Explicit Characterisation of Receding Horizon · PDF fileExplicit Characterisation of Receding...

Explicit Characterisation ofReceding Horizon Control

Marıa M. Seron

September 2004

Centre for Complex DynamicSystems and Control

Outline

1 Solution Using GeometrySolution for Arbitrary NExample

2 Implementation of the Explicit SolutionCalculation of the Explicit SolutionOn-line Implementation of the Explicit SolutionSuboptimal Solutions and Approximations


The Receding Horizon Optimal Control Problem

Let the system be given by

xk+1 = Axk + Buk , |uk | ≤ ∆, (1)

where xk ∈ Rn and ∆ > 0 is the input constraint level. Consider the

following fixed horizon optimal control problem

P2(x) : V2 (x) , min V2({xk }, {uk }), (2)

subject to:

xk+1 = Axk + Buk for k = 0, 1,

x0 = x ,

uk ∈ U , [−∆,∆] for k = 0, 1,

where the objective function in (2) is

V2({xk }, {uk }) ,12

x2Px2 +12

1∑

k=0

(

xk Qxk + uk Ruk

)

. (3)


The Receding Horizon Optimal Control Problem

In the objective function we select Q > 0, R > 0 and P satisfyingthe algebraic Riccati equation

P = A PA +Q − K RK , (4)

whereK , R−1BPA , R , R + BPB . (5)

Let the control sequence that minimises (3) be

{u0 , u

1 }. (6)

Then the RHC law is given by the first element of (6) (whichdepends on the current state x0 = x), that is,

K2(x) = u0 . (7)


Solution of the Associated QP

In the previous lecture we showed that the solution in active regionR8 is

u(x) =

[

−Gx − h∆

]

where

G =K + KBKA1 + (KB)2

h =KB

1 + (KB)2∆,

if

R8 :

[0 1]u

(x) ≥ ∆

H1∆[−1 1] ≤ [1 0]Hu

(x) ≤ H1∆[1 1]

where u

(x) = −H−1Fx.


Solution of the Associated QP

The above solution is valid whenever u

(x) = −H−1Fx belongs toR8, that is, it satisfies the equations

R8 :

[0 1]u ≥ ∆

H1∆[−1 1] ≤ [1 0]Hu ≤ H1∆[1 1].

Thus, setting u = u

(x) = −H−1Fx in the above equations weobtain a characterisation of R8 in the state space:

R8 :

−[0 1]H−1Fx ≥ ∆

H1∆[−1 1] ≤ −[1 0]Fx ≤ H1∆[1 1].


Geometric RHC Characterisation for N = 2

If we proceed in a similar way with the remaining regions, weobtain a characterisation of the RHC law in the form

K2(x) =

−Kx if x ∈ R,

∆ if x ∈ R1 ∪ R2 ∪ R3,

−Gx + h if x ∈ R4,

−∆ if x ∈ R5 ∪ R6 ∪ R7,

−Gx − h if x ∈ R8,

−8 −6 −4 −2 0 2 4 6 8−4

−3

−2

−1

0

1

2

3

4

PSfrag replacements

x1k

x2 k R

R1

R2

R3 R4 R5

R6

R7R8

where the regions R, R1, . . . , R8 in the state space are obtainedusing geometric arguments as described before.

We can see that the above characterisation coincides with the oneobtained previously using Dynamic Programming.


Geometric RHC Characterisation. More General Cases

For arbitrary horizon N, m ≥ 1 inputs, and more generalconstraints, as we showed in the previous lecture, the QP solutionin a particular active region corresponding to the face

Φ`u = ∆` − Λ`x ,

has the form

u(x) = H−1/2Φ

`[Φ`Φ

`]−1(∆` − Λ`x)

− H−1/2[I − Φ`[Φ`Φ

`]−1Φ`]H

−1/2Fx ,

, G`x + h`.



