Relaxations Applicable to Mixed Integer Predictive Control Comparisons and Efficient Computations

Transcription

1 echnical report from Atomatic Control at Linköpings niversitet Relaxations Applicable to Mixed Integer Predictive Control Comparisons and Efficient Comptations Daniel Axehill, Anders Hansson, Lieven Vandenberghe Division of Atomatic Control 3st Janary 28 Report no.: LiH-ISY-R-2839 Accepted for pblication in Proceedings the 46th IEEE Conference on Decision and Control Address: Department of Electrical Engineering Linköpings niversitet SE Linköping, Sweden WWW: AUOMAIC CONROL REGLEREKNIK LINKÖPINGS UNIVERSIE echnical reports from the Atomatic Control grop in Linköping are available from

2 Abstract In this work, different relaxations applicable to an MPC problem with a mix of real valed and binary valed control signals are compared. In the problem description considered, there are linear ineqality constraints on states and control signals. he relaxations are related theoretically and both the tightness of the bonds and the comptational complexities are compared in nmerical experiments. he relaxations considered are the Qadratic Programming (QP) relaxation, the standard Semidefinite Programming (SDP) relaxation and an eqality constrained SDP relaxation. he reslt is that the standard SDP relaxation is the one that sally gives the best bond and is most comptationally demanding, while the QP relaxation is the one that gives the worst bond and is least comptationally demanding. he eqality constrained relaxation presented in this paper often gives a better bond than the QP relaxation and is less comptationally demanding compared to the standard SDP relaxation. Frthermore, it is also shown how the eqality constrained SDP relaxation can be efficiently compted by solving the Newton system in an Interior Point algorithm sing a Riccati recrsion. his makes it possible to compte the eqality constrained relaxation with approximately linear comptational complexity in the prediction horizon. Keywords: Predictive control, Hybrid systems, Binary control, Integer programming, Semidefinite programming

3 Relaxations Applicable to Mixed Integer Predictive Control Comparisons and Efficient Comptations Daniel Axehill, Anders Hansson and Lieven Vandenberghe Abstract In this work, different relaxations applicable to an MPC problem with a mix of real valed and binary valed control signals are compared. In the problem description considered, there are linear ineqality constraints on states and control signals. he relaxations are related theoretically and both the tightness of the bonds and the comptational complexities are compared in nmerical experiments. he relaxations considered are the Qadratic Programming (QP) relaxation, the standard Semidefinite Programming (SDP) relaxation and an eqality constrained SDP relaxation. he reslt is that the standard SDP relaxation is the one that sally gives the best bond and is most comptationally demanding, while the QP relaxation is the one that gives the worst bond and is least comptationally demanding. he eqality constrained relaxation presented in this paper often gives a better bond than the QP relaxation and is less comptationally demanding compared to the standard SDP relaxation. Frthermore, it is also shown how the eqality constrained SDP relaxation can be efficiently compted by solving the Newton system in an Interior Point algorithm sing a Riccati recrsion. his makes it possible to compte the eqality constrained relaxation with approximately linear comptational complexity in the prediction horizon. I. INRODUCION In recent years the field of application of the poplar control strategy Model Predictive Control (MPC) has been broadened in several steps. From the beginning, MPC was only applicable to linearly constrained linear systems. oday, it is possible to se MPC for control of nonlinear systems and hybrid systems. In this work, the focs is on control of hybrid systems. In the basic linear setp, the MPC