LINEAR TIME VARYING TERMINAL LAWS IN MPQP

LINEAR TIME VARYING TERMINAL LAWS IN MPQP JA Rossiter Dept of Aut Control & Systems Eng University of Sheffield, Mappin Street Sheffield, S1 3JD, UK email: JARossiter@sheffieldacuk B Kouvaritakis M Cannon Dept of Eng Science Parks Road Oxford OX1 3PJ email: basilkouvaritakis@engoxacuk Abstract: This paper shows how changing the structure of the terminal control law can give significant reductions in the complexity of the MPQP solution to predictive control, at a small cost to performance 1 Introduction MPQP (multi parametric quadratic programming) solutions [1] are becoming a popular solution to the quadratic programming (QP) problems arising within model predictive control (MPC) [16] This interest is due to two main reasons: (i) the MPQP solution gives some transparency to the control action during constraint handling and (ii) there is potential for transfering much of the online computation to offline computation and hence improving speed One could also argue (not discussed here) that it is a natural framework for handling uncertainty Unfortunately, MPQP solutions can be very complex For instance the MPQP solution to the MPC based QP problem arising for a simple 2 state system could require the definition of 5 or more regions [1] One must then consider whether storage and implementation of the MPQP solution is actually more efficient than implementing an online QP solver Such considerations suggest that parametric programming solutions will often not be easy to implement on large dimensional problems [1] This paper takes the viewpoint that irrespective of general limitations in applicability, it is a worthwhile goal to ask how much one could simplify an MPQP solution foragivenmpcproblem[3,6,5,13] Themoresimplification in complexity that can be achieved, the wider the potential number of applications Broadly speaking MPC calls for the computation of an optimal control trajectory which in at most n steps, takes the system state into a target region For a quadratic cost and linear constraints, the optimum is piece wise affine (PWA) on the state Hence the optimal solution is made up of a coolection of linear state feedback laws each being associated with a specific region The smaller the targer region and/or the larger the number of steps allowed, the more regions that need to be defined 1 1 Itcanbeshownthatforsomeproblems[7],forn large Several authors have looked at improving efficiency For instance [6] looked at defining regions as hypercubes as this allows an efficient search, [5] looked at using a combination of one step sets and an overbounding lyapunov function and [13] uses interpolation with only the control laws associated to the facets Other authors [3] accept that there can be no reduction in the number of regions and investigate more efficient ways of identifying the active region We take a different approach which has some parallels with [9] The main insight derives from the observation that the maximal admissible set (MAS) [7] may be small, not just due to over tuning but also because effective constraint handling may require a linear time varying (LTV) control law Of course, LTV control is precisely what predictive control allows for thereby making significant increases in the reachable space possible The question that seems little discussed in the literature is whether the same LTV control law could be used for the entire space MPQP suggests that the optimal LTV control law changes with the state, however this need not imply theredoesnoexistasub-optimalltvcontrollaw with far wider applicability This paper gives some background to MPC and MPQP in section 2 and then section 3 shows how one could find a single LTV control law which has a large feasible region Section 4 discusses and illustrates how this larger region can be conbined with MPQP to generate a significantly simpler MPQP solution, at the price of only small suboptimality 2 Background 21 Notation This paper assumes a model and constraints x k+1 = Ax k + Bu k ; y k = Cx k u u k u; x x k x (1) where x, u, 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 enough the solution is equivalent to n = and hence the number of regions has a computable upper limit

