Explicit constrained nonlinear MPC - Nettsiden er flyttet formulation Nonlinear system: Optimization...

Explicit constrained nonlinear MPC

Tor A. Johansen

Norwegian University of Science and Technology, Trondheim, Norway

Main topics

� Approximate explicit nonlinear MPC based on orthogonal search trees

� Approximate explicit nonlinear MPC based on local mp-QP solutions

� Discussion and summary


1

Part I

Approximate explicit constrained nonlinear MPCbased on orthogonal search trees

� Nonlinear Constrained MPC formulation

� Theory: Stability, optimality, convexity

� Approximate mp-NLP via search tree partitioning

� Example


2

MPC formulation

Nonlinear system:

��

Optimization problem (similar to Chen and Allgower (1998)):

� ��

��

��

��

��

��

� ��

��

�

with � � �� subject to �� and

��

��

��

��

��


3

MPC formulation, assumptions

A1. �� .

A2. �� and �� .

A3. The function � is twice continuously differentiable, with �� .

The compact and convex terminal set is defined by

� ��

where � � � and � � � will be specified shortly.


4

Terminal set

Similar to Chen and Allgower (1998):

A4. �� is stabilizable, where

��

��

��

Let � denote the associated LQ optimal gain matrix, such that ��

is strictly Hurwitz.

Lemma 1. If � � � is such that �� is strictly Hurwitz, the Lyapunovequation

��

has a unique solution � �.


5

Terminal set, cont’d

Lemma 2. Let satisfy the conditions in Lemma 1. Then there exists aconstant � � � such that � �� satisfies

1. � � �� .

2. The autonomous nonlinear system��

is asymptotically stable for all �� , i.e. is positively invariant.

3. The infinite-horizon cost for the system in 2, given by

��

��

��

��

��

satisfies �� for all � � .


6

Multi-parametric nonlinear program (mp-NLP)

This and similar optimization problems can be formulated in a concise form� ��

�

�� subject to ��

This defines an mp-NLP, since it is an NLP in � parameterized by � � ��.

Define the set of � -step feasible initial states as follows

� � �� for some � � ��

We seek an explicit state feedback representation of the solution function

�� to this problem: Implementation without real-time optimization!


7

How to represent �� and ��?

mp-LP/mp-QP: Exact piecewise linear (PWL) solution, on a polyhedral parti-tion of � .

mp-NLP: No structural properties that can be exploited to determine a finiterepresentation of the exact solution.

Keep the PWL representation as an approximation, as it is convenient to workwith!


8

Orthogonal partitioning via a search tree

PWL approximation is flexible and leads to highly efficient real-time search.

How to construct the partition, and local linear approximations?


9

PWL approximate mp-NLP approach

How to construct the partition, and local linear approximations?

Construct feasible local linear approximation to the mp-NLP solution from NLPsolutions at a finite number of samples in every region of the partition.

Construct the partition such that the approximate solution has a cost close tothe optimal cost for all � � � .

This is easier under convexity assumptions.


10

mp-NLP and convexity

A5. � and� are jointly convex for all �� , where � � ��

is the set of admissible inputs.

The optimal cost function � � can now be shown to have some regularity prop-erties (Mangasarian and Rosen, 1964):

Theorem 1. � is a closed convex set, and � � � � � is convex andcontinuous. �

Convexity of � and � � is a direct consequence of A5, while continuity of � �

can be established under weaker conditions (Fiacco, 1983).


11

Optimal cost function bounds

Consider the vertices � � �� of any bounded polyhedron ��

� . Define the affine function � �� as the solution to thefollowing LP:

��

� ��

��

subject to � �� for all � � ��

Likewise, define the convex PWL function

� ��

��

� ��

Theorem 2. Consider any bounded polyhedron �� . Then � ��

� �� for all � � ��. �

These bounds can be computed by solving � NLPs.


12

Feasible local linear approximation

Similar to Bemporad and Filippi (2003):

Lemma 3. Consider any bounded polyhedron�� with vertices �� .If �� and �� solve the convex NLP ( � � arbitrary):

��

��

��

��

��

subject to ��

then �� is feasible for the mp-NLP for all � � ��. �

Can be computed from the solution of the same NLPs as above...


13

Sub-optimality bound

Since �� defined in Lemma 3 is feasible in ��, it follows that

��

is an upper bound on � �� in �� such that for all � � ��

� � �� !�

where

!� � � ��

��

��


14

Approximate mp-NLP algorithm1. Initialize the partition to the whole hypercube, i.e. � ��. Mark the hypercube � asunexplored.

