TuM03-5
Proceedings of the 13th Mediterranean Conference on Control and Automation Limassol, Cyprus, June 27-29, 2005
An Orthogonal Partitioning Approach to Simplify Robust Model Predictive Control M. T. Cychowski, B. Ding and T. O’Mahony, Member, IEEE
Abstract—The main limitation of many robust model predictive control (MPC) algorithms is the formidable on-line computational complexity. For linear discrete-time systems with a polytopic uncertainty description, a new efficient algorithm to determine an approximate explicit state feedback solution to robust MPC is developed. It is shown that this control profile is, in fact, a piecewise affine (PWA) function of the initial state defined on an orthogonal partition of the state space. Robust exponential stability of the controller is established in terms of satisfaction of certain linear matrix inequalities (LMIs). The real-time computational effort required to implement the approximate controller can be reduced to a simple search in a finite-dimensional tree.
M
I. INTRODUCTION
ODEL predictive control (MPC) of constrained uncertain systems has attracted considerable attention in the past few years as the technique enables constraints and model uncertainty to be taken into account in an explicit way. Practical difficulties associated with the implementation of stabilizing robust MPC laws are well known. The algorithms typically rely on the solution of a min-max optimization problem [1], [2] in which the worstcase performance cost is minimized over the control input while satisfying input and state constraints. The requirement to solve the min-max problem on-line greatly restricts the MPC application range to systems with relatively slow dynamics or high-performance computers. Despite the complex nature of the problem, several different approaches to reduce the computational complexity of robust MPC have been proposed. It is shown in [3] 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 Manuscript received April 13, 2005. This work was supported by Enterprise Ireland under the PATs Research Programme 2000-2004. M. T. Cychowski is with the Department of Electronic Engineering, Cork Institute of Technology, Cork, Ireland (phone: +353-21-432688; fax: +353-21-4326625; e-mail:
[email protected]). B. Ding is with the Institute of Automation, Hebei University of Technology, Tianjin, 300130, China and with the College of Information Science and Technology, Donghua University, Shanghai, 200052, China, (e-mail:
[email protected]). T. O’Mahony is with the Department of Electronic Engineering, Cork Institute of Technology, Cork, Ireland (e-mail:
[email protected]).
0-7803-8936-0/05/$20.00 ©2005 IEEE
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. For quadratic cost functions and parametric uncertainty, explicit (min-max) state feedback solutions are, in general, not available [4]. Therefore many authors [5], [6], [7] resort to nominal MPC 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 [8], [9]. The former obtains a sequence of suboptimal linear state feedback laws associated with invariant ellipsoidal regions of attraction while the latter is based on multi-parametric convex programming techniques. Recently, algorithms that determine an approximate explicit PWA state feedback solution by imposing an orthogonal search tree structure on the partition have been developed for linear constrained systems [10], and a general class of nonlinear systems [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 algorithm to robust MPC has been suggested for systems with additive disturbances and under the nominal model assumption [12]. However, to our knowledge, there is no technique to compute efficient offline solutions to the min-max MPC problem for systems with polytopic uncertainty other than our own in [13]. In this paper, a similar method is introduced that obtains a significant reduction both in real-time computational complexity and storage requirements. The technique, which has been inspired by the method in [11], employs a k-d tree structure [14] in the state space partition as a more flexible and powerful alternative to the generalized quad-tree structures used in [13]. Conditions for robust exponential stability of the closed-loop system are derived in terms of linear matrix inequality (LMI) constraints. Using this approach, an explicit description of a control law is obtained that assures robust constraint satisfaction, robust stability and low real-time computational complexity. Notation: for a vector x \n and positive-definite
877
2 matrix Q, the weighted norm x Q is denoted by xT Qx. The
symbol * will be used to denote the corresponding transpose of the lower block part of symmetric matrices. II. PROBLEM FORMULATION Consider a linear time varying uncertain system
xk 1
Ak xk Bk uk ,
(1)
[ Ak , Bk ] :,
where k t 0; u \m and x \ n are the input and the
weighting matrices and N ! 0 denotes the control horizon. The set k is typically chosen to be control invariant [15] with respect to Fk in the specified polytopic family (2). Since the optimization problem (4) is nonlinear and nonconvex in the uncertainty, exact solutions are, in general, not available [4]. Under some certain conditions however, the min-max robust MPC problem can be formulated as a semi-definite programming (SDP) problem involving linear matrix inequalities [16]. To see this, a parameterization of the robust cost index similar to [16] is adopted:
min J 1 J 2
and ( Ak , Bk ) is a controllable pair. It is assumed that : is a polytopic set
subject to
1
1
2
2
L
L
where
denotes
Co{}
convex
hull
and
J 1 t max U T U 2 xkT U ,
(6)
J 2 t max ( A N xk B NU )T (k ( A N xk B NU ),
(7)
[
(2)
: Co ^[ A , B ],[ A , B ],...,[ A , B ]` ,
(5)
U , Fk , (k
nun num measurable state respectively. Also Ak \ , Bk \
, ]:
[ AN , BN ]: N
l
l
[A ,B ]
A T ) A * ,
AT ) B and the
l $ {1, 2.., L} are vertices of the convex hull (see e.g.