[7] for several different control laws and show how a predicted trajectory, and corresponding invariant set, can be made developed througn a convex linear combination of these control laws The MAS for the system of (1) under the control law u = K i x is the largest set such that the use of control law u = K i x gives convergence and feasibility For appropriately defined M,d the MAS can be described as: S = {x : Mx d } (2) 22 Predictive control This paper is based on the algorithm of [14] implemented via the closed-loop paradigm [11], that is min C k n c i=1 c T k+iλc k+i st NC k + Mx k d (3) where C k = [c T k,, ct k+n c 1 ], u k = Kx k + c k and the definitions of Λ, N, M depend upon constraints and tuning weights in the predicted cost Under the assumption that u = Kx, is the unconstrained optimum it follows that for Mx d, the optimum solution is C = The maximum control admissible set (MCAS), or feasible region, is given by {x C, NC + Mx d } 23 MPQP for MPC Optimisation (3) is a quadratic program and is known [1] to have a piecewise affine solution That is, one can construct regions S i = {x : M i x d i } (4) such that the solution of (3) is given as: x S i C = L i x + p i (5) (Recall that u k = K(x k + c k )) Hence, instead of solving (3) using an online quadratic programming optimiser, one can find within which set S i the state lies and implement the corresponding control law Computing sets S i and the corresponding control parameters L i, p i can be a significant burden, but as this is an offline task it is of small consequence More importantly, the number of regions required to span the controllable space may be prohibitively large for some systems This implies that the data storage requirement and also the implied set membership tests are no longer efficient and this negates some of the benefit of an MPQP approach 24 Dual predictive mode interpretation of MPC The MPC law of [14] can be thought of as dual predictive mode (DPM) in that the predictions are constructed in two phases: (i) a LTV transient phase of n c steps in which u = Kx + c and (ii) a linear asymptotic phase thereafter in which u = Kx The control strategy could be reinterpreted as minimising the performance wrt C, over an infinite horizon, subject to the n c step ahead prediction being within the MAS of (2) The volume and shape of the stabilisable set is determined by two main tuning parameters: (i) the shape of the MAS (or terminal region) and (ii) the number of dof, that is, n c To increase the size of the stabilisable set one must either increase the volume of the MAS or increase n c However, as the shape of the MAS is dependent on the implied terminal control law, one can only change this shape by changing the control law, in essence that is by accepting some suboptimality in the predicted asymptotic behaviour This may be an acceptable compromise although there are no obvious systematic mechanisms for choosing a state feedback which balances the volume of the associated MAS with performance The option of increasing n c may also be undesirable as this can lead to an increase in the complexity of the QP of (3) and the number of regions in the MPQP solution 2 3 Fixed LTV terminal law This paper assumes that a DPM type of strategy is a well conditioned basis for setting up the MPQP problem, but asks the question whether a nonlinear asymptotic feedback gain could be used to give a systematic compromise between performance and the volume of the feasible region This section shows how such a feedback gain can be computed and integrated into an MPQP algorithm 31 LTV feedback gain It is assumed that the predicted control law is given as follows: u k Kx k u k+1 Kx k+1 u = k+n 1 Kx +C k ; C k = k u k+n Kx k+1 L 1 L 2 L n (6) It is noted that this feedback differs in structure from an optimal constrained feedback of (5) only due to lack of the constant term p The task in hand is to compute c k and hence L i,i=1, 2,, n such that the predicted cost (3) is minimised and the associated feasible set is maximised in volume 2 It is recognised that MPQP can be used to find infinite horizon solutions for some cases and hence there would be an upper limit on complexity However, for many practical cases this upper limit is prohibitively high x k

32 Evaluating the cost The cost function can be written as n J = x T i Qx i + u T i Ru i + x T n P x n (7) k=1 From (6) it is clear that the predictions x i are linear in the coefficients of L i Hence, it is easy to show that is quadratic in L i ; the proof is straightforward and will be omitted For SISO case) we can write J = L T Λ L L + c; L =[L T 1,L T 2, ] (8) where c does not depend upon L Clearly the unconstrained minimisation of J wrt L gives L = 33 Constraint inequalities Substitution of prediction (6) into constraints gives the inequalities {N L 1 L 2 L n + M}x d (9) For a given value of x, this can be rearranged to: F (x)l d(x) (1) Hence the inequalities have an explicit dependence upon the non-linear feedback terms L i Let v i,i = 1, 2, be points on the boundary of the MAS given in (2) It is known that for the choice L =, (1) must hold true for x = v i The target region is larger if one instead can satisfy F (v i )L d(v i ) e i (11) where e i > An obvious objective therefore is to max L, V = i e2 i as this, in some sense, maximises the volume of the feasible region implied by (9) 34 Choosing the best LTV feedback