The above QP solution is valid in the state space region

L`x ≤ W`,

where

L` =

[

[Φ`Φ`]−1(Λ` − Φ`H−1/2F)

ΦsΦ

`[Φ`Φ`]

−1(Λ` − Φ`H−1/2F) + (Λs − ΦsH−1/2F)

]

W` =

[

[Φ`Φ`]−1∆`

∆s + ΦsΦ

`[Φ`Φ`]

−1∆`

]



Thus, the optimal QP solution in an active region (corresponding toactive constraints with indices in `) is

u(x) , G`x + h`,

whenever x satisfiesL`x ≤ W`.

The RHC law KN(x) in the above region is then given by the first melements of the above solution:

KN(x) = [Im 0](G`x + h`) if L`x ≤ W`. (8)



To compute the complete RHC solution in the state space onerequires, for example, procedures that

enumerate all combinations of active constraints, or

recursively partition the state space searching for activeregions, or

enumerate all parameterised vertices based on doubledescription properties of polyhedra.

We will briefly comment on these computational issues later.


Example

Consider a system with matrices

A =

[

0.8955 −0.18970.0948 0.9903

]

, B =

[

0.09480.0048

]

.

In the objective function we take N = 4, Q =

[

0 00 2

]

and R = 0.01.

The terminal state weighting matrix P is chosen as the solution ofthe algebraic Riccati equation P = A PA +Q − K RK , whereK , R−1BPA and R , R + BPB.

We consider input constraints of the form |uk | ≤ 2.


Example

The state space partition for this case is shown in Figure (a). A“zoom” of this partition is shown in Figure (b). The region denotedby X0 is the projection onto the state space of the constraintpolyhedron; in regions X2, X3 and X4 only one constraint is active;in regions X5 and X6 two constraints are active; in region X7 threeconstraints are active; finally, X1 is the union of all regions wherethe control is saturated to the value −2.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

PSfrag replacements

x1

x 2 X0

X1

X2

X3

X4

X5

X6

X7

(a) Complete partition.

−1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.90.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

PSfrag replacements

x1

x 2 X0

X1 X2

X3

X4

X5

X6

X7

(b) Zoom.Centre for Complex DynamicSystems and Control

Example

The resulting RHC law (8) is

K4(x) = Gix + hi , if x ∈ Xi , i = 0, . . . , 7, (9)

where

G0 = −[4.4650 13.5974], h0 = 0,

G1 = [ 0 0 ], h1 = −2,

G2 = −[5.6901 15.9529], h2 = −0.7894,

G3 = −[4.9226 13.8202], h3 = −0.4811,

G4 = −[4.5946 13.3346], h4 = −0.2684,

G5 = −[6.6778 16.8644], h5 = −1.7057,

G6 = −[5.1778 13.4855], h6 = −0.9355,

G7 = −[7.4034 16.8111], h7 = −2.6783.

Similar expressions hold in the remaining unlabelled regions.These can be obtained by symmetry.


Example

The figures show the state state partitions for horizons N = 2,N = 3, N = 4 and N = 5.

−3 −2 −1 0 1 2 3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

PSfrag replacements

x1

x 2

N=2

N=3

N=4

N=5 −3 −2 −1 0 1 2 3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

PSfrag replacements

x1

x 2

N=2

N=3

N=4

N=5

−3 −2 −1 0 1 2 3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

PSfrag replacements

x1

x 2

N=2

N=3

N=4

N=5 −3 −2 −1 0 1 2 3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

PSfrag replacements

x1

x 2

N=2

N=3

N=4

N=5


Example

We next consider an initial condition x0 = [−1.2 0.53] andsimulate the system under the RHC (9).

The figure shows the resultingstate space trajectory.

The trajectory starts in region X4

and moves, successively, intoregions X6, X5, X1, X1, X0, andstays in X0 thereafter.

