Abstract—Recently, the approximate solutions to the infinite horizon min-max model predictive control with a time-varying terminal cost and constraint have been explicitly characterized in the form of a piecewise affine (PWA) state feedback law defined on an orthogonal partition of the state-space [13]. The advantage of this formulation is the reduction in on-line computational complexity which amounts to the evaluation of a PWA function in the control unit. This paper extends the authors work in [13]. In order to accommodate a wider class of systems, a nominal performance cost is chosen to substitute the “worst-case” one in the existing MPC technique. Robust exponential stability of the controller is established by means of satisfaction of certain linear matrix inequalities (LMIs). The proposed approach is applied to control a current loop of a permanent-magnet synchronous motor (PMSM) and an uncertain double integrator.

I. INTRODUCTION

RACTICAL difficulties associated with the implementation of stabilizing robust model predictive

control (MPC) laws are well known. The algorithms typically rely on the solution of a min-max optimization problem [1], [2], [3] in which the worst-case performance cost is minimized over the control input while satisfying input and state constraints. All such algorithms have in common large on-line computational burden and often suffer excessive conservativeness which may restrict the MPC application range to systems with relatively slow dynamics or with low-performance requirements.

Despite the complex nature of the problem, several different approaches towards reducing the conservativeness and/or computational complexity have been reported. It is shown in [4] that solutions to the min-max MPC problem based on a linear cost and parametric uncertainty have an explicit piecewise affine (PWA) state feedback representation defined on a (polyhedral) partition of the state space. The key advantage of this approach is that the on-line computation simply reduces to a function evaluation problem, thus potentially enlarging its scope of applicability to applications with fast dynamics. For quadratic cost functions and parametric uncertainty, explicit (min-max) state feedback solutions are, in general, not available [5]. Therefore many authors [6], [7], [8] resort to nominal MPC

Manuscript received February 18, 2005. This work was supported by Enterprise Ireland under the PATs Research Programme 2000-2004.

M. T. Cychowski is with Cork Institute of Technology, Cork, Ireland (fax: +353-21-4326625; e-mail: mcychowski@cit.ie).

T. O’Mahony is with Cork Institute of Technology, Cork, Ireland (e-mail: tomahony@cit.ie).

formulations in which the robustness is defined in terms of satisfaction of the input and output constraints under all possible uncertainty realizations.

On the other hand, approximate explicit algorithms have been suggested in [9], [10]. The former obtains a sequence of suboptimal linear state feedback laws associated with stable invariant ellipsoids while the latter is based on multi-parametric convex programming.

Recently, algorithms that determine an approximate explicit PWA state feedback solution by imposing an orthogonal search tree structure on the partition have been developed [11]. They allow a computationally demanding constrained optimization to be replaced by a more favorable search in a finite-dimensional tree. An extension of the results to systems with model uncertainty has also been suggested in [12]. However, to our knowledge, there is no technique to compute efficient off-line solutions to the min-max MPC problem with a time-varying terminal cost and constraint other than our own in [13].

The purpose of this paper is to extend the authors work in [13] by adopting a nominal performance cost in the MPC formulation. Specifically, in order to accommodate a wider class of systems, the nominal quadratic cost corresponding to the nominal local model is chosen to substitute the worst-case one as in [14]. The main benefit of the approach is that the resulting controller is of very low complexity since it can be represented as a finite-dimensional search tree and unlike most robust MPC schemes incorporating the nominal performance cost stability guarantees exist.

Notation: for a vector nx and positive definite matrix ,Q the weighted norm 2

Qx is denoted by .Tx Qx |k i kx is

the value of a vector x at a future time +k i predicted at time k. The symbol * will be used to denote the corresponding transpose of the lower block part of symmetric matrices.

II. ON-LINE ROBUST CONSTRAINED MPC

In this paper, the following class of state space systems is considered

( 1) ( ) ( ) ( ) ( ),[ ( ), ( )] ,

x k A k x k B k u k

A k B k (1)

where 0;k mu and nx are the control input and the measurable state vector respectively. For a polytopic uncertainty description, represents a polytope

Efficient Approximate Robust MPC Based on Quad-Tree Partitioning

Marcin T. Cychowski, and Thomas O’Mahony, Member, IEEE

P

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MB2.3

0-7803-9354-6/05/$20.00 ©2005 IEEE 239

1 1 2 2{[ , ],[ , ],...,[ , ]},L LCo A B A B A B (2)

in which {}Co denotes the convex hull and [ , ],l lA B

{1,2,..., }l L are vertices of the convex hull (see e.g.