The region of attraction and the predicted performance both depend upon the parameters of the proposed LTV feedback in a simple way Hence one can choose L by setting up a weighted average of predicted performance and set enlargement For instance using unit weights: J L = L T Λ L (x)l + i e 2 i (12) The implied constraint that e i could be included in addition to inequalities (11) and hence the overall optimisation is a quadratic program In summary, minimisation of J L subject to (11) and e i gives some compromise between the volume of the terminal region and the implied predicted performance To the authors knowledge, this compromise is more difficult to achieve (and hence less systematic) if one minimises wrt to linear feedback gains only [15] It should be noted that LMI methods tackling this, eg [8], are restricted to ellipsoidal sets The notable advantage of choosing the structure implicit in (6) is simplicity The target region is simply defined by (1) and one need only store a single LTV feedback that is applicable inside the region Moreover, as there is a single feedback law, even though LTV, the predicted cost (8) is known exactly - no overbounding is needed It is noted that the implied cost can be rewritten as: J L = x T Λ x (L)x (13) 4 Forming an MPQP solution with LTV feedback 41 The cost function Because the nonlinear feedback is suboptimal, it is necessary to redefine the cost from (3) so that it is applicable to a control law with three phases: Phase 1: Use u = Kx + c for n c steps Phase 2: Use nonlinear feedback (6) for steps n c +1ton c + n Phase 3: Use optimal feedback u = Kx for steps n c + n +1on We note that, using (13) for phases 2 and 3, the implied cost can be written as n c J = x T i Qx i + u T i Ru i + x T n c Λ x (L)x nc (14) k=1 Straightforward manipulation then gives J =[C C ] T W [C C ]+f; C = T x (15) and f is an unimportant scalar Notably, the optimum unconstrained C is no longer zero; a consequence of the fact that the LTV feedback is suboptimal 42 Constraints and the QP Taking the basic structure of the constraint equations to be NC + Mx d, if one increases n c by n, one could write [ ] C [N 1,N 2 ] + Mx d (16) C2 where C 2 was given by the C in (6) and C are the dof Hence substituting in for C 2 using the known values of L i and a prediction for x nc gives N t C + M t x d (17)

Hence the optimisation can be represented as: min J =[C C ] T W [C C ] st N t C+M t x d C (18) where C = T x and the control law to be implemented is given as u = Kx + c IfM t x d, then the LTV feedback (6) is feasible Optimisation takes the form of a conventional quadratic program and as such MPQP can be used to give a solution of the form given in (4,5) We will not repeat the steps here as they are well documented in the literature 43 Unusual cases Because the LTV feedback was formed to give a compromise between optimality and feasibility, for a fixed control strategy, the solution it gives will not, in general, be the same as the solution of (18) even on the boundaries of the region {x : M t x d} This is of little consequence except in the case where: MPQP algorithms with only minimal impact on performance First we will illustrate comparisons in the volume of terminal region, secondly the total number of regions required to cover a given region in space and third we will give some performance comparisons Due to space constraints, a single 2 state example, with input and state limits, is used [ 8 3 5 9 ] [ 25 x k + 5 ] ; y k =[1 2]x k x k+1 = (21) 1 u k 1; 5 x 1,2 5 (22) The optimal control law K =[ 494 122] is computed with weights Q = C T C, R =1 Forthis example we selected: n c = n c1 = n c2 = 6 (23) The MCAS is computed with n c = 6 dof, the nonlinear feedback with n c1 = 6 dof and the NLMCAS ({x C, N t C + M t x d }) withn c2 =6dof M t x d & N t C + M t x d (19) That is, the unconstrained optimal is feasible outside of the set S t The feasible space (NLMAS) of the unconstrained optimal to (18) is given as S f = {x :[N t T + M t ]x d} (2) 2 15 1 5 NLMCAS MCAS NLMAS MAS Lemma 41 A feasible predicted control trajectory can be computed by inspection if either x S t or x S f However, in general this feasible region will not be convex x 2 5 1 15 Proof: Feasibility is obvious as for x S t one can use feedback (6) and for x S f one can use C = T x The union of two convex sets is not convex in general unless one is a subset of the other 2 2 15 1 5 5 1 15 2 x 1 Figure 1 Feasible regions and target sets 44 MPQP with a non-convex terminal region The MPQP algorithms in the literature assume a convex terminal region and as such some modification is required The usual procedure inverts each facet of the terminal region (and subsequently other regions) in turn to search for new regions The first check is whether the proposed new point is a member