •   Implemented on a computer ... in 1967!
  •   2nd UKAC Convention, Bristol 1967




                                                      IC – p.5/30
Adaptive and self-tuning control

  •   System: A(δ)y(t) = B(δ)u(t) + ξ(t)
  •   Hot topic in 1970’s
  •   Many, many proposals
  •   Revolution: Astrom-Wittenmark 1973
       • Minimum variance control
       • Least squares estimation of parameters
       • Certainty equivalence
       • Magical result:


 IF parameters converge, they converge to
       minimum variance controller!
                                                  IC – p.6/30
Clarke’s adaptive controller

  •   Two shortcomings in min var adaptive controller
       • Agressive control (cancels expected error)
       • Stable zero dynamics required but
         • rapid sampling generates u/s zeros
  •   Clarke-Gawthrop solution (1975):
       • Replace minimum variance control by
       • arg minu(t) EI(t) {y(t + k)2 + λu(t)2 }

  •   Costing u reduces control activity with small increase
      in variance of o/p y
  •   CL stability requires OL stability and adjustment of λ
  •   Considerable impact (338 google citations)
                                                        IC – p.7/30
Generalised Predictive Control

  •   Clarke introduced GPC in 1987 as a general method
      of control for unconstrained linear systems that:
       • are non-minimum phase
       • are OL unstable
       • have unknown dead-time
       • have unknown order
  •   Method:
       • Minimize EI(t) { N (y(t + j)2 + λu(t + j)2 } wrt
                           j=0
         sequence u = {u(t), u(t + 1), . . . , }
       • Apply the first element of the minimizing sequence
         to the plant
       • This is receding horizon control
                                                     IC – p.8/30
Generalised Predictive Control

  •   Clarke’s two 1987 papers on GPC had a substantial
      impact
  •   First paper has 854 citations (Google)
  •   Papers contain rich set of extensions:
       • Tuning knobs (generalised output)
       • Rejection of constant disturbances
       • Adaptation
       • Ability to achieve wide range of control objectives
       • Terminal equality constraint (on y ) to ensure cl
         stability
       • Ability to handle control constaints (1988)

                                                        IC – p.9/30
Recent research


  •   Control loop tuning
  •   Performance monitoring
  •   Self-validating sensors and actuators
  •   Bounds on ultimate performance of sensors, and
  •   Design of sensors that approach these bounds




                                                     IC – p.10/30
WHERE IS MPC NOW?

 •   GPC restricted to linear systems
 •   In linear context had broad focus
      • eg adaptation, tuning
      • reflecting Clarke’s concern for application
 •   MPC: Constrained linear and nonlinear systems
      • Narrower focus in broader context
      • Sufficient conditions for stability
      • Suboptimal MPC for NL systems
      • Unreachable set points
      • Distributed MPC, etc
      • Improved optimization procedures
 •   Uncertain systems ... jury still out            IC – p.11/30
A CURRENT ISSUE: ROBUST MPC

 •   MPC successful for deterministic systems because
      • Solution of OL OC Pb (for given initial state)
      • is same as FB solution (via DP) for same state
      • Feedback not necessary for deterministic system
 •   To get similar properties for robust MPC
      • requires optimization over control policies:
        π = {µ0 (·), µ1 (·), . . . , µN −1 (·)}
      • subject to satisfaction of all constraints by all
        realizations of state and control trajectories
      • Impossibly complex
      • Implementation requires simplification


                                                        IC – p.12/30
     Robust MPC

        •   Pb simple to state – hard to solve
        •   Need implementable sol’n ≈ exact sol’n
        •   B’ded dist implies approx’n needed only over ’tube’


                     x
PSfrag replacements          Disturbed sol’n
                                    Nominal sol’n



                         0                           k
                                                           IC – p.13/30
Robust MPC