problem can be cast in the form of a Qadratic Programming (QP) problem. When a hybrid system is to be controlled, binary variables are introdced and the optimization problem is changed from a QP to a Mixed Integer Qadratic Programming (MIQP) problem, which is in general known to be NPhard, ]. MPC for hybrid systems is sometimes called Mixed Integer Predictive Control (MIPC). oday, there exist tailored optimization rotines with low comptational complexity for linear MPC. However, there is still a need for efficient optimization rotines for MIPC. A poplar method for solving MIQP problems is branch and bond, where the original integer optimization problem is solved as a seqence of smaller QP sbproblems. he sbproblems are ordered in a tree strctre, where one new integer variable is fixed at each level. Depending on the problem, sometimes a large nmber of QP sbproblems have D. Axehill and A. Hansson are with the Division of Atomatic Control, Linköpings niversitet, SE Linköping, Sweden, {daniel,hansson}@isy.li.se. L. Vandenberghe is with the Department of Electrical Engineering, University of California Los Angeles, Los Angeles, California , USA, vandenbe@ee.cla.ed. to be solved and the worst case complexity is known to be exponential. he efficiency of the branch and bond method highly relies on the possibility to efficiently compte good bonds on the optimal objective fnction vale. For MIQPproblems, sally QP relaxations (which are often called linear relaxations), where integer constraints are relaxed to interval constraints, are solved in the nodes to prodce these bonds. However, recent research has shown that it is possible to se Semidefinite Programming (SDP) in order to compte tighter bonds for the problem. Unfortnately, solving the SDP relaxation is generally mch more time consming than solving the corresponding QP relaxation. herefore, it is of greatest interest to investigate if it is possible to decrease the comptational complexity. In this work this is done by sing problem strctre. he SDP relaxations have previosly been considered in several contexts and they have sccessflly been applied to, e.g., the Max Ct problem 2] and the Mltiser Detection problem 3], 4]. In a previos work by the athors, i.e., in 5], the different relaxations have been compared for the special case withot ineqality constraints. In this work, the reslts are extended to the general case with ineqality constraints on states and control signals. Comptational reslts are shown for the general case with mixed real valed and binary valed control signals. Frthermore, it is shown how the proposed eqality constrained SDP relaxation can be efficiently solved by an Interior Point (IP) method where the Newton system is solved sing a Riccati recrsion. he comptational performance of the different relaxations are compared and the expected performance gain from the se of Riccati recrsions in the SDP solver are illstrated in nmerical experiments. In this paper, S n ++ (S n +) denotes symmetric positive (semi) definite matrices with n colmns, and R n ++ denotes positive real n-vectors. Sperscript is sed to denote vales of variables and fnctions at optimm. he set = {,...,N } is also freqently sed. II. INRODUCION O HE CONROL PROBLEM In this paper, an MIPC problem with a qadratic objective fnction, or performance measre, in the form J MPC = 2 N t= ( e(t) 2 Q e + (t) 2 Q ) + 2 e(n) 2 Q e () is considered, where v 2 Q = v Qv, Q e S p ++ and Q S m ++. he dynamical system to be controlled is in the form x() = x x(t +)=Ax(t)+B(t) (2) e(t) =Cx(t) r(t)

