An approach for Model Predictive Control of mixed integer-input linear systems based on convex relaxations

Transcription

1 5nd IEEE Conference on Decision and Control December -3, 3 Florence, Italy An approach for Model Predictive Control of mixed integer-input linear systems based on convex relaxations Marius Schmitt, Robin Vujanic, Joseph Warrington, Manfred Morari Abstract Model predictive control of systems with mixed discrete and continuous inputs usually requires the online solution of a mixed integer optimization problem Optimal solutions of such problems require methods whose worst-case complexity is exponential in the number of binary variables In this paper we propose an approximate approach in which the integer input constraints are initially relaxed A projection is then applied to the relaxed solution in order to obtain inputs satisfying the integer constraints Satisfaction of state constraints under the projected input sequence is to be guaranteed by applying a robust reformulation to the original relaxed problem We restrict our approach here to the practically important class of Pulse-Width Modulated power electronic systems, and present a suitable projection function for such systems We demonstrate an attractive trade-off between performance and computational cost, using the examples of a DC-DC buc converter and a single-phase AC-DC grid inverter I INTRODUCTION Model predictive control (MPC [] is an optimizationbased technique in which control inputs are applied in a receding horizon fashion At every time step, the input applied is the first of a sequence computed by solving a finite horizon optimal control problem for the current state The process is repeated at the next time step given a new measurement or estimate of the system state MPC most commonly employs a linearized model of the system, and convex cost and constraint functions In this case the corresponding finite horizon optimal control problem can be solved efficiently and reliably, since it is a convex optimization problem It is also possible to control systems with discrete state and/or input constraints by encoding a hybrid, rather than linear, system model into the MPC optimization problem [4] However, this results in a non-convex problem with integer constraints, typically a mixed-integer quadratic program (MIQP MIQPs must be solved using methods with exponential worst-case complexity, which limits the practical applicability In order to overcome this limitation, research so far has mainly focused on two approaches In the first one, tailored methods to efficiently solve the corresponding mixed integer problems using branch and bound methods have been proposed In particular, considerable research has been directed into efficiently computable and tight lower bounds to the optimal objective function, which are useful as they speed up the pruning of the B&B search tree [], [], [3] The The authors are with the Automatic Control Laboratory, Swiss Federal Institute of Technology (ETH Zurich, Physistrasse 3, 89 Zurich, Switzerland Contact: schmittm@eeethzch V in Fig i L (t L R L v C (t R C C Buc converter circuit R out v o (t other approach relies on the computation of explicit control laws [7] These require only very limited computations in real time, but the memory required to store the explicit law increases dramatically with the problem size In this wor we focus on the case in which integer constraints are imposed on the inputs only In our proposed approach we solve a single convex relaxation of the original control problem, and use a projection function to project the resulting inputs onto the set of inputs satisfying the integer constraints However, the new feasible input sequence also causes changes in the state trajectory In order to ensure that state constraints are also satisfied for the new input sequence, we use robust MPC techniques We thus guarantee that the solutions provided are feasible both in terms of input and state constraints Because our method only requires the solution to a convex optimal control problem followed by a computationally cheap projection, the time needed to compute a control input scales more modestly with the problem size than that needed to solve the associated MIQP exactly The ey ingredient of the approach is the projection function Such a function is often available in practice: we show how it can be designed for the important class of PWM systems, and we demonstrate the efficacy of the method by applying it to a constrained reference tracing problem for two different power converters Our solutions offer significantly better performance than solutions based on the linearized model typically used for converter control II MODELLING THE BUCK CONVERTER We use the buc converter (BC, Fig as an example to demonstrate the proposed idea In the results Section we will apply our controller also to another power converter having more switches The main tas of the BC is to convert a higher DC supply voltage V in by periodic switching to a lower DC output voltage v o (t The system equations ẋ(t = A c x(t+b c δ(t /3/$3 3 IEEE 648