where J 1 ! 0, J 2 ! 0,
[2]). The control objective is to regulate the uncertain system (1) to the origin while respecting the input and state constraints:
matrices A , B , AN , BN , ) and * are easily obtained
u min d u k i |k d u max ,
i t 0,
ymin d < xk i |k d ymax ,
i t 1.
(3)
from (1)-(2), ) , and * , (see [16] for details). The sets : and : N can be constructed as in [16]. Notice that (7) defines the inclusion condition i.e., being an ellipsoid k t 0 with k x k N |k k
k {x \n |xT (k x d J 2 } [2]. The triplet [J 2, (k , Fk ] in (5)-
where the bounds umin , u max , ymin , and ymax are vectors of appropriate dimensions satisfying
umin 0 umax ,
ymin 0 ymax elementwise. For the current state xk , a
typical robust constrained model predictive control algorithm [1] solves the following min-max optimization problem: U
max
[ Ak i , Bk i ]:
J (U , xk )
[ Ak i Bk i Fk ]T (k [ Ak i Bk i Fk ] (k FkT *Fk ) d 0, (8)
[ Ak i , Bk i ] :, i t N .
(4a)
The solution to the LMI-based min-max robust MPC problem is provided by the following lemma (see [16] for a complete description): Lemma 1. (On-line robust MPC) Consider the uncertain system (1)-(3). The min-max optimization problem (4) can be formulated as the following SDP problem:
min
(7) is chosen to satisfy
subject to (3) and
x k i |k k ,
i t N ,
(4b)
u k i |k
i t N ,
(4c)
i t 0,
(4d)
xk i 1|k
Fk xk i |k , Ak i xk i |k Bk i u k i |k ,
where the cost function is given by
(a) Invariant ellipsoid: There exist a symmetric positive definite matrix Q , two symmetric matrices {Z , * } and a matrix Y satisfying the following LMIs:
2 k N |k (k
J (U , xk ) ||x
N 1
2 k i |k )
|| ¦ [||x i 0
|| ||u
2 k i |k *
|| ].
(4e)
Here, U [ukT|k , ukT1|k ,..., ukT N 1|k ]T is the vector of control moves, ) , * and (k are symmetric positive definite 878
Q ª « Al Q B l Y « 1/ 2 « ) Q «¬ *1/ 2Y
* Q 0 0
* * º * * » » t 0, J 2I * » 0 J 2 I »¼
l $,
(9)
ª Z *º «Y T Q» t 0, ¬ ¼ Q *º ª «< ( Al Q Bl Y ) * » t 0, ¬ ¼
2 Z d diag(umax ),
2 * d diag( ymax ),
r {x \n |Hr x d dr },
(10)
Hr
(11)
ªI º « I » , dr ¬ ¼
ª xu º « l » , r ! , ¬ x ¼
(16)
1 1 where Q J 2 (k , Fk YQ . Then the state feedback controller u k i |k Fk xk i|k , i t N exponentially stabilizes
where I \nun denotes the identity matrix and the lower and upper limits x l and xu are real n-vectors satisfying xl xu in element wise. The index set of boxes is denoted ! . Moreover, let . {ȣ1 ,..., ȣM } represent a set of M
the system (1) for any xk N |k k while respecting the constraints in (3) for all i t N .
vertices of r such that r !