 •   Linear approx’n of opt policy over tube seems good
 •   For system x+ = Ax + Bu + w, quadratic cost, initial
     state x
      • opt control at (x, i) is v(i) + K(x − z(i))
      • {v(i)} is opt sol’n to nominal pb, initial state x
      • {z(i)} is resultant state sequence, v(i) = Kz(i)
 •   If w ∈ W , W compact, x(i) lies in z(i) + S where S is
     compact (K any stabilizing controller)
 •   Basis of tube sol’n for robust MPC for constrained
     linear systems
 •   that merely requires solving conventional MPC Pb
     with tightened constraints
                                                      IC – p.14/30
     Tube MPC for constrained linear systems

                          z(i)




          z(0)

ements
  x(0)

   x(i)

(i) ⊕ S
      S                                   IC – p.15/30
     Tube MPC for constrained linear systems

                       z(i)
                              z(i) ⊕ S




          z(0)
                 S
ements
  x(0)

   x(i)

                                          IC – p.15/30
    Tube MPC for constrained linear systems

                      x(i)
                             z(i)
                                z(i) ⊕ S
    x(0)

           z(0)
                  S
ements




                                           IC – p.15/30
    Tube MPC for constrained linear systems

            Original constraint          Tightened constraint
                           x(i)
                                  z(i)
                                     z(i) ⊕ S
     x(0)

ements      z(0)
                     S




                                                        IC – p.15/30
      Constrained linear system: output MPC

                         z(i)




ements     z(0)
  ˆ
  x(0)

   ˆ
   x(i)

(i) ⊕ S
      S
   x(0)
    x(i)                                      IC – p.16/30
    Constrained linear system: output MPC

                      ˆ
                      x(i)
                             z(i)
                                z(i) ⊕ S
    ˆ
    x(0)

ements     z(0)
                  S




  x(0)
  x(i)                                      IC – p.16/30
      Constrained linear system: output MPC


           x(0)

      ˆ
      x(0)

ements            z(0)
                         S

    ˆ
    x(i)
    z(i)
(i) ⊕ S


   x(i)                                       IC – p.16/30
     Constrained linear system: output MPC

           Original constraint   Tightened constraint
ements x(0)

      ˆ
      x(0)
    ˆ
    x(i)
           z(0)
    z(i)           S
(i) ⊕ S


   x(i)



                                                    IC – p.16/30
Tube MPC

 •   Tube MPC applicable to constrained linear systems:
      •   With additive bounded disturbance
      •   With parametric uncertainty
      •   With uncertain state (O/P MPC using observer +
          certainty equivalence)
 •   BUT can it be used for constrained NL systems?




                                                   IC – p.17/30
Tube MPC: constrained NL systems

  •   Can tube approach be extended to NL systems?
  •   System x+ = f (x, u) + w, w ∈ W
  •   At first sight, looks very difficult
  •   Need a control law valid in tube
  •   For NL systems, determination of control law (in
      contrast to control sequence) difficult
  •   In linear case, law is x → v + K(x − z) where
      v = κN (z), z and K easily determined
  •   NL case?


                                                      IC – p.18/30
Tube MPC: constrained NL systems

  •   Proposal: instead of determining control law, use
      second MP Controller to compute control action for
      each state
       • Compute {v(i)} and {z(i)}, solution of OC Pb for
         nominal system z + = f (z, v)
       • At each (x, z), solve ancillary pb PN (x, z) to
         determine u
  •   What should ancillary pb PN (x, z) be?




                                                    IC – p.19/30
What should ancillary pb be?

  •   To motivate: look at LQG Pb
  •   Suppose we have solution {z(i)}, {v(i)} to nominal
      OC Pb
  •   Then OC at any x is solution to ancillary nominal Pb
  •   in which cost is second variation cost (quadratic and
      zero at solution of nominal OC Pb)
  •   Ancillary controller steers trajectories towards the
      nominal solution.
  •   And bounds their deviation from the optimal nominal
      trajectory


                                                       IC – p.20/30
    Solutions of ancillary Pb

                 x(0)

          z(0)
                                nominal


ements                           ancillary




  x(1)
  z(1)
                                 IC – p.21/30
    Solutions of ancillary Pb

                 x(0)              x(1)

          z(0)
                                          nominal
                            z(1)
ements                                     ancillary