4 where t Z is the discrete time, x(t) R n is the state, x the initial state, (t) = c (t) b (t) ] is the control inpt and e(t) R p is the control error with reference signal r(t) R p. Frthermore, c (t) R mc, b (t) {, } m b and m = m c + m b. Note that the choice of partitioning of the control signal into real valed and binary valed components can be made withot any loss of generality. he system is also in each time instant sbject to c linear ineqality constraints in the form H x x(t)+h (t)+h, t (3) H x x(n)+h where H x R c n, H R c m and h R c. Remark : Even thogh a time-invariant description is chosen in the presentation of this work, all reslts also hold for the case with a time-varying problem description A(t), B(t), C(t), H x (t), H (t), h(t) and c(t). It is known from, e.g., 5], that this MIPC problem can be written in at least two different, bt eqivalent, forms. First, the eqality constraints representing the dynamics of the system, can be kept and the reslt is an MIQP in the form minimize J MIQP (, e) x,,e A B x s.t. C I] e ] ] b = r Hx H ] x e ] + h i {, }, i B (4) where J MIQP (, e) = 2 e Q e e + 2 Q and the set B contains the indices to the binary components in. Q e and Q are block diagonal matrices with Q e and Q, respectively, along the diagonal. A detailed explanation of the notation can be fond in 6, pp. 3 5]. Second, the eqality constraints in (2) can be sed to eliminate the states and control errors and the reslting optimization problem can be expressed as an MIQP problem eqivalent to the problem in (4) in the form minimize J MIQP2 () s.t. (H x S + H ) + h + H x S x x (5) i {, }, i B where J MIQP2 () = 2 ( E Q e E + Q ) + e Q e E + 2 e Q e e, E = CA B and e = CA b r, and the notation is similar to the one sed in 6]. he main idea sed in this paper is the fact that E Q e E + Q is dense while Q e and Q are block diagonal. he optimal objective fnction vales of the problems in (4) and in (5) coincide and the optimal objective fnction vale is defined as J J MIQP = J MIQP 2 = J MPC. III. RELAXAIONS An optimization problem is said to be a relaxation of another optimization problem if the feasible set is larger than the feasible set of the original problem and the objective fnctions are eqivalent in the two problems. In this work, the integer constraints are relaxed in different ways and the reslt is new convex optimization problems that are mch easier to solve. he QP relaxations of the problems in (4) and (5), can easily be derived by replacing the binary constraints with interval constraints. his means that the QP relaxation of the problem in (4) is minimize J QP (, e) x,,e ] A B x s.t. C I e ] b = r] Hx H ] x e ] (6) + h i, i B where J QP (, e) =J MIQP (, e) and the QP relaxation of the problem in (5) is minimize J QP2 () s.t. (H x S + H ) + h + H x S x x i, i B (7) where J QP2 () =J MIQP2 (). In recent years, the moment relaxation 7], 8] of problems with binary variables has been extensively stdied. he moment relaxation of the problem formlation in (4) is minimize J SDP (U, x,, e) U,x,,e ] A B x s.t. C I e ] b = r] Hx H ] x e ] + h (8) U ii = i,i B ] U where J SDP (U, x,, e) = 2 e Q e e + 2 tr (Q U), U S Nm +, x R (N+)n R Nm, e R (N+)p and where U ii denotes diagonal element i of the matrix U. he relaxation in (8) is in this work referred to as the eqality constrained SDP relaxation. Similarly, the moment relaxation of the problem in (5) can be fond to be minimize J SDP2 (U, ) U, s.t. (H x S + H ) + h + H x S x x U ii = i,i B (9) ] U where J SDP2 (U, ) = 2 tr (( ) ) E Q e E + Q U +e Q e E+ 2 e Q e e, R Nm and U S Nm +. he relation between the optimization problems in (4) (9) has previosly been thoroghly discssed in 5]. he reslt is JQP = JQP 2 JSDP JSDP 2 J () which holds similarly also in the case with ineqality constraints. One important conclsion is that the lower bonds on the optimal objective fnction vale from the SDP relaxation does in general depend on if the dynamics is expressed