(b) Robust convergence to k : Assume the following LMIs are satisfied:
{ x \ n |x
r
M
¦D ȣ , 0 d D h h
M
h
d 1,
h 1
J1 ª « U T x k ¬
*º » t 0, ¼
1 *º ª « A x B U Q » t 0, N ¬ N k ¼ G (U , xk ) d 0,
[ , ] : ,
(12)
[ A N , B N ] : N ,
(13)
where G (U , xk ) can be easily obtained from (3). Then U drives the state x k N |k into k while satisfying the constraints (3) for all 0 d i d N 1.
(c) Optimization: The min-max optimization problem (4) can be solved by
min
J 1 J 2 subject to (10)-(14)
h
1}.
(17)
h 1
The set of feasible regions 4( ) will henceforth be referred to as a partition. In each region of the partition 4( ) a local affine feedback controller
(14)
J 1 ,J 2 ,U ,Q ,Y
¦D
(15)
Uˆ r ( x) Fˆr x gˆ r , Fˆr \nu N , gˆ r \N , r ! ,
is defined. Notice that (18) implies that Uˆ : X ' o \ mN is a piecewise affine (PWA) function restricted to the set X ' *r! r which is the union of all hyper-rectangles in 4( ). The feedback parameters Fˆr and gˆ r can be
computed as suggested in [18], [10], by considering the optimal solutions to the on-line robust MPC problem (15) at vertices of a box only. Consider any given region
The considerable computation effort required to solve the min-max control problem (15) on-line is an obstacle that greatly restricts its application range to systems with relatively slow dynamics or high-performance computers. In the following, a new technique to obtain controllers of significantly lower computational complexity will be presented. The proposed algorithm off-line determines a sequence of approximate feedback laws defined on the state space partition of boxes. Subsequently, the constructed partition as well as the associated local controllers will be analyzed for robust stability. III. OFF-LINE SOLUTIONS TO ROBUST MPC
o {ȣ1 ,..., ȣM }, and let U h denote
X with vertices .
The min-max MPC algorithm guarantees exponential closed-loop stability, once a feasible solution is found.
(18)
the optimal control sequence computed at ȣh . Suppose that the feedback parameters Fˆ and gˆ satisfy the optimization problem:
min ˆ F , gˆ
M
ˆ gˆ ) ||2 , ¦ || U ho ( Fȣ h / h 1
0
/0 ! 0,
(19)
subject to G(U , ȣh ) d 0, ȣh . .
Lemma 2. (Feasible approximate controller). The least
ˆ gˆ , is robustly squares approximate solution Uˆ ( x) Fx feasible for all x and all uncertainty realizations [ Ak , Bk ] :.
Proof. Follows directly from convexity.
A. Feasible Robust Control Suppose the set X \n of feasible initial states can be decomposed into n-dimensional boxes or hyper-rectangles 4 ( ) {r }r ! X , given by polyhedra of the form
The accuracy of this approximation will be measured by the difference between the optimal and approximate solutions restricted to a hyper-rectangle
879
H
ˆ gˆ ) ||2 , ȣ . , max || U ho ( Fȣ h / h h
(20)
where / t 0 is the weighting matrix. A sensible choice for / is to weight only the components of the solution that correspond to the first m inputs [18]. Notice that satisfying the error bound (20) at the vertices does not necessarily imply that the bound will be satisfied for all x [11]. As a partial remedy, the following estimate is adopted:
Hˆ max H , . {ȣ1 , ȣ2 ,..., ȣM ,..., ȣM 1 , ȣM }, M t M , ȣh .