[2]). Moreover, let ˆ ˆ[ , ]A B denote the nominal model that is more likely to be the actual plant. For the current

( ),x k the robust constrained MPC solves the optimization problem:

12 2 2

| ( ) | |0

ˆ ˆmin || || [|| || || || ]N

N

k N k k k i k k i kU i

x x u (3a)

subject to | ( )k kx x k and

min | max ,k i ku u u 0,i (3b)

min | max ,k i kx Cx x 1,i (3c)

| |( ) ,k i k k i ku F k x ,i N (3d)

1| | |ˆ ˆˆ ˆ ,k i k k i k k i kx Ax Bu 0,i (3e)

1| | | ,k i k k i k i k k i k i kx A x B u ,i N (3f)

Here, | 1| 1|[ , ,..., ]T T T TN k k k k k N kU u u u is the vector of

control moves and 0N denotes the control horizon. Additionally, we require the terminal constraint

| ( ),k i kx k ,i N (4)

to be satisfied. The set ( )k is chosen to be control invariant [15] with respect to ( )F k in the specified polytopic family (2). Before going further the following assumptions are made:

(A1) min max0u u and min max0 ,x x

(A2) , , ( ) 0k

(A3) [ ( ), ( )]A k B k is stabilizable.

Remark 1. Problem (3) defines a robust constrained MPC problem with a time-varying terminal cost and constraint based on the nominal performance cost instead of the worst case cost for all models in . This may improve the optimality and feasibility of the algorithm when the nominal model is close to the actual system, and the resulting optimization problem may be significantly less complex.

The following lemma shows that problem (3) can be posed as a semi-definite programming (SDP) problem involving linear matrix inequalities (LMIs).

Lemma 1 (On-line robust MPC [14]). Consider the uncertain system (1) and let assumptions (A1)–(A3) hold.

(i) Suppose there exist a scalar 2 0, symmetric matrices

0,lQ Z, and a set of matrices ˆ{ , , }Q G Y satisfying the following LMIs:

1/ 22

1/ 22

ˆ * * *ˆ ˆˆ * * 0,

0 *0 0

TG G Q

AG BY QG IY I

(5)

*0,

Tl

l l t

G G QA G B Y Q

, ,l t (6)

*0,T T

l

ZY G G Q

2 ,jj jZ z ,l (7)

*0,

( )

Tl

l l

G G QC A G B Y

2 ,ss sx ,l (8)

where min, max,min{ , },j j jz u u min, max,min{ , }s s sx x x

Then the feedback controller | |( ) ,k i k k i ku F k x ,i N

where 1( ) ,F k YG exponentially stabilizes the system (1)for any | ( ),k N kx k i.e., ( )k is positively invariant.

(ii) If, in addition to (5)-(8) the following LMI’s are satisfied:

1 *0,ˆ ˆ( )N N N lx k U Q

,l (9)

1 *0,

( )N N N lx k U Q,l [ , ] ,N N N (10)

( , ( )) 0,NU x k (11)

where (11) incorporates the input and state constraints

before the switching horizon N and ˆ ,Nˆ ,N ,N ,N

N can be easily constructed as in [14]. Then NU drives

the state |ˆk N kx into ( )k while satisfying the constraints

(3b)–(3c) for all 0 1.i N

(iii) Further, define a scalar 1 0 and add an LMI:

11/ 2

1/ 2

* ** 0,

ˆ ˆ( ( ) ) 0N

N

U I

x k U I

(12)

240

where diag( ), diag( ) and ˆ , ˆ denote the nominal prediction matrices as in [14]. Then the optimization problem (3) can be solved by

1 2

21 2ˆ, , , , , ,

ˆmin || ( )||N lU Q Q Y G

x k subject to (5)-(12) (13)

Remark 2. The on-line implementation of the robust MPC algorithm (13) guarantees the closed-loop stability only for systems sufficiently close to the nominal model. This is however not a serious restriction for the algorithm developed in this paper as a-posteriori stability analysis will be applied to establish robust stability for all models in .