2. Select any unexplored hypercube �� . If no such hypercube exists, the algorithmterminates successfully.

3. Solve the NLP for � fixed to each of the vertices of the hypercube �� (some of these NLPsmay have been solved in earlier steps). If all solutions are feasible, go to step 4. Otherwise,compute the size of ��. If it is smaller than some tolerance, mark �� explored and infeasible.Otherwise, go to step 8.

4. If � � ��, choose �� and go to step 5 . Otherwise, go to step 6.

5. If �� , mark �� as explored and go to step 2. Otherwise, go to step 8.

6. Compute a feasible affine state feedback ��, as an approximation to be used in ��. If nofeasible solution was found, go to step 8.

7. Compute the error bound ��. If �� , mark �� as explored and feasible, and go to step2.

8. Split the hypercube �� into two hypercubes �� and �� using some heuristic rule. Markboth unexplored, remove �� from , add �� and �� to and go to step 2.


15

Example, Chen and Allg ower (1998)

�� "

�� "

The origin is an unstable equilibrium point, with a stabilizable linearization.We discretize this system using a sampling interval #� � ��. The controlobjective is given by the weighting matrices

� �

#�� #�

The terminal penalty is given by the solution to the Lyapunov equation

�

��

��

and a positively invariant terminal region is

� ��

The prediction horizon is � � ��"#� � ��.


16

Example, partition with 105 regions

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x1

x 2

Real-time computations: 14 arithmetic operations per sample.


17

Example, solution and cost

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

1−2

−1

0

1

2

x2x

1

u* (x 1,x 2)

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

1−2

−1

0

1

2

x2x

1

u(x 1,x 2)

Optimal solution �� to the left and approximate solution �� to the right.

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

10

10

20

30

40

50

x2x

1

V* (x 1,x 2)

−1

−0.5

0

0.5

1 −1

−0.5

0

0.5

10

10

20

30

40

50

x2x

1

V(x 1,x 2)

Optimal cost function � �� to the left and sub-optimal cost � �� to the right.


18

Example, simulations

0 1 2 3 4 5 6 7 8 9 10−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

t

x1(t)

0 1 2 3 4 5 6 7 8 9 10−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

t

x2(t)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

t

u(t)


19

Properties of the algorithm

The PWL approximation generated is denoted �� , where � � isthe union of hypercubes where a feasible solution has been found. It is aninner approximation to � and the approximation accuracy is determined bythe tolerance in step 3.

Theorem 3. Assume the partitioning rule in step 8 guarantees that the errordecreases by some minimum amount or factor at each split. The algorithmterminates with an approximate solution function �� that is feasible and satis-fies

� � �� !

for all � � � �. �


20

Stability

The exact MPC will make the origin asymptotically stable (Chen and Allgower,1998) . We show below that asymptotic stability is inherited by the approxi-mate MPC under an assumption on the tolerance !:

A6. Assume the partition � generated by Algorithm 1 has the property thatfor any hypercube �� that does not contain the origin

! � $ ��

��

��

where $ � �� is given.

Theorem 4. The origin is an asymptotically stable equilibrium point, for all

�� . �


21

Non-convex problems

If convexity does not hold, global optimization is generally needed if theoreticalguarantees are required:

1. The NLPs must be solved using global optimization in steps 3 and 6.

2. The computation of the error bound !� in step 7 must rely on global opti-mization, or convex underestimation.

3. The heuristics in step 8 may be modified to be efficient also in the non-convex case.


22

Key referencesT. A. Johansen, Approximate Explicit Receding Horizon Control of Constrained NonlinearSystems, submitted for publication, 2002

H. Chen and F. Allgower, A quasi-infinite horizon nonlinear model predictive control schemewith guaranteed stability, Automatica, Vol. 34, pp. 1205-1217, 1998

A. V. Fiacco, Introduction to sensitivity and stability analysis in nonlinear programming, Or-lando, Fl: Academic Press, 1983

T. A. Johansen and A. Grancharova, Approximate explicit constrained linear model predictivecontrol via orthogonal search tree, IEEE Trans. Automatic Control, accepted, 2003

A. Grancharova and T. A. Johansen, Approximate Explicit Model Predictive Control Incorpo-rating Heuristics, IEEE Conf. Computer Aided Control Design, Glasgow, 2002

A. Bemporad and C. Filippi, Suboptimal explicit RHC via approximate quadratic programming,J. Optimization Theory and Applications, Vol. 117, No. 1, 2003


23

Part II

Approximate explicit constrained nonlinear MPCbased on local mp-QP solutions.

