Interpolation based computationally efficient predictive control

9 downloads 0 Views 184KB Size Report
Feb 15, 2004 - advantage of convergence to an optimal control law. Furthermore, the ... of attraction could be expanded and computed as the convex hull of ...
INT. J. CONTROL,

15

FEBRUARY

2004, VOL. 77, NO. 3, 290–301

Interpolation based computationally efficient predictive control J. A. ROSSITERy*, B. KOUVARITAKISz and M. BACICz This paper investigates interpolation based predictive control and presents a study of the properties and therefore limitations of the approach. This understanding is used to develop an efficient algorithm with guarantees of recursive feasibility and stability.

1. Introduction Constrained MPC (Mayne et al. 2000) calls for the online optimization of a cost subject to a number of constraints. The usual choice for the cost is quadratic in the degrees of freedom (d.o.f.), whereas the constraints are: (i) input/state constraints which are usually linear; and (ii) stability constraints which can be linear or quadratic depending on whether the deployed terminal region is polytopic or ellipsoidal. The presence of quadratic constraints converts the optimization problem into a semi-definite program (SDP) which can be very demanding for large dimension systems and/or long prediction horizon. However, even with linear only constraints for which the online optimization reduces to a quadratic program (QP) (Tsang and Clarke 1988), the computation could still be excessive for systems with fast dynamics. The situation is made much worse if one wishes to invoke a guarantee of closed loop stability, which is possible through the imposition of quadratic stability constraints (Mayne et al. 2000) whose implementation converts the online optimization to an SDP. Restricting the number of d.o.f. then appears to be the only option, but this could lead to significant suboptimality and/or feasibility problems (restrictions to the size of the feasible region). A potentially effective solution to the above is a re-characterization of the d.o.f.: rather than considering individual predicted moves, one can instead interpolate between a set of predetermined predicted trajectories with desirable attributes (Rossiter et al. 1998, 2001). Thus the conventional approach considers projection of the predicted trajectory onto the set of pulse trajectories, ½1, 0, 0, . . ., ½0, 1, 0, . . ., etc., whereas the alternative projects onto fewer sets of especially chosen Received 1 July 2003. Revised and accepted 1 November 2003. *Author for correspondence. e-mail: j.a.rossiter@sheffield. ac.uk y Department of Automatic Control & Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK. z Department of Engineering Science, Parks Road, Oxford OX1 3PJ, UK.

predicted trajectories. Selecting one of these predicted trajectories to be the unconstrained linear quadratic (LQ) optimal (Kouvaritakis et al. 1998) affords the advantage of convergence to an optimal control law. Furthermore, the inclusion into the set of predicted trajectories of the ‘tail’, that is, the extension (shift) to current time of the trajectory computed at the previous time instant, enables the assertion of feasibility and closed loop stability (Mendez et al. 2000). Despite their advantages, these approaches require the online solution of a QP (albeit in a small number of degrees of freedom) and do not provide a characterization of the regions of attraction. A more general interpolation scheme is proposed in Bacic et al. (2003), whose region of attraction could be expanded and computed as the convex hull of invariant ellipsoids. However the implementation of this algorithm requires the online use of SDP and this can be computationally demanding. The remedy of this is to abandon ‘general interpolation’ and revert to co-linear interpolation which brings about a dramatic reduction in computational load but this creates difficulties in guaranteeing recursive feasibility and stability and restricts the size of the stabilizable set. The motivation behind this paper is the development of algorithms which are practicable for systems with fast dynamics, such as electromechanical systems requiring sample periods of the order of milliseconds or less. For such systems conventional MPC strategies (Mayne et al. 2000) are impracticably demanding unless one uses unreasonably short horizons thereby severely curtailing the size of the stabilizable set and leading to unacceptable degrees of suboptimality. Interpolation techniques provide an obvious and useful remedy, but amongst these: (i) the ones with guarantees of feasibility and stability either have no characterization of the stabilizable set (Mendez et al. 2000) or require (Bacic et al. 2003) the online implementation of an SDP which could still be computationally demanding; whereas (ii) the co-linear interpolation techniques (Rossiter et al. 2003) are not amenable to the recursive feasibility analysis. The aim of this paper is twofold: (i) to extend earlier general interpolation techniques in a way that retains the enlargement and explicit characterization of the

International Journal of Control ISSN 0020–7179 print/ISSN 1366–5820 online # 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00207170310001655327

291

Interpolation based predictive control stabilizable set but avoids the need for SDP; (ii) to extend the co-linear interpolation techniques in a way that retains the computational simplicity but provides a recursive guarantee of feasibility. The former is undertaken in } 2 where invariant ellipsoids are replaced by maximal invariant polytopic sets and where computational complexity is reduced to requiring the online solution of a QP, or (through the definition of an alternative cost) even a simple LP. MPC based on co-linear interpolation is reviewed in } 3 and suitably extended in } 4 where the relevant feasibility results are presented. A comparison of the algorithms proposed in this paper is undertaken in } 5 and some conclusions are given in } 6.