                                           IC – p.21/30
The ancillary problem

  •   The ancillary Pb is deterministic
  •   Uses nominal system x+ = f (x, u),
  •   Cost = deviation from optimal nominal trajectory:
  • VN (x, z, u) =     N −1
                       i=0    (x(i) − z(i), u(i) − v(i))
  • u = {u(0), u(1), . . . , (N − 1)}
  •   Ancillary OC Pb:
      u0 (x, z) = arg minu {VN (x, z, u | u ∈ UN , x(N ) = z(N )}
  • κN (x, z)=first   element of sequence u0 (x, z)
  •   Apply resultant control u = κN (x, z) to plant.
  • κN (x, z)   replaces v + K(x − z)
                                                           IC – p.22/30
Constrained NL systems
  • κN (x, z),     sol’n of ancillary OC Pb
  • V 0 (x, z)   is value fn of ancillary Pb)
     N

  •   Let Sd (z)           0
                     {x | VN (x, z) ≤ d}
  • Sd (z)                    0
             is level set of VN (x, z)
  •   There exists a d > 0 such that:
  • x(0) ∈ Sd (z(0)) =⇒ x(i) ∈ Sd (z(i)), u(i) ∈ U ∀i

  •   Similar to x(i) ∈ z(i) + S in linear case
  •   But sets Sd (z) cannot be predetermined
  •   Choosing tighter constraints for nominal OC Pb hard
                                                        IC – p.23/30
       Constrained NL systems

                         z(i)


       x(0)


            z(0)

ements



Sd (z(i))
om traj                         IC – p.24/30
       Constrained NL systems

                         z(i)


       x(0)

                                nom traj
            z(0)

ements



Sd (z(i))
                                           IC – p.24/30
       Constrained NL systems

                           x


                         z(i)
                                actual traj

       x(0)

                                       nom traj
            z(0)
ements



Sd (z(i))                                         IC – p.24/30
    Constrained NL systems

                        x    Sd (z(i))

                      z(i)     actual traj

         x(0)

                                         nom traj
           z(0)
ements




                                                    IC – p.24/30
Ancillary controller

  •   The nominal controller steers initial state to desired
      state, neglecting disturbances
       • Responds to changed in desired final state
  •   The ancillary controller reduces effect of disturbances
       • Can be tuned
       • Can have distinct cost function
       • Can have different sampling period
  •   Analagous to two-degree of freedom controller




                                                        IC – p.25/30
Example: Control of CSTR
                              Sampling Rate = 12s / Prediction Horizon = 360s
                         1

       Concentration
                       0.8
                       0.6
                       0.4
                       0.2
                         0
                          0      100          200           300        400      480
                              Sampling Rate = 8s / Prediction Horizon = 240s
                         1
       Concentration




                       0.8
                       0.6
                       0.4
                       0.2
                         0
                          0      100         200           300         400      480
                              Sampling Rate = 4s / Prediction Horizon = 120s
                         1
       Concentration




                       0.8
                       0.6
                       0.4
                       0.2
                         0
                          0       100         200         300          400      480
                                                                                      IC – p.26/30
Control of CSTR

                   100
                          Standard
                    80      MPC
       Frequency

                    60
                    40
                    20
                     0
                     20      25      30   35   40      45    50   55   60   65
                                                Cost

                   400    Tube−based
                             MPC
                   300
       Frequency




                   200

                   100

                     0
                     20      25      30   35   40       45   50   55   60   65
                                                    Cost

                                                                                 IC – p.27/30
FUTURE CHALLENGES

 •   Output MPC for nonlinear systems
 •   Adaptive MPC
      • Difficulty: uncontrollable subsystem modelling
        unknown parameters
 •   Distributed MPC
      • Cooperative vs non-cooperative
 •   Stochastic
      • Constraints?
 •   Computation
      • Fast systems


                                                   IC – p.28/30
CONCLUSION



 •   MPC can solve a wide range of control problems for
     deterministic or uncertain, linear or nonlinear,
     constrained systems


 •   Because it creates its own Lyapunov function


 •   There remain big challenges



                                                    IC – p.29/30
CONGRATULATIONS

     DAVID



              IC – p.30/30

								
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