� Nonlinear Constrained MPC formulation

� Multi-parametric Nonlinear Programming (mp-NLP)

� An approximate mp-NLP algorithm

� Example: Compressor surge control


24

Nonlinear Constrained Model Predictive Control

Nonlinear dynamic optimization problem�� # �� # ��

��

�� %�� &��#�� #�

subject to the inequality constraints for � � �� # �

��

��

and the ordinary differential equation (ODE) given by

%%��

with given initial condition �� .


25

Multi-parametric Nonlinear Programming

The input �� is assumed to be piecewise constant and parameterized by avector � � � � such that �� '�� is piecewise continuous.

The solution to the ODE is �� (�� , and the problem is reformu-lated as:

� ��

��

��(�� '�� %�� &�(�#� �� #�

subject to

��

��

� � ��

��

��

This is an NLP in � parameterized by ��, i.e. an mp-NLP. Solving the mp-NLP amounts to determining an explicit representation of the solution function

��.


26

Explicit solution approach

mp-LP and mp-QP solvers are well established: � �� is piecewise linear(PWL).

In mp-NLPs (unlike mp-QPs) there are no structural properties to exploit, suchthat we seek only an approximation to the solution � ��.

Main idea: Locally approximate the mp-NLP with a convex mp-QP. Iterativelysub-partition the space until sufficient agreement between the mp-QP andmp-NLP solutions is achieved.

This leads to a PWL approximation, which allows MPC implementation with aPWL function evaluation rather than solving an NLP in real time.


27

Assumptions

Consider any �� , and solution candidate �� with associated Lagrangemultipliers )� and optimal active set ��.

Assumption A1. � and � are twice continuously differentiable in a neighbor-hood of �� .

Assumption A2. The sufficient first- and second-order conditions (Karush-Kuhn-Tucker) for a local minimum at �� hold.

Assumption A3. Linear independence constraint qualification (LICQ) holds,i.e. the active constraint gradients ��

�� are linearly independent.

Assumption A4. Strict complementary slackness holds, i.e. �)��

� �.


28

Basic result (Fiacco, 1983)

The optimal solution �� depends in a continuous manner on �: It may bemeaningful with a PWL approximation to � ��!

Theorem 1. If A1 - A3 holds, then

1. �� is a local isolated minimum.

2. For � in a neighborhood of ��, there exists a unique continuous function

�� satisfying �� and the sufficient conditions for a local mini-mum.

3. Assume in addition A4 holds, and let � be in a neighborhood of ��. Then

�� is differentiable and the associated Lagrange multipliers )�� exists,and are unique and continuously differentiable. Finally, the set of active con-straints is unchanged, and the active constraint gradients are linearly inde-pendent at ��.


29

Local mp-QP Approximation

Minimize��

�� *��

��+�� ,�� -��

subject to�� .�� #�

The cost and constraints are defined by the matrices

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��


30

KKT conditions

Let the PWL solution to the mp-QP be denoted �� with associated La-grange multipliers )�� :

*��

��

� ,�� +��)��

diag�)��

�� .�� #��

� �

)��

��

��

�.�� #� � �

Assumption A5. For an optimal active set �, the matrix �� has full rowrank (LICQ) and /�

�� *�/�� , where the columns of /�� is a basis fornull�� .


31

QP-solution for fixed active set

Theorem 2. Let � be a polyhedral set with �� . If assumption A5 holds,then there exists a unique solution to the mp-QP and the critical region wherethe solution is optimal is given by the polyhedral set

��

�� )��

�� .�� #�

Hence, �� and )�� )�� if � � �� , andthe solution �� is a continuous, PWL function of � defined on a polyhedralpartition of �.


32

Comparing mp-QP and mp-NLP solutions

Local quadratic approximation makes sense!

Theorem 3. Let �� and suppose there exists a �� satisfying assump-tions A1 - A4. Then for � in a neighbourhood of ��

��

��

)�� )��

��Explicit constrained nonlinear MPC

33

Approximation criteria

How small neighborhoods �� are necessary?

The solution error bound is defined as (0 is a weight)

1 � ��

��

�0� �'�� '��

The cost error bound is defined as

! � ��

��

��

The maximum constraint violation is (2 is a weight)

Æ � ��

��

2�� Explicit constrained nonlinear MPC

34


Consider the vertices � � �� of any bounded polyhedron ��.Define the affine function � �� as the solution to the LP

��

� ��

� ��

subject to

� �� for all � � ��

Likewise, define the convex piecewise affine function

� ��

��

� ��

Theorem 4. If � and � are jointly convex (in � and �) on the boundedpolyhedron ��, then � �� for all � � ��.