2. Linear programming formulation of general interpolation Invariant feasible sets (Blachini 1999) provide a convenient means of deriving a recursive guarantee of feasibility in that membership is equivalent to testing for predicted constraint satisfaction of closed-loop trajectories. With the view to enlarging the size of such regions of attraction, interest has been focused on maximal admissible sets (MAS) (Gilbert and Tan 1991) which are convex and polytopic. The weakness of such sets is that they do not extend to the case of models which are subject to polytopic uncertainty. This consideration as well as the emergence/development of efficient LMI toolboxes has shifted interest from maximal polytopic sets to ellipsoidal inner approximations (Kothare et al. 1996). However, the difference in volume between the maximal and ellipsoidal approximations can be considerable and this formed the motivation behind Bacic et al. (2003) that uses general interpolation in order to replace ellipsoidal sets with the convex hull of ellipsoidal sets. The condition for membership of this convex hull turns out to be an LMI and thus, the implementation of the interpolating strategy within MPC leads naturally to an algorithm which calls for the online solution of an SDP. Nevertheless, this can be very demanding and in this section we propose a simple extension which is based on polytopic rather than ellipsoidal invariant sets and leads to an MPC implementation which calls for the online solution of a QP. 2.1. Notation and feasible regions This paper assumes a state space model and constraints 9 xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ; yk ¼ CxðkÞ > = u  uðkÞ  u ð1Þ > ; x  xðkÞ  x

where x, u and y are the state, input and output respectively and where the above inequalities apply on an element-by-element basis. Next we define the MAS for several different control laws and show how a predicted trajectory, and corresponding invariant set, can be made up by a convex linear combination of these control laws. Lemma 1: The MAS (Gilbert et al. 1991), for the system of (1) under the control law u ¼ Ki x is defined by a linear set of inequalities of the form S i ¼ fx: Mi x  di  0g

ð2Þ

Proof: For systems with strictly stable closed-loop poles and the asymptotic point not on a constraint, it is only necessary to invoke constraints over a known finite prediction horizon . Over such a horizon the vector of closed-loop state and input predictions assume the form 2 3 2 3 xðk þ 1jkÞ uðkjkÞ 6 xðk þ 2jkÞ 7 6 uðk þ 1jkÞ 7 6 6 7 7 7 6 7 ¼ P x¼6 xðkÞ; u ¼ i .. .. 6 7 6 7 5 5 ~ 4 ~ 4 . . uðk þ   1jkÞ ¼ Qi xðkÞ ð3Þ

xðk þ jkÞ

Thus invoking the constraints in (1) we get (2) with     NPi n Mi ¼ ; d¼ ð4Þ l LQi Expressions for Pi , Qi , N, L, n, l can be obtained through straightforward algebra. œ Theorem 1: The convex hull of S i , i ¼ 1, . . . ,  is invariant and feasible under the control law uðk þ jjkÞ ¼ 

 X

i Ki Fji xi ðkÞ

i¼1

where

9   > >  xðkÞ ¼ i xi ðkÞ;  x ðkÞ 2 S > > i i> >  > i¼1 >  =   X    0 i ¼ 1; >  i > >  > i¼1 >  > >  > ;  Fi ¼ A  BKi  X

ð5Þ

Proof: Every initial condition xðkÞ in the convex hull of S i can be decomposed as per (5). Then under (5) the state vector predictions are given by xðk þ jjkÞ ¼ 

 X

i xi ðk þ jjkÞ;

xi ðk þ jjkÞ ¼ Fij xi ðkÞ

i¼1

ð6Þ

292

J. A. Rossiter et al.

and by the invariance of S i , xi ðkÞ 2 S i , implies that xi ðk þ jjkÞ 2 S i and hence from (6) xðk þ jjkÞ will lie in the convex hull of S i . Corresponding to predictions (6), thePcontrol law of (5) can be rewritten as uðk þ jjkÞ ¼  i¼1 i Ki xi ðk þ jjkÞ which, by the triangle inequality implies (comparison is taken to be element by element)  X i¼1

min ði Ki xi ðk þ jjkÞÞ  uðk þ jjkÞ



i¼1

max ði Ki xi ðk þ jjkÞÞ

ð7Þ

xi 2S i

However the feasibility of S i implies  X i¼1  X i¼1

max ði Ki xi ðk þ jjkÞÞ  xi 2S i

 X

i u ¼ u

X

i¼1

min ði Ki xi ðk þ jjkÞÞ 

xi 2S i

 X

i u ¼ u

X

i¼1

9 > > i ¼ u > > = > > > i ¼ u > ; ð8Þ

which asserts the feasibility of the convex hull of S i under the control law of (5). œ Corollary 1:

The predicted control law

uðk þ jjkÞ ¼ 

 X

i Ki F j x^ i ,

xðkÞ ¼

X

x^ i

ð9Þ

i¼1

results in feasible predicted trajectories iff there exist x^ i , i such that X i ¼ 1; i  0 ð10Þ Mi x^ i  i d; Proof: Let xi ¼ x^ i =i , then (9) is equivalent to xi 2 S i . The rest follows from Theorem 1. œ 2.2. Interpolation based predictive control Interpolation is most effective when the introduction of a new Ki , S i results in a new convex hull which is in fact bigger (Rossiter et al. 2001) than would arise from adding an extra d.o.f. to a more conventional MPC algorithm such as in Scokaert and Rawlings (1998). It is well known that retuning (Kothare et al. 1996) can be an effective way of enlarging feasible regions whereas adding a single d.o.f. to the input trajectory is not. Hence, in this section, a predictive control algorithm is derived which uses the predictions of (9) and thus whose region of attraction is the convex hull of all the associated MAS. 2.2.1. Quadratic performance index. First define an index of performance by the quadratic cost J¼

1 X k¼1

xðkÞT QxðkÞ þ uðk  1ÞT Ruðk  1Þ

Theorem 2: The predicted cost under the control law of (9) is quadratic in x^ i and is given as J ¼ x~ T Px~ ;

xi 2S i

 X

The aim is to compute and minimize this J subject to predictions/constraints of (5) and (9). The cost function is shown to be quadratic in the state decomposition and hence can be minimized by using the decomposition variables as the d.o.f.

ð11Þ

x~ ¼ ½x^ T1 , x^ T2 , . . . , x^ T T

ð12Þ

where P is defined as the positive definite solution of the Lyapunov equation P ¼ GTu RGu þ CT GTx QGx C þ CT PC 8 C ¼ diag ½F1 , . . . , F  > < Gx ¼ ½I, I, . . . , I > : Gu ¼ ½K1 , K2 , . . . , K 

ð13Þ

Proof: Use of Lyapunov equations to sum the cost of infinite horizons is standard. Hence, given the substitution of predictions (6) and (9) into (11), the proof is obvious. œ Algorithm 1: An interpolation MPC algorithm is given by minimizing the cost of (12) w.r.t. x~ given in (12) and subject to the corresponding constraints of (5) X 8 > xðkÞ ¼ x^ i > > < min x~ T Px~ s:t: ð14Þ Mi x^ i  i d > x^ i , i¼1,...,  > X > : i ¼ 1, i  0 Theorem 3: Algorithm 1 has guaranteed recursive feasibility and closed-loop convergence. Proof: Assume feasibility at time k, then, because the decomposition of x into components x^ i is free, at sampling instant k þ 1 we can choose a decomposition so that x~ ðk þ 1jk þ 1Þ ¼ x~ ðk þ 1jkÞ

ð15Þ

which is known from Theorem 1 to be feasible. This implies that Jðk þ 1Þ ¼ JðkÞ  xðkÞT QxðkÞ  uðk  1ÞT  Ruðk  1Þ is an upper limit on the cost function. Therefore J is Lyapunov and the control strategy is both stabilizing and convergent. œ Numerical illustrations in } 5 demonstrate how effective the proposed interpolation of this section is in increasing the volume of the feasible region and moreover the good performance of Algorithm 1. 2.2.2. Linear performance index. When interpolating between two trajectories (x1 , x2 ) of which one (say x1 ) is generated by the LQ optimal, it may be convenient to replace the online optimization of Algorithm 1 by an LP. This is not strictly equivalent to the QP solution

293

Interpolation based predictive control in general, but nevertheless it could be argued that it makes good practical sense and also has a guarantee of convergence/feasibility through the monotonicity of 2 . Algorithm 2: For the case that  ¼ 2, 2 ¼ 1  1 , x^ 2 ¼ x  x^ 1 , an efficient LP interpolation MPC algorithm is given by 8 > < M1 x^ 1  ð1  2 Þd min 2 s:t: M2 ½x  x^ 1   2 d ð16Þ > x^ 1 , 2 : 0  2  1 Section 5 will show that despite the use of the suboptimal (but of course computationally efficient) LP Algorithm 2, the performance is nevertheless very good. 3. Co-linear interpolation Algorithm 2 gives a significant reduction in online computation when compared with conventional algorithms which require SDP or QP solvers. Furthermore restricting the number of interpolation trajectories to only two requires only n þ 1 d.o.f. Nevertheless, for large dimension systems with fast dynamics, even this computational burden could be excessive. Under such circumstances there is a need for further computational reductions, and a convenient way for achieving that is offered through the use of co-linear interpolation (Rossiter et al. 1998, 2001) according to which x1 , x2 and x are all taken to be in the same direction. With this restriction (5) becomes x^ 1 ¼ ð1  ÞxðkÞ; uðk þ jjkÞ ¼

K1 F1j x^ 1

x^ 2 ¼ xðkÞ;