5 as eqality constraints or if it is eliminated and inclded in the objective fnction. IV. REDUCING COMPUAIONAL COMPLEXIY A. Utilizing block strctre he main advantage with the problem in (8) compared to the one in (9) is that the objective fnction Hessian for the problem in (8) is block diagonal. Specifically, Q is block diagonal. his property can be sed as described in 5] to rewrite the problem in the eqivalent form minimize U(t),x(t),(t),e(t) s.t. J SDP (U(t),x(t),(t),e(t)) U ii (t) = i (t), i {m c +,...,m} ] U(t) (t) (t) x() = x x(t +)=Ax(t)+B(t) e(t) =Cx(t) +D(t) r(t) e(n) =Cx(N) r(n) H x x(t)+h (t)+h H x x(n)+h where t nless stated differently and J SDP (U(t),x(t),(t),e(t)) = 2 N t= ( e(t) Q e e(t)+tr(q U(t)) ) + 2 e(n) Q e (N)e(N) () For completeness, a direct term from (t) to e(t), with coefficient matrix D, has been introdced in the dynamics of the system in (). With the problem in the form in (), the dynamics is clearly visible and the large matrix variable U has been replaced by several smaller matrix variables U(t) S m +. he nmber of variables (matrix elements) now grows linearly in the prediction horizon N. Remark 2: If Q is diagonal, then the problem in () can be formlated as a QP and this makes it tractable to replace the ordinary QP relaxation sed in branch and bond with the SDP relaxation in () (compted as a QP). his is frther described in 5] for the case withot ineqality constraints. B. Efficient comptation of search directions In this section, it will be shown how the search directions sed in an IP solver applied to the problem in () can be compted efficiently sing Riccati recrsions. For notational simplicity, the derivation is performed for the special case with only binary valed control signals, i.e., m = m b. Using the log-determinant barrier fnction, the associated centering problem to the SDP problem in () can be written as minimize U(t),e(t),x(t),(t) J c (U(t),e(t),x(t),(t),s) s.t. U ii (t) = i (t), i {,...,m} x() = x x(t +)=Ax(t)+B(t) e(t) =Cx(t)+D(t) r(t) e(n) =Cx(N) r(n) w(t) = (2) where t nless stated differently and J c (U(t),e(t),x(t),(t),s)=sJ SDP (U(t),x(t),(t),e(t)) + N t= ( log det ( Ũ(t) ) log ( h i H x,i x(t) H,i (t) )) log ( h i H x,i x(n) ) and where s is the barrier ] parameter, U(t) S m +, (t) R m and Ũ(t) = U(t) (t) (t) w(t). After introdcing the Lagrangian mltipliers y(t), p(t), λ(t) and ν(t), and after forming the Lagrangian, the KK conditions are fond to be e(t) = sq ee(t) p(t) =,t=,...,n x(t) = y(t)+a y(t +)+C p(t) V x,i (x(t),(t)) = x(n) = y(n)+c p(n) V x,i (x(n),(n)) = (3a) (3b) (3c) U(t) = s 2 Q (Ũ(t) ) +diag(λ(t)) = (3d) (Ũ(t) ) (t) = 2 + B y(t +)+D p(t) 2 λ(t) V,i (x(t),(t)) = (3e) (Ũ(t) ) w(t) = + ν(t) = 22 (3f) x() = x (3g) x(t +)=Ax(t)+B(t) (3h) e(t) =Cx(t) +D(t) r(t) (3i) e(n) =Cx(N) r(n) (3j) U ii (t) = i (t), i {,...,m} (3k) w(t) = (3l)

6 where t nless stated differently and the fnctions V x,i and V,i are defined in (3). ( ) ij denotes sb block (i, j). In the KK system in (3), the eqations in (3b) (3f) are nonlinear. he Newton system is fond by linearizing the KK system with respect to U(t), x(t) and (t) arond U (t), x (t) and (t) respectively. After linearization according to (28), (3d) can approximately be written as s 2 Q Z (t)+z (t)δu(t)z (t)+z (t)δ(t) Z (t) + Z (t)δ(t)z (t) +diag(λ(t)) =, t (4) where Z (t) S m ++ and z (t) R m are defined in (29). By mltiplying the eqations in (4) from left and right by Z (t), ΔU(t) can be fond as ΔU(t) =Z (t) s 2 Z (t) Q Z (t) Z (t) z (t)δ(t) Δ(t)z (t) Z (t) Z (t) diag(λ(t))z (t),t (5) By considering the diagonal of (5), and sing the eqations in (3k), (3) and (32), the following reslt is obtained Δ(t) = 2diag ( Z (t) z (t) ) Δ(t) Z (t)λ(t) +diag ( Z (t) s 2 Z (t) Q Z (t) ) (6) By sing (6) and that Z (t) (see Appendix), λ(t) can be fond as λ(t) = Z (t) { ( I +2diag ( Z (t) z (t) )) Δ(t) +diag ( Z (t) s 2 Z (t) Q Z (t) )} (7) After linearization according to (28) of the eqation in (3e), the reslt is 2 ( z (t)z (t) + v (t)z (t) ) Δ(t)+B y(t +) + D p(t) λ(t)+2z (t)δu(t)z (t) 2z (t) + V,i (x (t), (t))v,i (x (t), (t)) Δ(t) (8) = V,i (x (t), (t)), t where the eqations in (29), (33) and (34) have been sed, and where v (t) R ++ is defined in (29). By sing the expression for ΔU(t) in (5), the eqation in (8) can be written as 2 ( v (t) z (t) Z (t) z (t) ) Z (t)δ(t)+b y(t +) + D p(t) λ(t) 2diag(λ(t))Z (t) z (t) + V,i (x (t), (t))v,i (x (t), (t)) Δ(t) = sq Z (t) z (t)+ V,i (x (t), (t)) (9) By inserting (7) into (9), an expression in