of a known region In the case of non-convex terminal regions one would invert, in turn, both the facets of S f and S t The check for membership of known regions will eliminate any double computation that could arise due to non-convexity 5 Examples This section will illustrate the potential of the proposed non-linear feedback in reducing complexity of 51 Terminal regions S and S t The terminal regions MAS/NLMAS and corresponding feasible regions MCAS/NLMCAS are plotted in figure 1 Figure 2 overlays the NLMAS, the MCAS and the MPQP regions for solving (18); this is to illustrate the number of regions required by the proposed algorithm to obtain the same (or larger) feasible region as the original MPQP algorithm The inner region is S and the outer region is S t 52 Complexity of MPQP For the QP algorithm of (18), we allow the same number of dof as given to the algorithm (5) as this seems to be as fair as can be allowed Then, we find the feasible region for (18) and find out how many regions of the new algorithm are needed to cover the same space Illustrations are given in figure 2 and also summarised in table 1

Example 1 Regions for (5) 117 Regions for (18) 29 Table 1 No of MPQP regions x 2 2 15 1 5 5 1 15 2 2 15 1 5 5 1 15 2 x 1 Figure 2 Feasible regions 53 Closed-loop performance It is necessary to compare the closed-loop performance of the algorithms to demonstrate that despite the significant reduction in complexity, there has been only a small loss in optimality For this, points were taken on the boundary of S (one per facet) and closed-loop simulations were performed for each point; marked with circles in figure 1 The runtime cost was computed for each simulation and summed and then divided by the runtime cost for the algorithm of (3) with a large n c (the global optimum) The comparisons are given in table 2 Example 1 Sum of run time costs (5) 15 Sum of run time costs (18) 12 Table 2 Normalised runtime costs 54 Summary and conclusions It is clear that the NLMAS may be far larger than the MAS Hence it is also unsurprising that the NLM- CAS is larger than the MCAS, for the same n c More significantly, the MPQP solution to (18) needs fewer regions to cover the MCAS, that is, the complexity is much reduced The closed-loop simulations demonstrate that rather than causing a deterioration in performance, on average the performance is also better! For this example, deploying a fixed LTV feedback simplifies MPQP and gives better performance References [1] Bemporad, A, M Morari, V Dua, and EN Pistikopoulos The explicit linear quadratic regulator for constrained systems Automatica, 38(1):3 2, January 22 [2] Blachini, F, 1999, Set invariance in control, Automatica, 35, 1747-1767 [3] Borrelli, F, M Baotic, A Bemporad, and M Morari Efficient on-line computation of constrained optimal control In Proc 4th IEEE Conf on Decision and Control, December 21 [4] Clarke, DW, Mohtadi, C and Tuffs, PS, 1987, Generalised predictive control, Parts 1 and 2, Automatica, 23, 137-16 [5] Grieder, P, F Borrelli, FD Torrisi, and M Morari Computation of the constrained infinite time linear quadratic regulator ACC 23 [6] Johansen, TA and A Grancharova, Approximate explicit constrained linear model predictive control via orthogonal search tree, IEEE Trans AC, 48, 5, 81-815, 23 [7] Gilbert, EG and K T Tan, 1991, Linear systems with state and control constraints: the theory and application of maximal output admissable sets, IEEE Trans AC, 36, 9, 18-12 [8] Kothare, MV, Balakrishnan, V and Morari, M, 1996, Robust constrained model predictive control using linear matrix inequalities, Automatica, 32, 1361-1379 [9] Kouvaritakis, B, JA Rossiter and J Schuurmans, 2, Efficient robust predictive control, IEEE Trans AC, 45, 8, pp1545-1549 [1] Mayne, DQ, Rawlings, JB, Rao, CV and Scokaert, POM, 2, Constrained model predictive control: stability and optimality, Automatica, 36, 789-814 [11] Rossiter,JA, MJ Rice, and B Kouvaritakis A numerically robust state-space approach to stable predictive control strategies Automatica, 38(1):65 73, 1991 [12] Rossiter, JA, Kouvaritakis, B and Cannon, M, 21, Computationally efficient algorithms for constraint handling with guaranteed stability and near optimality, IJC, 74, 17, 1678-1689 [13] Rossiter, JA and Grieder, P, Using Interpolation to Simplify Explicit Model Predictive Control, to appear ACC4 [14] Scokaert, POM and Rawlings, JB, 1998, Constrained linear quadratic regulation, IEEE Trans AC, 43, 8, pp1163-1168 [15] Tan, KT and EG Gilbert, 1992, Multimode controllers for linear discrete time systems with general state and control constraints, Optimisation techniques and applications (World Scientific Pub Co, Singapore), pp433-442 [16] Tsang, TTC and Clarke, DW, 1988, Generalised predictive control with input constraints, IEE Proceedings Pt D, 6, 451-46