35


Theorem 4 gives the following bounds on the cost function error�!� � � �� !�

where

!� � ��

��

��

!� � ��

��

��


36

Approximate mp-NLP algorithm

Step 1. Let �� .

Step 2. Select �� as the Chebychev center of ��, by solving an LP.

Step 3. Compute �� by solving a NLP.

Step 4. Compute the local mp-QP problem at �� .

Step 5. Estimate the approximation errors !, 1 and Æ on ��.

Step 6. If ! � !, 1 � 1, or Æ � Æ, then sub-partition �� into polyhedralregions.

Step 7. Select a new �� from the partition. If no further sub-partitioning isneeded, go to step 8. Otherwise, repeat Steps 2-7 until the tolerances !, 1

and Æ are respected in all polyhedral regions in the partition of �.

Step 8. For all sub-partitions ��, solve the mp-QP.


37

Example: Compressor surge control

Consider the following 2nd-order compressor model with �� being normalizedmass flow, �� normalized pressure and � normalized mass flow through aclose coupled valve in series with the compressor

�� !��

��

��

The following compressor and valve characteristics are used

�!�� 3"��*�

� � ��

4

� ��

� ��

4

� ��

�� $sign��

��

with $ � ��, � � �, * � ��, 3"� � �� and 4 � ��, (Gravdahl andEgeland, 1997).


38

Example, cont’d

The control objective is to avoid surge, i.e. stabilize the system. This may beformulated as

��

&��

with �� 1 � � and the setpoint �� , �� corresponds toan unstable equilibrium point.

We have chosen � � �, � �, and � � ��. The horizon is chosen as

# � �, which is split into � � 5 � �� equal-sized intervals.

Valve capacity requires the constraint � � �� to hold, and the pres-sure constraint �� avoids operation too far left of the operatingpoint.