the following form will be obtained R(t)Δ(t)+B y(t +)+D p(t) =l(t), t (2) where R(t) and l(t) are defined as R(t) =2 ( v (t) z (t) Z (t) z (t) ) Z (t) + Z ˆ (t) Z (t) Z ˆ (t) + V,i (x (t), (t))v,i (x (t), (t)) l(t) =sq Z (t) z (t)+ Z ˆ (t) Z (t) diag (Z (t) s 2 Z (t) Q Z (t) ) + V,i (x (t), (t)) (2) where ˆ Z (t) = ( I +2diag ( Z (t) z (t) )). For what follows, it is important to show that R(t). Since U (t) and (t) are interior points that satisfy the positive semidefinite constraints in the problem strictly, it follows from the definition in (29) that Z (t) and Z (t). herefore, it follows from the Schr complement formla that Z (t) v (t) z (t) Z (t) z (t) (22) which implies that the first term in R(t) in (2) is positive definite. Since the other terms are qadratic and Z (t), the sm is positive definite. Hence, R(t) is positive definite. Now, by sing x(t) =x (t)+δx(t) and (t) = (t)+ Δ(t), the eqations in (3h) and (3i) can be expressed in Δx(t) and Δ(t) as follows Δx(t +)=AΔx(t)+BΔ(t) x (t +)+Ax (t)+b (t) (23) e(t) =CΔx(t) +DΔ(t) r(t) + Cx (t)+d (t) If the eqations in (3a), (3b), (2) and (23) are collected blockwise, the reslt is I D C p(t) I sq e e(t) D R(t) P Δ(t) C i Vx,i(x (t), (t))v x,i(x (t), (t)) I B A + B A I y(t+) Δx(t+) I Q(t) K(t) K(t) Q 2(t) A B ] = y(t) Δx(t) Δ(t) r(t) Cx (t) D (t) l(t) P i Vx,i(x (t), (t)) x (t+) Ax (t) B (t) A B I Δx(t) y(t),t (24) where the sms over i range from to c. After e(t) and p(t) have been eliminated the reslt can be written in the form ] ] ] ] y(t+) + Δx(t+) = q(t) r(t) b(t+) ],t (25)

7 Algorithm Riccati recrsion P (N) =Q (N) Ψ(N) = q(n) for t =to N do G(t +)=Q 2 (t)+b P (t +)B H(t +)=K(t)+A P (t +)B L(t +)=G(t +) H(t +) P (t) =Q (t)+a P (t +)A H(t +)L(t +) Ψ(t) = q(t) L(t +) ( B(t) Ψ(t +)+ r(t) ) + A Ψ(t +) end for x() = b() for t =to N do ũ(t) = L(t +) x(t) +G(t +) ( B Ψ(t +)+ r(t) ) x(t +)=A x(t)+bũ(t) b(t +) y(t) = Ψ(t)+P (t) x(t) end for y(n) = Ψ(N)+Q (N) x(n) where Q (t), Q 2 (t), K(t), q(t), r(t) and b(t) are defined in (35). Frthermore, at time instants t =and t = N the following eqations hold x() = x, y(n)+q (N)Δx(N) = q(n) (26) Since R(t), the system defined by the eqations in (25) and (26) are in a form which can be solved very efficiently sing Riccati recrsions. A derivation of the Riccati recrsions can be fond in, e.g., 9], ], and the reslting recrsions can be fond in Algorithm. Finally, by sing (28), the linearized version of the eqation in (3f) is fond to be v (t)+z (t) ΔU(t)z (t)+v (t)δ(t) z (t) + z (t) Δ(t)v (27) (t)+ν(t) = As a smmary, all comptations necessary to solve the Newton system efficiently are presented as Algorithm 2. All comptations, inclding the Riccati recrsion, can be performed with linear comptational complexity in the prediction horizon. In each iteration in an IP method, sally one or two Newton (like) systems of eqations have to be solved. Since in practice the nmber of iterations is roghly independent of problem size, the overall cost of solving the SDP is roghly proportional to the cost of solving the Newton eqations, ]. It shold be mentioned that even thogh the proposed Riccati method for simplicity is illstrated on a basic barrier method, it is also applicable in a primal-dal IP method as the one described in ]. Conseqently, by sing the method presented in this section, the optimization problem in () can be solved with approximately linear comptational complexity in the prediction horizon. V. NUMERICAL EXPERIMENS In this section, the relaxations in (6), (9) and () are compared in nmerical experiments. Frthermore, it is shown how large performance gain, compared to a generic solver, that can be expected when the problem in () is solved Algorithm 2 Efficient soltion of the Newton system : Precompte Z, ( Z ), Z, ( Z), z, v, V x, V, Q, Q 2, K, q, r and b for all time instants necessary. 2: Compte Δx(t), Δ(t) and y(t) sing Algorithm with the definitions in (35). 3: Compte e(t) according to (23) and p(t) according to (3a). 4: Compte λ(t) according to (7). 