Despite the fact that SDP problems can be solved in polynomial time using interior point algorithms, the computation effort required to solve the robust MPC problem (13) on-line can be quite prohibitive for many real-time applications. In the following section, we will present an algorithm for reducing the computational complexity of the optimization problem (13), by computing approximatepiecewise affine state feedback solutions defined on an orthogonal partition of the state space.

III. EFFICIENT APPROXIMATE ROBUST MPC

A. The Local Control Laws

Let N n denote the set of initial conditions ( )x k for which the optimal control problem (13) is feasible. Suppose

N can be decomposed into a collection of closed boxes or hyper-cubes { } ,r rX given by polyhedra of the form

{ : },nr r rX x H x d ,r (14)

where 2 ,n nrH 2n

rd and denotes the index set

of hyper-cubes. Denote .Nrr

X Moreover, let

1{ ,..., }M represent a set of 2n vertices of rX such that { } ,rCo X .r

In each region of the partition, a local affine feedback law

( ) ,r r rU x F x g ,rx X (15)

is defined. The parameters m nrF and m

rg will be computed by considering the optimal solutions to the on-line robust MPC problem (13) at the vertices of .rX The following lemma is an extension of the results in [16] to systems affected by polytopic uncertainty.

Lemma 2 Consider a hyper-cube NrX with vertices

1{ ,..., },M and let *0 ( )hU denote the first m elements

of the optimal control sequence computed at .h If there

exist parameters rF and rg solving the optimization problem:

* 20, 1

min || ( ) ( ) || ,r r

M

h r h rF g h

U F g (16a)

subject to h

( , ) 0,r h r hF g (16b)

then the approximate optimizer ( )r r rU x F x g is robustly feasible for all rx X and all [ ( ), ( )] .A k B kProof. Follows directly from convexity of (16).

B. Main Algorithm The objective of the algorithm presented in this section is

to compute, off-line, a partition of hyper-cubes defined inside a given box 0

nX along with the associated

explicit controllers ( )rU x such that the approximation accuracy satisfies some prescribed tolerance. The set 0X is introduced as an artificial bound of the state-space to make sure is bounded.

The accuracy of the approximation will be measured by the difference between the first m components of the optimal and approximate solutions restricted to a hyper-cube 0X :

* 20max || ( ) ( ) || , ,h r h hh

U U (17)

Moreover, the error in (17) for each region of the partition is required to satisfy

* 20max( , min || ( ) || ),a r hh

U (18)

where 0a and 0r can be interpreted as absolute and relative tolerances respectively [13]. In view of the results of the previous sections, the following recursive off-line algorithm is proposed:

Algorithm 1 (Approximate explicit robust MPC).

1. Initialize partition with 0{ },X {};2. Select 0 .X If then terminate;3. Solve (13) for every 0h X {1, 2,..., }h M to obtain

the optimizers * *0 1 0( ),..., ( ).MU U If any solution is

infeasible go to step 6;

241

4. Compute an approximation ( )U x using (16). If a feasible solution is not found, go to step 6;

5. Compute the error ˆ using (17). If ˆ , then

0},X{ 0\{ },X and go to step 2.6. Compute the size of 0 .X If is smaller than some

tolerance, then 0\{ },X and go to step 2;7. Partition 0X into M equal boxes 1,..., .MX X Perform

1{ ,..., },MX X and go to step 2.

The algorithm terminates after a finite number of steps with the piecewise affine approximation : mNUrestricted to the set which is the union of all hyper-rectangles in and an inner approximation to the feasible set .N As a side product, the algorithm also determines a binary search tree structure of the partition. This explicit structure is exploited for efficient real-time implementation via quad-trees trees [17].

C. The Question of Stability Since the approximate feedback controller defined in the

previous section does not guarantee stability by design, aposteriori stability analysis is required to ensure that the feedback is also robustly stabilizing. In addition, any nonzero tolerance imposed on the approximation error renders the asymptotic convergence to the origin impossible. In order to circumvent this problem, a strategy proposed in [13] can be applied where a locally stabilizing robust control law is used in a close neighborhood of the origin.