(21)
where the set . contains, in addition to . , a finite number of arbitrary points in . Moreover, it is assumed that for all regions in the partition, the error bound (21) should respect the following tolerance:
H
2 max(H a , H r min || U ho ||/ ), h
2. Select any unexplored region 1 P. If P then the algorithm terminates successfully. 3.
where H a ! 0 and H r ! 0 can be interpreted as absolute and relative tolerances respectively [11]. B. Approximate Off-line Algorithm An immediate consequence of enforcing orthogonal structures (16) on the state space partition is that the partition can be organized as a multidimensional binary search tree (quad-tree or k-d tree [14]) the search complexity of which is logarithmic with respect to the number of regions. In this paper, k-d trees are adopted as more flexible and powerful alternatives to quad-trees. A k-d tree is constructed hierarchically. At the current tree level, a splitting plane is selected that subdivides a given box into two equal-sized boxes. The process is repeated recursively until certain termination criteria are reached. The choice of the splitting plane at each tree level may be arbitrary or, as suggested in [11], the plane can be selected to be orthogonal to the axis for which the change of the approximation error (21) is maximal. In the following, an algorithm for computing the approximate explicit solution to the robust MPC problem
xk
in (12), (13) and (14) by
ȣh
h {1, 2,..., M } and solve (15) to obtain the set of
optimizers {U1o ,...,U Mo }. If all solutions are feasible, go to step 4. Otherwise, compute the size of 1 measured by the largest Euclidean distance between any pair of vertices. If it is smaller than some tolerance, mark 1 explored and infeasible and go to step 2. Otherwise, go to step 6. 4. Compute an approximation Uˆ ( x) using (19). If no feasible solution is found, go to step 6. 5. Compute the error in the solution Hˆ using (20), (21). If Hˆ d H , add the region 1 to 4( ) and go to step 2. 6.
(22)
Substitute
Partition 1 into two equal hyper-rectangles 2 , 3
using the method proposed in [11]. Remove 1 from P and add 2 , 3 . Go to step 2.
The algorithm terminates after a finite number of steps with the piecewise approximation Uˆ ( x) and a partition 4( ) inside of which this approximation is valid. This does not imply that the algorithm will terminate in finite time, though in extensive simulations the termination was always attained. Notice that the off-line computational complexity and real-time storage requirements may grow very quickly with the system dimension.
IV. STABILITY
Since the approximate feedback controller defined in the previous section does not guarantee stability by design, a posteriori stability analysis is required to ensure that the feedback is also robustly stabilizing. In addition, any nonzero tolerance H 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, various classes of Lyapunov functions may be considered (see [19] for an excellent review). In this paper, the attention is restricted to the following common quadratic Lyapunov functions:
Algorithm 1. (Off-line approximate robust MPC).
(4) defined on a feasible set X \n will be presented. Let the initial box 0 X (root of the tree) be defined as a minimal bounding box containing the set X . Considering [13], [11] the following recursive off-line algorithm is suggested:
1. Let the set of all unexplored hyper-rectangles be denoted as P. Initialize the partition to the region 0 X i.e. P
{0 }.
V ( x) xT PLP x, x 4( ),
(23)
nun where PLP \ ! 0. Suppose the following design
880
requirement is satisfied for all regions in 4( ) and all uncertainty realizations l $
3
V ( xk 1 ) V ( xk ) d U||xk ||2 , k t 0,
2
(24)
1
x2
where the scalar U ! 0 is introduced to enforce exponential stability. It is easy to verify [2], [5] that the function V ( x ) is a Lyapunov function over the box
-1
partition 4( ). The condition (24) can be reformulated along the lines of [5] to obtain a description suitable for LMI solvers. To see this, define for all l $ the variation of the Lyapunov function associated with region r as follows
'Vrl ( xk )
xkT 'Qrl xk 2 xkT 'lrl 'crl ,
(25)
0
-2
-3
-0.2
0 x1
0.2
0.4
0.6
O 2.5 (dashed).
where
1 d u d 1, 0.5 d x1 d 0.5.
'lrl
( Al Bl Fˆr0 )T PLP ( Al Bl Fˆr0 ) PLP , ( Al Bl Fˆ 0 )T P Bl gˆ 0 ,
'crl
( Bl gˆ r0 )T PLP Bl gˆ r0 .