5: Compte ΔU(t) according to (5). 6: Compte w(t) according to (3l). 7: Compte ν(t) according to (27). efficiently with an IP solver that solves the Newton system sing Riccati recrsions. All tests of the comptational times were performed on a compter with two processors of the type Dal Core AMD Opteron 27 sharing 4 GB RAM (the code was not written to tilize mltiple cores) rnning CentOS release 4.4 Kernel ELsmp and MALAB 7.2. In all experiments with SDP problems, the dal SDP problems were derived manally and given to YALMIP, 2], and they were finally solved by SDP3 version 4., 3]. he optimal soltion (MIQP) and the QP relaxation were compted by sing CPLEX version.. In the first experiment, the relative gaps of the different relaxations are compared for different prediction horizons. he relative gap is here defined as J J R J, where J R is replaced by the optimal objective fnction vale of the relaxation of interest. he reslts are presented in Figre a and are for each prediction horizon fond as the average of 5 problems generated by the MALAB fnction drss with 4 states, 2 real valed control signals, 2 binary valed control signals and fll cost matrices Q x and Q. he two real valed control signals were constrained by a random pper and random lower bond, chosen sch that the problems are feasible and sch that the constraints are reasonable active along the prediction horizon. he reslt from the experiment clearly confirms the theoretical reslt in (). It shold be mentioned that the tightness is problem dependent, and the qality of the bonds may vary. he reslt shows that there are practical control problems where the eqality constrained relaxation is sefl, i.e., the bond is tighter compared to the QP relaxation and the comptational time is less than for the ordinary SDP relaxation. Note that all approaches actally seem to prodce rather good bonds, and that the practical difference seems fairly small in the examples considered. In 5], it has been shown in simlations that rather small improvements in the bonds can actally ct down the branch and bond tree significantly and that this, at least in the case with diagonal Q, can decrease the comptational time. In the second experiment, the respective comptational times for the different relaxations are compared. he reslt is presented in Figre b and it was fond by sing the MALAB command cptime and it does not inclde the time spent in YALMIP. he conclsion is that the standard SDP relaxation is rather slow to compte. he eqality

8 Problem formlation and linear system soltion time QP Relaxation gap Soltion time SDP3 Riccati.5 SDP 2 SDP Relative gap %] Soltion time s] 2 QP SDP 2 SDP Soltion time s] Prediction horizon steps] (a) his figre shows the average relative gaps between the optimal objective fnction vale and the optimal objective fnction vales from the relaxations. 2 Prediction horizon steps] (b) In this figre, the average comptational times for the relaxations are shown. Fig. : he figres present the nmerical reslts illstrating the relative tightness of the relaxations in (6), (9) and () and the corresponding comptational time. constrained SDP relaxation is mch less comptationally demanding for large vales of N and the complexity grows significantly slower as N grows. he QP relaxation is the least comptationally demanding relaxation to compte. Note that, if Q is diagonal, the reslt in Remark 2 applies and it is then possible to compte the eqality constrained SDP relaxation to a comptational cost similar to the one of the QP relaxation. In the third experiment, the comptational times for solving the Newton system are investigated. As previosly mentioned, a system of eqations of this type has to be solved in each iteration in an IP method and the soltion of this system is the major cost in the soltion process. In the reslts in Figre b, the generic state-of-the-art SDP solver SDP3 is sed to solve the eqality constrained SDP relaxation. he prpose with this third experiment is to show that it wold be possible to improve