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An Algorithm for Multi-Parametric Quadratic Programming and Explicit MPC Solutions P TuP11-4 An Algorithm for Multi-Parametric Quadratic Programming and Explicit MPC Solutions P. Tøndel1, T.A. Johansen1, A. Bemporad2 Abstract PWL solution to linear quadratic MPC problems is addressed in [11, 12], and efficient on-line computation schemes of ex- Explicit solutions to constrained linear MPC problems can plicit MPC controllers are proposed in [13]. This paper ex- be obtained by solving multi-parametric quadratic programs tends the theoretical results of [1], by analyzing several prop- (mp-QP) where the parameters are the components of the erties of the geometry of the polyhedral partition and its rela- state vector. We study the properties of the polyhedral parti- tion to the combination of active constraints at the optimum tion of the state-space induced by the multiparametric piece- of the quadratic program. Based on these results, we derive a wise linear solution and propose a new mp-QP solver. Com- new exploration strategy for subdividing the parameter space, pared to existing algorithms, our approach adopts a differ- which avoids (i) unnecessary partitioning, (ii) the solution to ent exploration strategy for subdividing the parameter space, LP problems for determining an interior point in each new avoiding unnecessary partitioning and QP problem solving, region of the parameter space, and (iii) the solution to the QP with a significant improvement of efficiency. problem for such an interior point. As a consequence, there is a significant improvement of efficiency with respect to the 1 Introduction algorithm of [1]. Our motivation for investigating multi-parametric quadratic programming (mp-QP) comes from linear Model Predictive 2 From Linear MPC to an mp-QP Problem Control (MPC). This refers to a class of control algorithms The main aspects of formulating a linear MPC problem as a that compute a manipulated variable trajectory from a lin- multi-parametric QP will, for convenience, be repeated here. ear process model to minimize a quadratic performance in- See [1] for further details. Consider the linear system dex subject to linear constraints on a prediction horizon. The first control input is then applied to the process. At the next x(t +1)=Ax(t)+Bu(t) (1) sample, measurements are used to update the optimization y(t)=Cx(t) problem, and the optimization is repeated. In this way, this becomes a closed-loop approach. There has been some limi- where x(t) ∈ Rn is the state variable, u(t) ∈ Rm is the input tation to which processes MPC could be used on, due to the variable, A ∈ Rn×n, B ∈ Rn×m and (A; B) is a control- computationally expensive on-line optimization which was lable pair. For the current x(t), MPC solves the optimization required. There has recently been derived explicit solutions problem to the constrained MPC problem, which could increase the area of use for this kind of controllers. Independent works T minU {J(U; x(t)) = x Pxt+N|t by [1], [2], [3] and [4] has reported how a piecewise linear P t+N|t + N−1 xT Qx + uT Ru } (PWL) solution can be computed off-line, while the on-line k=0 t+k|t t+k|t t+k t+k effort is limited to evaluate this PWL function. In particu- s:t:ymin ≤ yt+k|t ≤ ymax;k=1; :::; N u ≤ u ≤ u ;k =0; :::; M − 1 lar, in [1] and [2] such a PWL function is obtained by treat- min t+k max (2) ing the MPC optimization problem as a parametric program. ut+k = Kxt+k|t;M≤ k ≤ N − 1 Parametric programming is a term for solving an optimiza- xt|t = x(t) tion problem for a range of parameter values. One can dis- xt+k+1|t = Axt+k|t + But+k;k≥ 0 tinguish between parametric programs, in which only one yt+k|t = Cxt+k|t;k≥ 0 parameter is considered, and multi-parametric programs, in 0 which a vector of parameters is considered. The algorithm with respect to U , {ut; :::; ut+M−1},whereR = R 0, reported in [1] is the only mp-QP algorithm known to the au- Q = Q0 0, P = P 0 0. When the final cost matrix P and thors for solving general linear MPC problems, while single gain K are calculated from the algebraic Riccati equation, parameter parametric QP is treated in [5]. Multi-parametric under the assumption that the constraints are not active for LP (mp-LP) is treated in [6] and [7], mp-LP in connection k ≥ N, (2) exactly solve the constrained (infinite-horizon) with MPC based on linear programming is investigated in LQR problem