39

Partition of the state space

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x1

x 2

R1 R2

R3

R4

R5

R6

R7

R8

R9R10

R11

R12R13R14

R15 R16R17R18R19

R1

R2

R3

R4R5R6R7R1

R2

R3R4

R5R6

R1

R2

R3R4

R5R6 R7

R8

R9 R10R11R12R13R14R15R16R17

R1

R2

R3

R4

R5

R6

R7

R8

R9

R10R11R12

R1

R2

R3

R4

R5R6

R7

R8R9

R10

R11

R12R13R14R15R16

R1R2

R3

R4

R5R6

R7

R8 R9R10R11

R12R13

R1

R2

R3R4R5R6

R1 R2

R3 R4

R1R2R3

R4R5

R6

R7

R8R9

R1

R2 R3

R4R5

R1R2

R3R4

R5

R1R2R3

R4R5R7R8

R9

R10R11

R1R2

R3

R4R5

R6

R1

R2R3

R4

R5R6 R7R8

R9

R10

R1

R2R3

R4

R5R6R7

R1R2

R3

R4

R5

R6

R7

R1

R2

R3

R4

R5

R1R2R3R4 R5R6

R1

R2

R3

R4

R5

R6

R7

R8

R9

R10

R1R2 R3R4R5

R1

R3

R4

R5

R6

R7

R8

R9

R10R11

R1 R2

R1

R2

R1

R2

R3

R4

R5

R6R7

R8

R9

R10

R11R12

R13

R14R15R16R17R18R19

R1

R2R3

R4R5

R6

R7

R1

R2

R4R5R6R7

R1R2 R3

R4

R5

R6

R7R8

R1

R2

R3

R4R5

R6

R8

R1

R2R3R4

R5

R1

R2

R3

R4R5

R6

R7

R8

R9R10R11R12

R1

R2

R3R4

R1R2

R3

R4R5

R6R7

R8R9

R10R11

R1

R2R3

R4

R5R6

R7

R8

R9

R10

R11

R12

R13R14R15R16R17

R1

R2

R3

R4

R5

R6R7

R8

R9R10

R11

R12R13

R14

R15R16

R1

R2

R3R4

R5R6

R1R2R3R4R5R6 R7

R8R9

R10R11R12

R1R2

R3

R4

R5R6

R7

R1R2

R3

R4

R5

R6R1

R2R1

R2R1

R2

R3R4

R5R6

R7

R8

R9R10R11

R12

R13

R14

R15R16R17

R18

R19R20R21

R22

R23R24R25R26

R1

R2R3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x1

x 2

R1

R2

R3R4

R5

R6

R7

R8

R9R10R11R12

R13

R14

R15

R16R17

R18

R19 R20

R21R22

R23

R24R25

R26R27

R28

R29R30

R31R33

R34

R35

R36 R37

R38

R39

R41

R42

R43R44

R45

R46 R47

R48

R49

R50

R51

R53R54R55

R56

R57R58

R59

R60

R62

R63

R64 R65R66R67

R68 R69R70

R71

R72

R73

R74R75

R76

R77

R78R79R80R81R82R83R84

R85

R86

R87

R88R89R90R91R92

R93R94R95

R96

R97

R98R99 R100

R101

The PWL mapping with 379 regions can be represented as a binary searchtree with 329 nodes, of depth 9. Real-time evaluation of the controller there-fore requires 49 arithmetic operations, in the worst case, and 1367 numbersneeds to be stored in real-time computer memory.


40

Optimal solution and cost

Approximate solution Exact solution

00.2

0.40.6

0.8

00.2

0.40.6

0

0.1

0.2

0.3

x1

x2

u(x1,x 2)

00.2

0.40.6

0.8

00.2

0.40.6

0

0.1

0.2

0.3

x1

x2

u* (x 1,x 2)

0

0.5

1

0

0.5

10

0.5

1

1.5

2

2.5

x1

x2

V(x 1,x 2)

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.80

0.5

1

1.5

2

2.5

x1

x2

V* (x 1,x 2)


41

Simulation with controller activated at � � ��

0 10 20 30 40 50 60−0.2

0

0.2

0.4

0.6

0.8

t

x1(t)

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

x2(t)

0 10 20 30 40 50 60−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

t

u(t)


42

Key referencesT. A. Johansen, On Multi-parametric Nonlinear Programming and Explicit Nonlinear ModelPredictive Control, IEEE Conf. Decision and Control, Las Vegas, NV, 2002

A. V. Fiacco, Introduction to sensitivity and stability analysis in nonlinear programming, Or-lando, Fl: Academic Press, 1983

P. Tøndel, T. A. Johansen and A. Bemporad, An algorithm for multi-parametric quadratic pro-gramming and explicit MPC solutions, Automatica, accepted, March, 2003

A. Bemporad and M. Morari and V. Dua and E. N. Pistikopoulos, The Explicit Linear QuadraticRegulator for Constrained Systems, Automatica, Vol. 38, pp. 3-20, 2002

P. Tøndel, T. A. Johansen and A. Bemporad, Evaluation of Piecewise Affine Control via BinarySearch Tree, Automatica, accepted, May, 2003

A. Bemporad and C. Filippi, Suboptimal explicit RHC via approximate quadratic programming,J. Optimization Theory and Applications, Vol. 117, No. 1, 2003


43

Part III – Discussion and summary

Explicit linear MPC (including robust/hybrid)

� Solved using mp-LP/mp-QP/mp-MILP

� Exact piecewise linear representation

� Solvers efficient for constrained MPC problems with a few states and in-puts and reasonable horizons

� Real-time implementation as binary search tree: 's computation timesusing fixed-point arithmetic


44

Discussion and summary, cont’d

Explicit non-linear MPC - emerging mp-NLP theory/technology based on mp-QP/mp-LP and NLP solvers.

Bad news: Non-convexity is challenging (not unexpected...)

Good news: i) Approximation theory (including stability) seems to be fairlystraightforward to establish. ii) Real-time computational complexity of PWLapproximation is comparable to linear MPC!


45

Advantages and disadvantages

+ Efficient real-time computations for small problems+ Simple real-time SW implementation: Verifiability, reliability, safety-criticalapplications, small systems+ Fixed-point arithmetic sufficient

- Requires more real-time computer memory- Does not scale well to large problems- Reconfigurability is not straightforward

Explicit solution is probably most interesting for nonlinear MPC problems:They are often smaller scale, and SW reliability and computational complexityissues are more pronounced.


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Applications

� Embedded systems - low-level control, small systems, may be safety-critical (automotive, medical, marine etc.)

� Control allocation problems - static/quasi-dynamic, over-actuated mechan-ical systems (aerospace, marine, MEMS,...)

� Low-level process control - MPC-like functionality at unit-process levelreplacing PID-loops???


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Explicit constrained nonlinear MPC - Nettsiden er flyttet formulation Nonlinear system: Optimization...

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