the comptational times presented in Figre b for the eqality constrained SDP relaxation by solving the Newton like eqation systems in the IP solver sing the Riccati approach in Algorithm 2. he expected performance is evalated by comparing the comptational time of certain parts of the code in the primal-dal IP solver SDP3 with the comptational time to perform a similar operation sing the Riccati approach in Algorithm 2. he time presented for SDP3 is the time it takes to form the Schr complement system, 3], and to solve it (similar to solving a Newton system). his is compared to the time it takes to (implicitly) form and solve a similar Newton system sing the Riccati approach for a problem of the same size. Each Newton system is solved times and the presented comptational time is an average from these experiments. From Figre 2, it can be fond that for large vales of N, SDP3 and the Riccati approach solve the system with comptational complexities of abot O(N.9 ) and O(N) respectively. he absolte comptational time is not of major interest here since the Riccati approach is implemented in m-code, while the corresponding operations in SDP3 are implemented in fast C-code. Or nmerical experiments have shown that the nmber of iterations sed in an IP solver applied to the problem in () is, as expected, roghly 2 2 Prediction horizon steps] Fig. 2: he reslt in this figre compares the comptational time for solving the Newton (like) system in the predictor step in SDP3 for a problem of the type in () and the comptational time for solving a similar Newton system sing the Riccati approach presented in this paper for a problem with similar properties and size. he comptational complexity for large vales of N grows as O(N.9 ) and O(N) respectively. independent of the prediction horizon. In or tests, arond 5 iterations were needed to solve the problem. Hence, the overall comptational complexity is approximately O(N.9 ) for the state-of-the-art generic solver SDP3 bt can be redced to approximately O(N) if the Riccati approach presented in this paper is applied. ACKNOWLEDGMENS Johan Löfberg is acknowledged for help with YALMIP. he research has been spported by the Swedish Research Concil for Engineering Sciences nder contract Nr VI. CONCLUSIONS In this paper, the QP relaxation, the standard moment relaxation and an eqality constrained moment relaxation have been applied to an MPC problem with mixed real valed and binary valed control signals sbject to linear ineqality constraints on states and control signals. he respective tightness and comptational complexity of the relaxations have been compared. he conclsion is that the best lower bond is achieved by the standard moment relaxation, which is also the most comptationally demanding relaxation. Frthermore, the QP relaxation gives the worst lower bond, bt is also the least comptationally demanding relaxation. he eqality constrained moment relaxation presented in this paper gives a bond at least as good as the bond from the QP relaxation and it is less comptationally demanding compared to the standard moment relaxation. his is mainly becase of the fact that the nmber of variables are less and they grow linearly in the prediction horizon. In the special case that the cost matrix for the control signal Q is diagonal, the eqality constrained SDP relaxation can be formlated and solved as a QP. Frthermore, the comptational complexity for compting the soltion to the eqality constrained SDP relaxation is expected to be possible to redce frther by solving the Newton eqations in an IP solver by sing