'Qrl
-0.4
Fig. 1. Orthogonal partition of the approximate robust MPC and the state trajectories corresponding to O 0.5 (solid), O 1.5 (dotted) and
-0.6
r
LP
r
(26)
0 0 Here, Fˆr and gˆr denote the first m components of the
approximate solution (18) associated with a region r . Lemma 3. There exists a quadratic function (23) meeting the requirement (24) if for all r ! and l $ there exist l symmetric matrices Nr t 0 with a proper dimension and a
nonnegative scalar U satisfying the following LMI:
(27)
Theorem 1. If there exists a quadratic function (23) such that (27) is satisfied, then the off-line approximate robust MPC guarantees exponential closed-loop stability.
V. SIMULATION EXAMPLE For simplicity of illustration we use an uncertain double integrator example widely studied in the literature. The model takes the form of (1) where
ª 1 0º , B k «¬ Ok 1 »¼
'x2
0.015 and
0.06. The tolerance on the approximation error is 0.0001 and H r
0.1.
A. Complexity Analysis The off-line solution computed with Algorithm 1 is depicted in Fig. 1 and consists of 314 regions and 12 levels of search. The computational complexity of the approximate approach consists, in the worst-case, of a total of 16 arithmetic operations per sample (12 comparisons, 2 multiplications and 2 additions). For comparison, on a Pentium IV machine (1.8 GHz and total memory 500 MB) the average time for the on-line robust MPC algorithm (15) to compute a solution is 0.9 s, which indicates that millions of arithmetic operations are required in real-time to solve the LMI problem. B. Performance Properties The approximate controller is robustly stabilizing as the LMI problem in Lemma 3 provides a Lyapunov function
Proof. Follows trivially from Theorem 4 in [5].
Ak
regions are restricted to be larger than 'x1 chosen according to (22) with H a
ª ( H r )T N rl H r 'Qrl U I * º « » d 0. l l T 'crl ¼ ¬ d r N r H r ('lr )
The weighting matrices are selected as ) I and * 1 and the control horizon is N 4. The state space to be partitioned is defined by 0 [0.75, 0.75] u [3,3] and the
ª1 º , k t 0, «¬ 0 »¼
and Ok [0.5, 2.5] is an uncertain time-varying parameter. The following input and state constraints are imposed
matrix PLP
> 0.8985 0.2711
0.2721 0.2707
4 @ and a decay rate of U 6 u10 .
The performance of the closed-loop system was simulated
[0.75 1]T and for three values of the uncertain parameter O {0.5,1.5, 2.5} . The resulting for the initial condition x0
closed-loop responses are depicted in Fig. 1. It can be observed that the proposed controller brings the system to the origin despite of the model uncertainty, and keeps the state evolution within its limits. Furthermore, assuming
881
description is proposed. It is shown that the approximate solution to this problem can be pre-computed off-line in an explicit PWA state feedback form defined on the partition of boxes. This explicit structure is exploited for efficient real-time implementation via binary search trees avoiding on-line optimization. This makes the presented method an attractive alternative to the existing robust MPC schemes.
3
2.5
2
1.5
x2 1
REFERENCES [1]
0.5
x1
0
[2] -0.5
0
2
4
6
8
10
12
14
16
18
20
Time (samples)
[3]
0.4
[4] 0.2
[5]
0
-0.2
[6]
-0.4
[7] -0.6
[8]
-0.8
-1
0
2
4
6
8
10
12
14
16
18
[9]
20
Time (samples)
Fig. 2. The state (top) and input (bottom) trajectories for the exact (dashed) and approximate (solid) solutions corresponding to Ok 1.5 sin( k ).
Ok
1.5 sin(k ), the state and input trajectories for the
exact (Lemma 1) and approximate algorithms are shown in Fig. 2. Notice that the approximate robust MPC controller achieves nearly the same performance as its exact (on-line) counterpart. 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, based on simulations for 2525 initial states:
Hu
[10] [11] [12] [13] [14] [15]
| Uˆ ( x) U o ( x)| u100%. umax umin
[16]
The total average performance decrease for the considered example is around 0.45% which is not prohibitive as indicated by the simulation results depicted in Fig. 2.
[17] [18] [19]
VI. CONCLUSIONS A new off-line algorithm to address robust model predictive control of systems with a polytopic uncertainty
882
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