−1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

PSfrag replacements

x1

x 2


Recapitulation

The geometric characterisation of QP of the form

u(x) = Φ`[Φ`Φ

`]−1(∆ − Λ`x) − [I − Φ`[Φ`Φ

`]−1Φ`]H

−1/2Fx (10)

= H1/2(G`x + h`)

if

[

[Φ`Φ`]−1(Λ` − Φ`H−1/2F)

ΦsΦ

`[Φ`Φ`]

−1(Λ` − Φ`H−1/2F) + (Λs − ΦsH−1/2F)

]

︸︷︷︸

L`

x ≤[

[Φ`Φ`]−1∆`

∆s + ΦsΦ

`[Φ`Φ`]

−1∆`

]

︸︷︷︸

W`

,

(11)

leads to the RHC characterisation

KN(x) = [Im 0](G`x + h`) if L`x ≤ W`.

The same characterisation can be obtained using the KKToptimality conditions, as we showed in the previous lecture.


Explicit vs Implicit

Recall what we defined as explicit procedures as opposed toimplicit, or numerical, procedures:

Explicit

x

PSfrag replacements

fp(z) = z2+ 2apz + bp

z ∂fp∂z = 0⇒ z(p) = −ap

Implicit (numerical)

PSfrag replacements

fp(z) = z2+ 2apz + bp

zz0pzk

pzk+1p

. . .

We have now obtained an explicit characterisation of RHC, whichcan replace the traditional numerical RHC solution in onlineimplementations.


Numerical Implementation of RHC


Explicit Solution


Implementation of the Explicit Solution

Key issues:

Off line

How to efficiently calculate KN(x) for all x of interest (regionpartition and corresponding control law).

How to store KN(x) as a function of x (look-up table).

On line

How to retrieve the value of KN(x) given x.


Calculation of the Explicit Solution

Naive approach: enumeration of all possible combinations ofactive constraints.

Semi-naive approach: combine the above with dynamicprogramming.

More efficient: use algorithms to implement multiparametricquadratic programming (Dua, Pistikopoulos, Bemporad,Tondel, Johansen).

Better: combine the above with reachability analysis (Grieder,Borrelli, Torrisi, Morari).

New approach: enumeration of all parameterised verticesbased on double description properties of polyhedral regions(Olaru and Dumur).


On-line Implementation of the Explicit Solution

Once the look-up table is precomputed and stored, the on-linecomputation is reduced to:

(i) identifying the “current region”, that is, deciding in whichregion the current state belongs and

(ii) computing the control input using the affine control lawcorresponding to that region.

The first step is the most computationally demanding.


Searching the Look-up Table

Naive approach: sequential search through the regions of thepolyhedral partition.

Efficient approach: organise the partition in a binary searchtree: each level, or “node”, is associated with one hyperplaneinequality. When the tree is used “on line” for the current state,at each node one inequality is evaluated, its sign checked,and the left or right subtree selected based on the sign.Traversing the tree from the “root” to a “leaf” node, the controllaw corresponding to the current state is found.

Tøndel, Johansen and Bemporad (2002) proposed an algorithm toconstruct a binary search tree with search time logarithmic in thenumber of polyhedral regions.


Binary Search Algorithm


Example: A Simple Binary Search

Consider the double integrator with matrices

A =

[

1 10 1

]

, B =

[

0.51

]

, C =[

1 0]

,

|u(k )| ≤ 1, Q = I2, R = 0.01, and P chosen as the solution of thealgebraic Riccati equation.

For N = 5, for example, the state-space partition and RHC law are:

−10 −8 −6 −4 −2 0 2 4 6 8 10−5

−4

−3

−2

−1

0

1

2

3

4

5

PSfrag replacements

x1

x 2 X0

X1X2

X3X4

X5

K5(x) = G`x + h` if x ∈ X`

X` G` h`X0 -[0.9653 1.3895] 0X1 -[0.6154 1.2870] -0.4156X2 -[0.4390 1.2121] -0.7982X3 -[0.3399 1.1665] -1.1746X4 -[0.2771 1.1367] -1.5495X5 0 -1



The “look-up table” implementation of RHC checks, at each timeinstant, in which region the current state is and picks thecorresponding control law from a stored table.

We first consider a simple Matlab code that implements this searchin one step. This has only a few lines of code.