In order to establish stability of the approximate explicit robust MPC algorithm, we exploit a common quadratic Lyapunov function of the form:

( ) ,TV x x x (19)

where T is positive definite. This function is required to satisfy:

2( + ( )) ( ) || || ,l lV A x B U x V x x ,x ,l (20)

where the scalar 0 is introduced to enforce exponential stability. The conditions (19)-(20) can be transformed into an LMI description by analogy to [6]. Define

, ( )r l l l rA B F and ,r l l rB g as corresponding parts of the approximate closed-loop dynamics associated with a region r and the l-th uncertainty realization. The requirement (20) can then be rewritten as

, ,

, , , ,

0*,

0 0

Tr l r lT T

Tr l r l r l r l

IP Px x x x

P P

,l .r (21)

where [ 1] .Tx x The application of the S-procedure yields the following result.

Theorem 1 (Stability test). If there exist symmetric matrices 0l

rN and a scalar solving the optimization problem:

find , , lrN subject to ,r ,l

, ,

, , , ,

+ + *0

( ) +

T l Tr r r r l r l

l T T l Tr r r r l r l r r r r l r l

H N H P P I

d N H P d N d P(22)

then the approximate explicit robust MPC guarantees exponential closed-loop stability.

Proof. The proof is a simple extension of the results in [6] which are based on systems with bounded additive uncertainty.

IV. SIMULATION EXAMPLES

In this section, we present two examples that illustrate the implementation of the proposed approximate explicit approach.

A. Permanent Magnet Synchronous Motor (PMSM) The stator voltage equations of a PMSM drive in the

synchronous reference frame are described as follows [18]:

( )( ) ( ) ( ) ( ),d

d s d s e s q

di tv t R i t L t L i t

dt (23)

( )( ) ( ) ( )( ( ) ),q

q s q s e s d m

di tv t R i t L t L i t

dt (24)

where ,sR sL are the stator resistance and inductance, e

denotes the rotor electrical angular velocity and m is the flux linkage established by the permanent magnet. The stator voltages dv and qv are the control variables and the d-q

currents are the system states. The values of ,sR sL and e

are taken from [18] and are assumed to belong to the intervals 1.5 3.6,sR 3 34.0 10 6.1 10 ,sL

3 37.3 10 7.3 10 .e The nominal values of these

coefficients are 2.55,sR 35.08 10sL and 0.e

We constrain the inputs and states within the range ,187.63 187.63d qv and ,12.7 12.7.d qi The

model (23), (24) is discretized using a sampling time 30.25 10 .sT

The approximate explicit robust MPC approach described in section 2 is applied with the weighting matrices I

and 310 I and the control horizon 1.N The initial

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hyper-cube is defined by 0 [ 13,13] [ 13,13]X and the regions are restricted to be larger than 0.13. The tolerance on the approximation error is chosen according to (18) with

0.02a and 0.05.r

The approximate off-line solution computed with Algorithm 1 is depicted in Fig. 1 and consists of 676 hyper-cubes and 7 levels of search. The computational complexity of the approximate approach consists, in the worst-case, of a total of 22 arithmetic operations per sample (14 comparisons, 4 multiplications and 4 additions). For comparison, on a Pentium IV machine (1.8 GHz and total memory 500 MB) the average time for the exact algorithm (13) to compute the solution is 0.96 s, which is well beyond the sampling rate of the considered PMSM drive.

The generated quad-tree has 224 nodes and there are 673 unique affine control laws representing the PWA function. The implementation of the approximate explicit controller on-line requires the storage of 4504 real numbers and 896 integers.

Given an initially disturbed state (0) [12,7]Tx the state and input trajectories corresponding to the nominal and extreme uncertainty realizations for the approximate algorithm are shown in Fig. 2 and Fig. 3, respectively. It can be observed that the proposed controller keeps the input and state evolutions within the constraints despite the model uncertainty.