for (1) with weights Q, R (see also [14], [15] [8], and multi-parametric mixed-integer linear programming and [16]). This and related problems can by some algebraic [9] is used in [10] for obtaining explicit solutions to hy- manipulation be reformulated as brid MPC. The problem of reducing the complexity of the 1 T 1 Vz(x(t)) = min z Hz (3) Department of Engineering Cybernetics, Norwegian Uni- z 2 versity of Science and Technology, 7491 Trondheim, Norway, [email protected], Tor.Arne.Johansen@itk. s:t: Gz ≤ W + Sx(t) (4) ntnu.no. 2 Dipartimento di Ingegneria dell’Informazione, University of Siena, z , U + H−1F T x(t) U = uT ; :::; uT T 53100 Siena, Italy, [email protected]. ETH Zentrum, 8092 where , t t+M−1 ,and Zurich, Switzerland x(t) is the current state, which can be treated as a vector of 1199 parameters. Note that H 0 since R 0. The number of inequalities is denoted by q and the number of free variables X n n ×n q×n is nz = m · N: Then z ∈ R z , H ∈ R z z , G ∈ R z , q×1 q×n n×q R0 R0 W ∈ R , S ∈ R , F ∈ R . The problem we con- R1 sider here is to find the solution of the optimization problem (3)–(4) in an explicit form z∗ = z∗ (x(t)). Bemporad et. al. [1] showed that the solution z∗(x(t)) (and U ∗(x(t))) is a con- tinuous PWL function1 defined over a polyhedral partition of a) b) the parameter space, and Vz(x(t)) is a convex (and therefore continuous) piecewise quadratic function. R2 R0 R0 R3 3 Background on mp-QP R1 R1 As shown in [1], the mp-QP problem (3)-(4) can be solved R5 R4 by applying the Karush-Kuhn-Tucker (KKT) conditions CRi T q Hz + G λ =0,λ∈ R ; (5) c) d) i i i λi G z − W − S x =0;i=1; :::; q; (6) Figure 1: State-space exploration strategy. λ ≥ 0; (7) Gz − W − Sx ≤ 0: (8) We have now characterized the solution to (3)-(4) for a given For ease of notation we write x instead of x(t). Superscript optimal active set A∗ ⊆{1;:::;q},andafixedx.How- ∗ i on some matrix denotes the ith row. Since H has full rank, ever, as long as A remains the optimal active set in a neigh- (5) gives borhood of x, the solution (11) remains optimal, when z is ∗ z = −H−1GT λ (9) viewed as a function of x. Such a neighborhood where A is optimal is determined by imposing that z must remain feasi- Definition 1 Let z∗(x) be the optimal solution to (3)-(4) for ble (8) x a given . We define active constraints the constraints with GH−1G~T (GH~ −1G~T )−1(W~ + Sx~ ) ≤ W + Sx:; Giz∗(x) − W i − Six =0, and inactive constraints the con- (12) straints with Giz∗(x) − W i − Six<0.Theoptimal active ∗ and that the Lagrange multipliers λ must remain non- set A (x) is the set of indices of active constraints at the negative (7) optimum A∗(x)= i | Giz∗(x)=W i + Six .Wealsode- fine as weakly active constraint an active constraint with an −(GH~ −1G~T )−1(W~ + Sx~ ) ≥ 0:: (13) associated zero Lagrange multiplier λi, and as strongly ac- tive constraint an active constraint with a positive Lagrange Equations (12) and (13) describe a polyhedron in the state λi multiplier . space. This region is denoted as the critical region CR0 corresponding to the given set A∗ of active constraints, is Let λ˘ be the Lagrange multipliers of the inactive constraints, a convex polyhedral set, and represents the largest set of pa- ∗ λ˘ =0,andλ~ the Lagrange multipliers of the active con- rameters x such that the combination A of active constraints straints, λ~ ≥ 0. Assume for the moment that we know which at the minimizer is optimal [1]. constraints are active at the optimum for a given x. We can The recursive algorithm of [1] can be briefly summarized as i x now form matrices G;~ W~ and S~ which contains the rows G , follows: Choose a parameter 0. Solve a QP to find the op- i i A x W and S corresponding to the active constraints. timal active set 0 for 0, and then use (10)-(13) to charac- terize the solution and critical region corresponding to A0. Definition 2 For an active set, we say that the linear inde- Then divide the parameter space as in Figure 1 by reversing pendence constraint qualification (LICQ) holds if the set of one by one the hyperplanes defining the critical region. Iter- atively subdivide each new region Ri in a similar way. The active constraint gradients are linearly independent, i.e., G~ main drawback of this algorithm is that the regions Ri are has full row rank. not related to optimality, as they can split some of the critical CR CR ~ regions like 1 in Figure 1d.
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