We will use the double integrator example with N from 2 to 10, inthe area −20 ≤ x1 ≤ 20, −8 ≤ x2 ≤ 8, to compare the performanceof the ‘look-up table” implementation of RHC and Matlab’s QPalgorithm.

The total data stored for the look-up table uses 20448 bytes.



We grid the state space with 3600 points and, for each point of thegrid we evaluate the controller using each method, and for eachmethod

we measure computation time

we measure number of floating point operations.

We then average the latter two measures over the grid.

We repeat the same for horizons N = 2, . . . , 10.



The left figure is a plot of computation times, and right figure showsthe number of floating point operations (flops), for the “look-uptable” method (solid line) and QP (dashed-dotted line) as afunction of the horizon N.

2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

N

seco

nds

Figure: Computation time

2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105

N

flops

Figure: Flops



The performance of both algorithms is comparable in relationwith computation speed, although the number of flops is muchlarger for the “look-up table” method.

This is expected since, as the horizon increases, there aremore regions in the state space partition that have to bechecked at each instant.

This problem can be easily mitigated by splitting the regionsinto an orthogonal binary search tree; this increases the totalnumber of regions (as some of the original regions are dividedinto smaller subregions) but the number of checks at eachsample time is smaller and so there are less flops.



To accomplish this task we divide the area of interest into thesmaller subregions of a grid defined by Nh separatinghyperplanes parallel to the axes.

The original RHC partition is then intersected with the grid toform a new partition.

The maximum number of inequalities to check at each sampletime is then Nh plus the maximum number of inequalities ofthe new RHC regions inside each subregion of the grid.



For the double integrator example, the total number of regionsof the original RHC partition in the area of interest−20 ≤ x1 ≤ 20, −8 ≤ x2 ≤ 8 is N0 = 165 and the total numberof hyperplane inequalities to check at each time is 632.

We next grid the area using using five horizontal lines andseven vertical lines (that is, Nh = 12), defining a grid with 48subregions, as shown in the following figure.The intersection of the original RHC partition with the gridyields a total of 672 regions, but the maximum number ofinequalities to check at each sample time is Nh+96=108.

The data stored in this case use 164352 bytes (compare with20448 bytes used before).



−20 −15 −10 −5 0 5 10 15 20−8

−6

−4

−2

0

2

4

6

8

x1

x2

Figure: Binary partition using 12 hyperplanes



The inclusion of the binary search tree in the algorithm fortable look-up does not add too much complexity to the Matlabcode, which has approximately 30 lines.

To compare the “on-line” performance of the new tablelook-up algorithm and QP, we proceed as before, averagingcomputation time and number of floating point operations overa state space grid of 3600 points.

In all cases, the binary search tree has Nh = 12 separatinghyperplanes.



The left figure is a plot of computation times, and the right figureshows the number of floating point operations (flops), for the“binary search look-up table” method (solid line) and QP(dashed-dotted line) as a function of the horizon N.

2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03

N

seco

nds

Figure: Computation time

2 3 4 5 6 7 8 9 100

1

2

3

4

5

6x 10

4

N

flops

Figure: Flops



We can see from these figures that the table look-up usingbinary search has much better performance than the standardQP algorithm.

Both measures (computation time and number of floatingpoint operations) stay approximately constant for all horizons.


Suboptimal Solutions and Approximations

For RHC problems of large dimension (large state and inputdimensions and long constraint horizons) it may be unrealisticto compute the exact explicit solution, and even if that werepossible, its implementation would be impractical due to thelarge amount of memory that would be required to store acomplex region partition.

On the other hand, on-line optimisation to solve theassociated QP may also be impractical from the computationtime perspective.


Suboptimal Solutions and Approximations

Possible simplifications:

Regional solutions.

Approximate partition for example, using orthogonalhyperplanes in the tree nodes.

Approximate partition using neural networks.

Input trajectory parameterisation to reduce degrees offreedom.

Allow certain degree of suboptimality and constraint violation.

More ... on-going research and growing literature.


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