 K2 F2j x^ 1 ;

01

ð17Þ

where for convenience 1 and 2 have been substituted by ð1  Þ and . The predicted state and input trajectories have the same form as that of (6) and thus it is easy to show from (2) that the feasibility of the predictions can be ensured if and only if  qðkÞ ¼ ðM2  M1 ÞxðkÞ qðkÞ  pðkÞ  0; ð18Þ pðkÞ ¼ d  M1 xðkÞ Note that M1 and M2 must be constructed so that the inequality implied by the kth row of each corresponds to the same constraint. Algorithm 3 (Rossiter et al. 2001, 2003): At each sampling instant, define the constraints given in (18) and perform the minimization  qðkÞ  pðkÞ  0 min  s:t: ð19Þ  01 and compute the current control action from uðkÞ ¼ ½ð1  ÞK1 þ K2 xðkÞ

ð20Þ

Remark 1: J is proportional to 2 , hence minimizing  is equivalent to minimizing J for the prediction class of (17). Nevertheless, all MPC algorithms (e.g. Clarke et al. 1987, Scokaert and Rawlings 1998) are suboptimal with small numbers of d.o.f. although numerical examples (Rossiter et al. 2001) show that this suboptimality is often negligible for Algorithm 3 (as is also true with many other MPC algorithms). However, the significant advantage of Algorithm 3 is its efficiency with the main computational burden being the update of qðkÞ and pðkÞ which requires just two matrix/vector multiplications. The downside of Algorithm 3 is the same as its strength: the co-linearity requirement! This poses some serious control theoretic problems. Given x^ 1 ðkÞ ¼ ð1  ÞxðkÞ, x^ 2 ðkÞ ¼ xðkÞ which clearly are co-linear, in general this will not be the case for x^ 1 ðk þ 1jkÞ ¼ ð1  ÞF1 xðkÞ and x^ 2 ðk þ 1jkÞ ¼ F2 xðkÞ. Thus unlike the case of general interpolation which allows one to use (if desired) the previously computed predicted trajectory (time shifted by one instant), co-linearity requires a new decomposition at each time instant. Therefore one cannot (Mendez et al. 2000) assert a monotonicity property for the cost (as in Theorem 2) or for 2 (as in Algorithm 2). Even worse it is not even possible to guarantee that feasibility at initial time ensures feasibility at all future instants. Therefore, the aim of the next section is to explore and overcome these problems, while retaining the very significant computational advantages of co-linear interpolation.

4.

Extensions of co-linear interpolation

For Algorithm 3 xðkÞ 2 [i S i does not imply xðk þ 1Þ 2 [i S i , and hence, in general, feasibility cannot be guaranteed. Recursive feasibility could however be assured by requiring that at each time instant the implied poles of interpolation x^ 1 and x^ 2 are so adjusted to satisfy the constraints of (16). This is easy to accomplish through a process of scaling. First, it is convenient to introduce some further assumptions and notation. . Assume that the constraints (2) are normalized; that is, scale the rows of Mi , d such that each element of di ¼ 1. . Given eTj is the jth standard basis vector, define the notation i ¼ jMi xj  max eTj Mi x j

ð21Þ

This gives an alternative definition of the MAS in (2) as S i ¼ fx: jMi xj  1g

or

S i ¼ fx: i  1g

ð22Þ

294

J. A. Rossiter et al.

. Define s1 and s2 as any positive numbers such that: x ¼ x^ 1 þ x^ 2 ; si   i ,

x^ 1 ¼

x , s1

x^ 2 ¼

x , s2

ð23Þ

i ¼ 1, 2

Note that this implies x^ 1 2 S 1 and x^ 2 2 S 2 . . Due to co-linearity, feasibility is restricted to the region where either 1  1 or 2  1. Remark 2: The value  ¼ jMxj determines the relative closeness to a constraint. So if  ¼ 1, then the predictions hit a constraint, whereas, for example, if  ¼ 0:5, then the current state could be doubled in magnitude before the predicted trajectories reach a constraint. 4.1. Ensuring feasibility of co-linear interpolation predictions The result below suggests a way of selecting  of (17) that ensures feasibility. Lemma 2: The condition that either 1  1 or 2  1 implies the existence of si, i ¼ 1, 2 satisfying (23) and in turn implies that there exists a : 0    1 and co-linear decomposition x ¼ ð1  Þx1 þ x2 ;

x1 ¼

x , s1

x2 ¼

x s2

ð24Þ

for which the state and input predictions are feasible under the control law uðk þ jjkÞ ¼ ð1  ÞK1 F1j x1  K2 F2j x2

ð25Þ

Proof: The condition si  i in combination with (22) and (24) ensures that xi 2 S i , i ¼ 1, 2 which, as proven in Theorem 1, establishes feasibility of the predictions for 0    1. It therefore remains only to show the existence of a least one pair s1 , s2 satisfying (23). Rewriting (24) requires x¼