2 (A AVERAGED (B HYBRID Parameter Value Load R out = Ω Parasitic Resistances R L, R C = Ω Capacitor C = 5 µf Inductor L = mh Supply voltage V = V Switching period T p = ms TABLE I BUCK CONVERTER: VALUES OF THE PARAMETERS cycle 3 4 Fig The averaged model (A assumes a constant v = d over one switching period (dashed curve, and this control is transformed into a / signal (solid curve by a PWM The hybrid model (B generates a better approximation of the v (dashed curve, and this control must again be transformed into an appropriate on/off signal (solid curve and the output equation v o (t = C c x(t can be readily obtained from first principles, with x(t = ( i L (t v C (t (II and ( A c L = (RL + R C R out R C +R out ( R out L R C +R out ( R out C R C +R out ( C R C +R out ( Vin ( B c = L, C c = ( R C R out (II L R C +R out ( R out C R C +R out The switch position δ(t {, } t is the control input The parameter values used in the simulations are given in Table I A Averaged Model The standard way of controlling not only the BC but a wide range of switched systems in power electronics is by deriving an averaged model and apply Pulse Width Modulation (PWM at the input In PWM, the time horizon is divided into switching periods and the δ(t is replaced by d [, ] called the duty cycle The duty cycle defines the time during which the switch is closed in the corresponding switching period and in PWM, an on-off signal is generated from it as depicted in figure (A The discrete-time dynamics can then be approximated linearly: x + = Ax + Bd, d y = Cx (II3 For the BC, the matrices A, B and C are obtained by discretizing (II This simple model approximates the average values of the states over one switching period, but does not reflect the (hybrid system dynamics within one cycle at all This shortcoming is well nown [5] and results in several main drawbacs: constraint violations may occur, as depicted eg in Figure 3, the unmodeled hybrid dynamics of the system may lead to steady state deviations which is highlighted in Section IV and no objective on the switching and in particular on the switching losses can be formulated An approach that has received attention lately [5], [8], [3] is the use of more refined system models that capture the hybrid dynamics explicitly These models divide the switching periods further into samples and approximate the system dynamics over these samples instead Feasibility of the input, in particular allowing at most one on/off switch per switching period, is ensured using logical constraints We describe this procedure in more detail in the next section B Hybrid Model If the approximation provided by the averaged model is inadequate, or if the switched behavior has to be taen into account explicitly, a model capturing the hybrid nature of PWM inputs is needed We use an approach similar to the one described in [5] We divide each switching period of length T p into M intervals of equal length T s = T p /M To model the switching, we introduce new inputs δ {, }M and δ + {, }M for each switch A value δ +,i = denotes that the switch is closed at some point during sample i in switching period, δ,i = means that is is opened The exact time when the switch is operated between sample i and i+ is modeled by a continuous input d R M restricted to δ d δ + in which is interpreted elementwise Thus the continuous signal can be different from only during samples when a decision to switch on or off is made For notational convenience, we introduce for each PWM operated switch the v R M defined as v := (δ + δ + d (II4 }{{} :=D which is illustrated in Figure (B The value v,i indicates the fraction of sample i in switching period during which the switch is closed We wish to control the state during the switching periods, and therefore introduce the staced state vector x := ( x, x, x,m (II5 in which x,i = x ( T p + i T s denotes the states at the i th sample during the th switching period The system dynamics are given by x + = A x + B v (II6 649

3 in which the matrices A, B are given by B A A A =, B = AB B A M A M B AB B (II7 with matrices A, B obtained by discretizing the system dynamics (II3 with sample time T s In order to reproduce the switched behavior in our system model, constraints need to be imposed on the input In the input constraints v (II8a δ + (II8b δ (II8c δ d δ + (II8d (II8a ensures feasibility of the input, (II8b and (II8c ensure that we switch on resp off at most once per cycle and (II8d models the exact switching time as described before Last, an integrality constraint needs to be imposed on the switching inputs δ +, δ {, }M (II9 Together (II8 and (II9 are referred to as input constraints and we denote an input u which satisfies the input constraints as input feasible In addition to the input constraints, we allow for general polytopic constraints on the states E x x e (II It should be highlighted that the maximal switching frequency of the hybrid model is the same as in the averaged model Liewise, the control algorithm based on the hybrid model is evaluated only once per switching period A system model which is updated at every sample is also possible using the more general Mixed Logical Dynamical (MLD framewor [4], but such an approach leads to a time varying system model In addition, we explicitly now the set of input feasible inputs at each step for our model, hence it will be easy to find a suitable projection that provides inputs satisfying the integrality constraints This is not possible for general MLD systems III MPC FORMULATION In model predictive control, the control objective to be minimized is defined stagewise as J (x, u, (III = where each function J is the cost at stage, and N is the number of stages in the problem In the simulations of the BC we control the output voltage and thus define an objective for the averaged system model J avg (x, u = ( x ˆx C C ( x ˆx (III i L /i L u /u out out Fig 3 Step response under MPC for the buc converter based on the averaged model Depicted are the states and the inputs, ie the precomputed input by the controller (green dashed and the that is actually applied to the system The system behavior is only roughly approximated and the constraint on i L (dashed is violated significantly in which C denotes the system output matrix and ˆx a reference A similar objective J hyb (x, u can be defined for the hybrid system In MPC, an infinite horizon optimal control problem is approximated by a number of finite horizon optimal control problems, which are solved in receding horizon fashion We can thus define MPC controllers for the averaged resp hybrid system model by posing the following finite horizon constrained optimal control problems for the averaged system: min x,u st = and the hybrid system: min x,u st J avg (x, u averaged dynamics (II3 state constraints (II = J hyb (x, u hybrid dynamics (II6 state constraints (II switching constraints (II8 integrality constraints (II9 (III3 (III4 The performance of the MPC controllers for the buc converter on a step response with a state constraint on the maximal current is depicted in Figure 3 for the MPC based on the averaged model and in Figure 4 for the hybrid system model Violation of the state constraints is clearly visible under the action of the controller based on the averaged model In addition, the model based on the duty cycle leads to steady state inaccuracies (see Section V IV RELAXATION BASED HYBRID MPC In this section we state the relaxation of the hybrid optimal control problem (III4 We propose a projection function for PWM inputs, and a method to establish a priori the influence of the projection operation on the states trajectories This information can be used to robustify the relaxed optimization problem, which ensures that the projected sequence of inputs (which is input feasible by 643

