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
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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: PN 2 + λu(t + j)2 } wrt • Minimize EI(t) { (y(t + j) 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)
lacements x(0) x(i) z(i) ⊕ S S
IC – p.15/30
Tube MPC for constrained linear systems z(i) z(i) ⊕ S
z(0)
S
lacements 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
lacements
IC – p.15/30
Tube MPC for constrained linear systems Original constraint x(i)
Tightened constraint z(i) z(i) ⊕ S
x(0)
lacements
z(0)
S
IC – p.15/30
Constrained linear system: output MPC z(i)
lacements xˆ(0)
z(0)
xˆ(i) z(i) ⊕ S S x(0) x(i)
IC – p.16/30
Constrained linear system: output MPC xˆ(i)
z(i) z(i) ⊕ S
xˆ(0)
lacements
x(0) x(i)
z(0)
S
IC – p.16/30
Constrained linear system: output MPC
x(0) xˆ(0)
lacements
z(0)
S
xˆ(i) z(i) z(i) ⊕ S
x(i)
IC – p.16/30
Constrained linear system: output MPC Original constraint
Tightened constraint
lacements x(0) xˆ(0) xˆ(i) z(0) z(i) z(i) ⊕ S
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 7→ 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
lacements
ancillary
x(1) z(1) IC – p.21/30
Solutions of ancillary Pb
lacements
x(0)
x(1)
z(0)
nominal z(1)
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: PN −1 • VN (x, z, u) = 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) N
is value fn of ancillary Pb)
•
Let Sd (z) , {x | VN0 (x, z) ≤ d}
• Sd (z) •
is level set of VN0 (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)
lacements
Sd (z(i)) nom traj
IC – p.24/30
Constrained NL systems z(i)
x(0)
z(0)
nom traj
lacements
Sd (z(i)) IC – p.24/30
Constrained NL systems x
z(i)
actual traj
x(0)
z(0)
nom traj
lacements
Sd (z(i))
IC – p.24/30
Constrained NL systems x
z(i)
Sd (z(i))
actual traj
x(0) z(0)
nom traj
lacements
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
Concentration
Concentration
Concentration
Example: Control of CSTR 1 0.8 0.6 0.4 0.2 0 0 1 0.8 0.6 0.4 0.2 0 0 1 0.8 0.6 0.4 0.2 0 0
Sampling Rate = 12s / Prediction Horizon = 360s
100 200 300 400 Sampling Rate = 8s / Prediction Horizon = 240s
480
100 200 300 400 Sampling Rate = 4s / Prediction Horizon = 120s
480
100
200
300
400
480
IC – p.26/30
Control of CSTR 100 Frequency
80
Standard MPC
60 40 20 0 20
25
30
35
40
45
50
55
60
65
45 Cost
50
55
60
65
Cost
Frequency
400 300
Tube−based MPC
200 100 0 20
25
30
35
40
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