B. Uncertain Double Integrator Consider the discrete-time double integrator:

1 0.2 0.1 ( ) 0( 1) ( ) ( ),

0 1 1.5 0.5 ( )k

x k x k u kk

(25)

where 0 ( ) 1k is an uncertain time-varying parameter. The following input and state constraints are imposed:

1 ( ) 1,u k [ 8, 5] ( ) [8,5] ,T Tx k (26)

The weighting matrices are selected as diag{1,0.01}and 3 and the control horizon is 4.N The initial box is defined by 0 [ 8,8] [ 5,5]X and the regions are restricted to be larger than 0.05. The tolerances in (18) are chosen as 0.0001a and 0.1.r

The application of Algorithm 1 with the nominal value of the uncertain parameter 0.5 yields the state-space partition of 582 regions. The approximate controller is robustly stabilizing as the LMI test in Theorem 1 provides a Lyapunov function matrix 0.7593 0.3736

0.3736 0.4202 and a decay rate

of 0.001. To verify performance and stability of the

closed-loop system, 20 trajectories are calculated, for two initial states (0) [7.5, 2]Tx and (0) [ 7.5,2] ,Tx and with ( )k randomly chosen at each time instant. The

-10 -5 0 5 10

-10

-5

0

5

10

id

i q

Fig. 1. Orthogonal partition of the approximate robust MPC for the PMSM example.

0 1 2 3 4 5-2

0

2

4

6

8

10

12

Time (samples)

x(k)

Fig. 2. The nominal (solid) and extreme (dotted) state trajectories for the approximate robust MPC of the PMSM example.

0 1 2 3 4 5-200

-150

-100

-50

0

50

Time (samples)

u(k)

Fig. 3. The nominal (solid) and extreme (dotted) input trajectories for the approximate robust MPC of the PMSM example.

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resulting closed-loop responses and the level curves of the associated Lyapunov function are depicted in Fig. 4. It can be observed that the proposed controller brings the system to the origin despite the model uncertainty, and keeps the state evolution within its limits.

The accuracy of the approximation is validated by computing the relative deviations (measured in percentage) of the approximate control input values from the exact ones:

* *0

max min

| ( ) ( )|100%.u

U x U xu u

(27)

Fig. 5 shows the corresponding error evaluated based on simulations for 4600 initial states in the feasible region of the state-space. It is evident from the results that the maximum performance decrease for the considered example does not exceed 3.64% and the total average error is around 0.20%.

V. CONCLUSIONS

A new off-line synthesis approach to robust model predictive control of systems with a polytopic uncertainty description is proposed. It is shown that the approximate solution to this problem can be pre-computed off-line in an explicit form as a piecewise affine state feedback law defined on the partition of boxes. The proposed algorithm allows a computationally demanding constrained optimization to be replaced by a simpler search in a finite dimensional tree. This makes the presented method an attractive alternative to the existing robust MPC schemes.

REFERENCES

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[12] A. Grancharova, and T. A. Johansen, “Explicit min-max model predictive control of constrained nonlinear systems with model uncertainty”, in Proc. 16th IFAC World Congress, Prague, 2005.

[13] M. T. Cychowski and T. O’Mahony, “Efficient off-line solutions to robust model predictive control using orthogonal partitioning”, in Proc. 16th IFAC World Congress, Prague, 2005.

[14] B. Ding, P. Yang, H. Sun, and S. Li, “Synthesizing on-line constrained robust model predictive control based-on nominal performance cost”, in Proc. 5th World Congress on Intelligent Control and Automation,Hangzhou, 2004, pp. 610A–614.

[15] F. Blanchini, “Set invariance in control”, Automatica, vol. 35, pp. 1747–1767, Nov. 1999.

[16] A. Bemporad, and C. Filippi, “Suboptimal explicit MPC via approximate quadratic programming”, in Proc. IEEE Conf. Decision and Control, Orlando, 2001.

[17] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry. Berlin: Springer-Verlag, 2000.

[18] M. T. Cychowski, R. Nalepa, and T. O’Mahony, “Explicit model predictive control of a permanent magnet synchronous motor drive”, 40th Universities Power Engineering Conf., to be published.

-8 -6 -4 -2 0 2 4 6 8-5

-4

-3

-2

-1

0

1

2

3

4

5

x1

x 2

-8-4

04

8 -5

0

50

1

2

3

4

x2x

1

Erro

r [%

]

Fig. 4. 20 random state trajectories for Example 2 and the level curves of Fig. 5. The approximation error for Example 2 computed on a grid. the associated Lyapunov function.

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