ð1  Þ  xþ x s1 s2

ð26Þ

Hence, decomposition (26) and (24) require that ð1  Þ  þ ¼ 1; s1 s2

Remark 3: For convenience one may want to reintroduce the variable  of (17) and hence define x^ 1 ¼ ð1  Þx, x^ 2 ¼ x. This can be done by noting the equivalence between (17) and (26) which gives ð1  Þ ¼ 1  ; s1

 ¼ s2

ð28Þ

4.2. Optimizing performance over feasible predictions Lemma 2 demonstrates the existence of an interpolation of the form (25) such that the predictions are feasible. However, it does not show how the variables , , s1 and s2 might be chosen. The dependence of  on  and s2 is bilinear and this would appear to complicate matters. However, it is possible (Rossiter et al. 2001) to prove that the predicted cost (10) under the predicted control trajectory of (25) is proportional to 2 . In light of this observation, it is possible to determine explicitly the optimal values of s1 and s2 and hence from (28) that of . Lemma 3: The smallest  in (25) is either 0 for the case x 2 S 1 or otherwise ¼

1 2  2 1   2

ð29Þ

Proof: If x 2 S 1 , then (25) holds true for  ¼ 0; this is the smallest value that  can assume. Simple geometric arguments on the other hand should suffice to convince us that for x 62 S 1 , the smallest value for  will be achieved for an x1 which lies as close to x as is possible and an x2 which lies as far away from x as is possible, namely for both x1 and x2 lying on the boundaries of S 1 and S 2 respectively. From definition (21) and Remark 2, this condition is obtained for si ¼ 1=i , which values imply (from 27), that ð1  Þ  þ ¼1 1 2

ð30Þ

Given (30), equation (29) is now obvious. Furthermore, substituting (29) into (28) and s2 ¼ 2 gives ¼

1  1  1  2

ð31Þ

i ¼ 1, 2 ð27Þ

œ

By assumption one must have that either 1  1 or 2  1; these are dealt with in turn. If 1  1 then one could choose  ¼ 0, s1 ¼ 1, s2 ¼ 1. Otherwise, if 1 > 1, 2 < 1, one option is to set s2 ¼ 1,  ¼ 1, s1 ¼ 1. Clearly, in general, other values s1 , s2 ,  which satisfy (23) and (26) will exist (and will lead to better performance). œ

Remark 4: An alternative insight comes from drawing a straight line between x1 ¼ x=1 and x2 ¼ x=2 noting that then xi is on the boundary of S i so any convex linear combination (as in (24)) of these predictions must be feasible. It then remains to find a  such that x ¼ ð1  Þx1 þ x2 ; such a  is given in (29).

0    1;

s i  i ,

295

Interpolation based predictive control Algorithm 4: Let the control law be definedy as jM1 xðkÞj > 1 ) uðkÞ ¼

1  2  1 K xðkÞ þ 1 K xðkÞ 2   1 2 2  1 1

jM1 xðkÞj  1 ) uðkÞ ¼ K1 xðkÞ

ð32Þ

The control law is undefined if 2 > 1 and 1 > 1. This section has demonstrated that the choice of  given in (31) guarantees feasibility of the predictions (17). It remains to show that moreover one can guarantee recursive feasibility; that is, a prediction being feasible implies that a feasible choice is in the class of predictions at the next sampling instant. 4.3. Recursive feasibility for Algorithm 4 A proof of feasibility reduces to proving that [ [ xðkÞ 2 ðS 1 , S 2 Þ ) xðk þ 1Þ 2 ðS 1 , S 2 Þ ð33Þ To do this we need to consider two cases separately: (i) the case S 1  S 2 and (ii) the case S 1 6 S 2 . 4.3.1. Recursive feasibility with S 1  S 2 . This section will demonstrate that xðkÞ 2 S 2 ) xðk þ 1Þ 2 S 2 under the condition that S 1  S 2 . Lemma 4:

Define, with 0    1, the two convex sets ð34Þ

T2 ¼ fx^ 2 : M2 x^ 2  d  0g For the choice of  given in (31) and x^ 1 ¼ ð1  Þx, x^ 2 ¼ x, it follows that )

x^ 1 2 T1 ,

x^ 2 2 T2

ð35Þ

Proof: The choice of  is that of (30) and hence the proof follows automatically from Lemma 2. œ Theorem 4: With x ¼ ð1  Þx1 þ x2 , 0    1 and x1 2 S 1 , x2 2 S 2 , the condition S 1  S 2 is sufficient to ensure that xðkÞ 2 S 2

)

xðk þ 1Þ 2 S 2

ð36Þ

and hence that control law of (32) has a recursive feasibility guarantee. Proof: It is known that x^ 1 ðkÞ 2 T1 ) x^ 1 ðk þ 1Þ 2 T1 . Moreover, as S 1  S 2 one can state that: T1  T2,  ;