9 Riccati recrsions. his approach has been described in detail and promising reslts from nmerical experiments have been presented. A complete implementation of a solver sing the proposed Riccati approach and efficient comptation of the soltion to the standard SDP relaxation are left as ftre work. APPENDIX In this appendix some constants are defined and some help ] U(t) (t) eqations are presented. Recall that Ũ(t) = (t) w(t) and let U(t) =U (t)+δu(t), (t) = (t)+δ(t) and w(t) =. For small deviations ΔU(t) and Δ(t) from U (t) and (t), and if time indices are neglected, the following reslt can be fond from the aylor expansion (Ũ ) Z Z ΔUZ z Δ Z Z Δz (Ũ ) z Z ΔUz z Δ z Z Δv 2 (Ũ ) v z ΔUz v Δ z z Δv 22 (28) where the constants Z (t), z (t) and v (t) are defined as ] ] Z Z (t) = (t) z (t) U z (t) v = (t) (t) (t) (t) (29) Note that the inverse in (29) always exists since the IP solver generates a seqence of U (t) and (t) that always satisfies the positive semidefinite constraint in the problem strictly. Component i of the vector valed fnctions V x and V respectively are defined as Hx,i V x,i (x(t),(t)) = h i + H x,i x(t)+h,i (t),t Hx,i V x,i (x(n),(n)) = h i + H x,i x(n) H,i V,i (x(t),(t)) = h i + H x,i x(t)+h,i (t),t (3) he following eqalities hold diag ( Z (t) z (t)δ(t) +Δ(t)z (t) Z (t) ) ) =2diag ( Z (t) z (t) ) Δ(t) (3) diag ( Z (t) diag(λ(t))z (t) ) = Z (t) Z (t) λ(t) Z (32) (t)λ(t) where the constant Z (t) has been defined and is the Hadamard (elementwise) prodct. Note that Z (t) since Z (t), 4, p. 458]. Frthermore, z (t)δ(t) z (t) =z (t)z (t) Δ(t) (33) since Δ(t) z (t) is a scalar. since v (t) is a scalar. Z (t)δ(t)v (t) =v (t)z (t)δ(t) (34) he following definitions are sed in order to se Algorithm 2 to solve the Newton system. b() = x, b(t) =x (t +) Ax (t) B (t) b() = b(), b(t) = b(t) q(t) =sc ( Q e r(t) Cx (t) D (t) ) + V x,i (x (t), (t)) q(t) = q(t)+a (t)p (t +) b(t +), q(n) = q(n) r(t) =l(t)+sd ( Q e r(t) Cx (t) D (t) ) r(t) = r(t)+b (t)p (t +) b(t +) x(t) =Δx(t), ũ(t) =Δ(t) Q (t) =sc Q e C + V x,i (x (t), (t))v x,i (x (t), (t)) Q 2 (t) =R(t)+sD Q e D, REFERENCES K(t) =sc Q e D (35) ] L. A. Wolsey, Integer Programming. John Wiley & Sons, Inc., ] M. X. Goemans and D. P. Williamson,.878-approximation algorithms for MAX CU and MAX 2SA, in Proceedings of the 26th Annal ACM Symposim on heory of Compting, SOC 94, Montréal, Qébec, Canada, May 994, pp ] W. K. Ma,. N. Davidson, K. Wong, Z. Q. Lo, and P. Ching, Qasi-maximm-likelihood mltiser detection sing semi-definite relaxation, IEEE rans. Signal Process., vol. 5, no. 4, pp , 22. 4] J. Dahl, B. H. Flery, and L. Vandenberghe, Approximate maximmlikelihood estimation sing semidefinite programming, in IEEE International Conference on Acostics, Speech, and Signal Processing 23, vol. 6, Apr. 23, pp. VI ] D. Axehill, L. Vandenberghe, and A. Hansson, On relaxations applicable to model predictive control for systems with binary control signals, Atomatic Control, ISY, ech. Rep. 277, Jan. 27. Online]. Available: 6] D. Axehill, Applications of integer qadratic programming in control and commnication, Licentiate s hesis, Linköpings niversitet, 25. Online]. Available: 7] J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optimiz., vol., no. 3, pp , 2. 8] H. Wolkowicz, R. Saigal, and L. Vandenberghe, Eds., Handbook of Semidefinite Programming heory, Algorithms and Applications. Klwer, 2. 9] L. Vandenberghe, S. Boyd, and M. Noralishahi, Robst linear programming and optimal control, Department of Electrical Engineering, University of California Los Angeles, ech. Rep., 22. ] M. Åkerblad, A second order cone programming algorithm for model predictive control, Licentiate s hesis, Royal Institte of echnology, 22. ] L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and. Roh, Interior-point algorithms for semidefinite programming problems derived from the KYP lemma, ser. Lectre notes in control and information sciences. Springer, Feb. 25, vol. 32, pp ] J. Löfberg, Yalmip: A toolbox for modeling and optimization in MALAB, in Proceedings of the CACSD Conference, aipei, aiwan, 24. Online]. Available: joloef/yalmip.php 3] K. C. oh, R. ütüncü, and M. J. odd, On the implementation and sage of SDP3 a MALAB software package for semidefiniteqadratic-linear programming, version 4., Department of Mathematics, National University of Singapore, ech. Rep., Jl ] R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge University Press, 985.