T2,  ¼ fx: M2 x  ð1  Þd  0g

4.3.2. Recursive feasibility with S 1 6 S 2 . In this case one cannot use the assertion that x^ 1 ðk þ 1Þ 2 T1 ) x^ 1 ðk þ 1Þ 2 T2,  . As such, except for the trivial cases of  ¼ 0, 1, we cannot prove that the prediction set of (25) will ensure xðk þ 1Þ 2 [ ðS 1 , S 2 Þ. This is because the recursive feasibility proof deployed in } 2 used the convex hull, not the union of the two sets. However, it is not difficult to add an extra trivial computation to ensure that xðk þ 1Þ 2 [ ðS 1 , S 2 Þ. Theorem 5: Find the minimum i (if feasible) such that xðk þ 1Þ 2 S i " # " # 9 M1 ðF1 ð11 ÞþF2 1 Þ d > > 1 ¼min 1 s:t: xðkÞ 0 > > > 1 = M1 ð11 ÞþM2 1 d " # " # > > M2 ðF1 ð12 ÞþF2 2 Þ d > > xðkÞ 0 > 2 ¼min 2 s:t: ; 2 M1 ð12 ÞþM2 2 d ð39Þ and define o as the  from (31). Then if  is chosen as  ¼ min ðmax ½o , 1 , max ½o , 2 Þ

T1 ¼ fx^ 1 : M1 x^ 1  ð1  Þd  0g;

 ¼ 2

and hence x^ 1 ðk þ 1Þ 2 T1 ) x^ 1 ðk þ 1Þ 2 T2,  . Finally, it is also obvious from definitions (34) and (37) that  x^ 2 2 T2 ð38Þ ) x^ 1 þ x^ 2 2 S 2 x^ 1 2 T2,  œ

ð37Þ

then prediction set (25) will have recursive feasibility. Proof: It is obvious that condition (40) ensures that the choice of  is such that xðk þ 1Þ 2 [ ðS 1 , S 2 Þ. The condition xðkÞ 2 [ ðS 1 , S 2 Þ ensures at least one of 1 and 2 exists. œ 4.4. Stability and discussions The interest here is to explain how the proposed Algorithm 4 differs from that of Algorithm 3. There are three main observations. 1. For the case S 1  S 2 , Algorithm 4 guarantees that xðkÞ 2 S 2 ) xðk þ 1Þ 2 S 2 . This is not the case for Algorithm 3, although it does have the advantage that it may also be defined for points outside [ ðS 1 , S 2 Þ; that is, it has a potentially larger region of attraction. 2. Algorithm 4 is more cautious than Algorithm 3 in that it will often choose a larger value of . This is because optimization (19) does not insist that x^ i 2 S i , i ¼ 1, 2. Instead it places the emphasis on the worst case combined row sum, notably j½ð1  ÞM1 þ M2 xðkÞj  1

y This takes the same form as (20) with  given as in (31), that is  ¼  (1  1)/(2  1).

ð40Þ

6) jM1 ð1  ÞxðkÞj þ jM2 xðkÞj  1

ð41Þ

296

J. A. Rossiter et al.

Algorithm 4 computes 1 with no regard to 2 and vice versa. Hence the index of the corresponding maximizing rows (see (21)) may be different. 3. The formulation of (32) facilitates staightforward implementation as the optimization is removed. Due to co-linearity one may not establish for either algorithm a monotonicity of cost proof, because it is not possible to use the shift of the previously computed predicted trajectory (Mendez et al. 2000, Rossiter et al. 2003). Hence the user is left needing an alternative approach. A convenient LMI approach was presented in Rossiter et al. (2003); that paper did not tackle feasibility and hence is complemented by this contribution. If the tests of Rossiter et al. (2003) should fail then alternative approaches are needed. Simple alternativesy are to restrict the choice of  even further (Rossiter and Kouvaritakis 1998) or to use solely K2 until such time that predictions (25) give a reduction in the cost. 5. Examples This section will illustrate the use of algorithms developed in this paper. Several aspects will be demonstrated: (i) the potential increase in the feasible region using interpolation; (ii) the good performance achieved by Algorithm 4 and (iii) a comparison of performance/ feasibility with the global optimal (Scokaert and Rawlings 1998)—denoted OMPC hereafter. The following double integrator model will be used for the numerical study     1 0:1 0 xðk þ 1Þ ¼ xðkÞ þ uðkÞ; 0 1 0:0787 yk ¼ ½1 0xðkÞ ð42Þ with input and state limits 1  uðkÞ  1;

 2  ½1

1xðkÞ  2

ð43Þ

The optimal control law for various weights Q and R are given in table 1. It is assumed that the preferred choice is K11 . 5.1. Feasible regions The corresponding MAS for Kij are given in figure 1. The feasible region is determined by the selection of K1 and K2 . This paper uses K1 ¼ K11 , K2 ¼ K12 in order to maximize the width of the feasible region; other pairings will lead to feasible regions with different shapes (for instance K2 ¼ K21 gives a long thin region). y The reader is reminded that rigorous guarantees of stability, as with robustness, often come at the price of a sacrifice in performance, especially where, as here, there is also a demand for computational simplicity.

 Q¼  R¼  R¼

0:1 0 1 0

0 1

0 0:1 

1 0

0 0



 Q¼

0:04 0:2 0:2 1





Table 1.

K11 ¼ ½2:8282 2:8260

K12 ¼ ½0:5539 3:0153

K21 ¼ ½0:9392 1:5926

K22 ¼ ½0:1906 1:1821

Optimal feedback K for different weight combinations.

Figures 2–4 show the feasible regions that arise for Algorithms 1–4 and OMPC with nc ¼ 1, 2, 3, 4, 5, 20 (nc denotes the number of d.o.f.). . Figure 2 shows that OMPC has a very small feasible region unless nc is large. . Figure 3 shows that interpolation (i.e. Algorithms 1 and 2) is more effective at increasing the feasible region than increasing nc in OMPC. . Figure 4 shows that Algorithms 3 and 4 have smaller ‘guaranteed’ feasible regions (dark shading), that is [ ðS 1 , S 2 Þ. Although Algorithm 3 has a larger feasible region (light shading) than Algorithm 4, it is non-convex, not straightforward to define and therefore difficult to utilize effectively. 5.2. Closed-loop simulations A number of initial conditions are used to illustrate the variability of closed-loop behaviour that is possible around the state space; these points and the corresponding closed-loop state trajectories are shown in figure 5. Simulations are performed only where the initial state is clearly inside the feasible region shown in figures 2–4. OMPC is infeasible with small nc for all these points, so the cost and the state trajectories are for nc ¼ 20; this is to benchmark the other algorithms. The corresponding evolutions of  are given in figure 6 and the closed-loop costs, corresponding to J are given in table 2. The key observations follow. . Algorithm 1 gives the largest ‘guaranteed’ feasible region and excellent performance (close to OMPC) and here with only 3 d.o.f. . Algorithm 3 gives excellent performance (in fact closer to OMPC than Algorithm 1 when feasibley). However, although for this example it retained recursive feasibility, as yet no proof (or counter proof) exists. . Algorithm 4 maintains the state within the union of the two MAS and hence ensures feasibility. y Perhaps because it does not carry restrictions implicit to give guarantees.

297

Interpolation based predictive control 2

K11 K12 K 21 K

1.5

1

22

0.5

0

−0.5

−1

−1.5

−2 −4

−3

−2

−1

Figure 1.

0

1

2

3

4

MAS for Example 1.

3

2

1

0

−1

−2

−3 −5

−4

−3

Figure 2.

−2

−1

0

1

2

3

Feasible regions for OMPC with nc ¼ 0, 1, . . . , 5, 20.

4

5

298

J. A. Rossiter et al. 3

OMPC Algorithm 2.1 2

1

0

−1

−2

−3 −5

Figure 3.

−4

−3

−2

−1

0

1

2

3

4

5

Feasible regions for Algorithms 1 and 2 (shaded) and OMPC with nc ¼ 5, 20 (dotted lines).

2

Algorithm 3.1 Algorithm 4.1 1.5

1

0.5

0

−0.5

−1

−1.5

−2 −4

−3

Figure 4.

−2

−1

0

1

2

3

Feasible regions for Algorithms 3 (light shade) and 4 (dark shade).

4

299

Interpolation based predictive control 1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−3

−2

−1

0

1

2

−3

3

−2

Algorithms 2.1, 2.2 1

1

0.5

0.5

0

0

−0.5

−0.5

−1

−1

−3

−2

−1

0

1

2

−1

−3

3

−2

−1

Algorithm 3.1 Figure 5.

Algorithm 4.1

Algorithm 2.2

3

0

1

2

3

1

0.6 0.4 0.2

0

10

20

30

40

0.8 0.6 0.4 0.2 0

50

0

Sampling instants

10

20

30

40

50

Sampling instants 1

Algorithm 2.1

1

Algorithm 3.1

2

State trajectories for different initial conditions.

0.8

0.8 0.6 0.4 0.2 0

1

OMPC

1

0

0

Algorithm 4.1

0

10

20

30

40

Sampling instants Figure 6.

50

0.8 0.6 0.4 0.2 0

0

10

20

30

40

Sampling instants

Variation in  for different initial conditions.

50

300

J. A. Rossiter et al. OMPC (nc ¼ 20) 25.8 2.07 6.12 60.2 16.6 10.82 10.23 46.62 18.93 Table 2.

Algorithm 1

Algorithm 3

Algorithm 4

Algorithm 2

28.2 2.07 6.12 93.6 16.6 11.02 10.23 48.89 19.21

26.0 0 6.12 62.0 0 10.86 10.23 46.63 18.93

0 0 7.14 123.8 0 0 11.19 70.64 32.21

29.4 2.80 6.22 225.3 40.3 11.02 14.90 160.66 21.12

Closed-loop run-time costs corresponding to J (0 implies infeasibility).

It gives reasonable performance but the restriction, to guarantee feasibility, of (16) clearly affects performance. . Algorithm 2 has the worst performance. The price of extending the feasibility region without using a large computational load is that the performance criteria must be modified away from the most desirable one. It is interesting to note from figure 4 that the Algorithms 1, 3 and 4, which do not require  to be monotonic, have faster convergence of . Algorithm 4 has the slowest convergence, again probably due to the restrictions implicit from (16). Algorithm 2 guarantees that  changes monotonically and yet has poor convergence.

6. Conclusions This paper has shown the potential benefits of using interpolation to generate predictive control algorithms as opposed to the more usual technique (Scokaert et al. 1998) of allocating individual control values as the d.o.f. With interpolation one can achieve: (i) larger feasibility regions for the same number of d.o.f./computational loading, and (ii) performance that is surprisingly close to the global optimum with a far smaller online computation. The traditional weakness of interpolation algorithms is that it is less straightforward to formulate recursive guarantees of feasibility and stability, and hence convergence can also be hard to guarantee. This paper has derived algorithms such that: (i) by allowing the number of d.o.f. to be one greater than the state dimension, this problem can be removed and (ii) even with just one d.o.f. one can give guarantees (within a more restricted region). In applications with fast dynamics where speed of computation is a limiting factor, the algorithms of } } 3 and 4 provide what appears to be the only way forward. This is so because the online optimization can be

performed explicitly and the complexity of the algorithms grows only linearly with the order of the system. Earlier co-linear interpolation results lacked the guarantee of feasibility and stability, and the development here overcomes this difficulty. However, to date it has not been possible to establish a monotonic decrease property for the interpolation variable, and thus guarantees of convergence can only be given through reliance on the convergence properties of the detuned control law u ¼ K2 x. Though convenient for the purposes of analysis, monotonicity is only a sufficient condition for convergence and future research will seek to relax this, for instance by requiring a decrease in the mean over a number of prediction steps. It is further pointed out that the computational efficiency of Algorithm 4 together with its enlarged region of attraction makes it an ideal candidate for a terminal law within the more usual MPC paradigm (Mayne et al. 2000) and the further development of this forms the object of future research. References Bacic, M., Cannon, M., Lee, Y. I., and Kouvaritakis, B., 2003, General interpolation in MPC and its advantages. IEEE Transactions on Automatic Control, 48, 1092–1096. Blachini, F., 1999, Set invariance in control. Automatica, 35, 1747–1767. Clarke, D. W., Mohtadi, C., and Tuffs, P. S., 1987, Generalised predictive control, Parts 1 and 2. Automatica, 23, 137–160. Gilbert, E. G., and Tan, K. T., 1991, Linear systems with state and control constraints: the theory and application of maximal output admissable sets. IEEE Transactions on Automatic Control, 36, 1008–1020. Kothare, M. V., Balakrishnan, V., and Morari, M., 1996, Robust constrained model predictive control using linear matrix inequalities. Automatica, 32, 1361–1379. Kouvaritakis, B., Rossiter, J. A., and Cannon, M., 1998, Linear quadratic feasible predictive control. Automatica, 34, 1583–1592. Mayne, D. Q., Rawlings, J. B., Rao, C. V., and Scokaert, P. O. M., 2000, Constrained model predictive control: stability and optimality. Automatica, 36, 789–814. Mendez, J. A., Kouvaritakis, B., and Rossiter, J. A., 2000, State Space approach to interpolation in MPC.

Interpolation based predictive control International Journal of Robust Nonlinear Control, 10, 27–38. Rossiter, J. A., and Kouvaritakis, B., 1998, Reducing computational load for LQ optimal predictive controllers. Proceedings of the UKACC, pp. 606–611. Rossiter, J. A., Kouvaritakis, B., and Cannon, M., 2001, Computationally efficient algorithms for constraint handling with guaranteed stability and near optimality. International Journal of Control, 74, 1678–1689. Rossiter, J. A., Kouvaritakis, B., and Cannon, M., 2003, Stability proof for computationally efficient

301

predictive control in the uncertain case, Proceedings of the ACC. Rossiter, J. A., Rice, M. J., Schuurmanns, J., and Kouvaritakis, B., 1998, A computationally efficient constrained predictive control law. American Control Conference. Scokaert, P. O. M., and Rawlings, J. B., 1998, Constrained linear quadratic regulation. IEEE Transactions on Automatic Control, 43, 1163–1168. Tsang, T. T. C., and Clarke, D. W., 1988, Generalised predictive control with input constraints. IEE Proceedings Pt. D, 6, 451–460.