4 i L /i L u /u out out Fig 4 Step response under MPC for the buc converter based on the hybrid, mixed integer model Depicted are the states and the inputs, ie the precomputed input by the controller (green dashed and the that is actually applied to the system The system behavior is approximated very closely virtue of the projection function does not violate state constraints either The most straightforward relaxation is the replacement of any integrality constraint x {, } with the convex relaxation x [, ] The relaxation of the hybrid MPC optimal control problem is thus defined by (III4 with a relaxed integrality constraint (II9 A Projection function We propose a suitable projection function for system with PWM inputs next, in which δ +, δ denote the relaxed counterparts of δ +, δ and d is the counterpart to d All relaxed inputs are collected in ũ = ( δ + δ d Following (II4, we also introduce ṽ := D ( δ + δ + d We chose a projection such that the average duty cycle D = T p v(tdt = M ṽ and the first moment of the Tp input M = t v(tdt = ( / 3 / (M / ṽ over one switching period are preserved The only control actions we are allowed to mae during one cycle is when to close the switch and when to open it, so intuitively speaing we have only two degrees of freedom The switching times t on/off = can thus be computed explicitly M D D (IV The input defined by the switching times t on,off can be encoded in a hybrid input u and we denote by p (ũ : R M R M R M {, } M {, } M R M the projection function for PWM inputs that projects the relaxed input ũ to a hybrid input u such that the switching times are defined as above This hybrid input is input feasible by construction Projected inputs u = p (ũ satisfy the following constraints: M v = M ṽ (IVa ( / 3 / (M / (v ṽ = (IVb input feasibility (II8, (II9 (IVc 5 T p Fig 5 The projection function for the BC, with M = 5 samples of length T s per switching period T p Depicted are the and the control signal ṽ For each switching period, the area below the graphs is the same (IVa and that the first moments of the inputs coincide (IVb Examples of the projection of several relaxed inputs ũ are given in Figure 5 Note that by using this projection, the switch will be open at the end of every cycle B Robust feasibility We want to implement a controller which solves the QP-relaxation of the MIQP, uses the previously defined projection and applies the projected inputs to the plant Applying the projected inputs u = p(ũ instead of the relaxed inputs ũ to the system will yield a different state evolution in general We refer to the set of all possible state deviations as the induced uncertainty set W, ie W := { x x satisfies ( for some ũ, x } ( x = x + (p(ũ, x x + (ũ, x T s (IV3 For any relaxed input ũ the state deviation can be written as x + (p(ũ, x x + (ũ, x = A x + B p(ũ (A x + B ũ = B (p(ũ ũ (IV4 ie the uncertainty set is independent of the state x since we assume linear system dynamics We can interpret the state deviation induced by the projection as unnown disturbances which are bounded within the uncertainty set, ie x + = Ax + Bp(ũ! = Ax + Bũ + w (IV5 for some w W But this equivalence maes use of robust MPC approaches for additive polytopic uncertainty possible in order to achieve robust feasibility of the computed inputs We use a constraint tightening approach [4] in order to achieve robust feasibility In this approach, the nominal feasible set {x E x x e}, ie all states that satisfy (II, is tightened for subsequent iterations in the prediction horizon to account for the disturbances For open loop robust control (without the usage of a prestabilizing controller constraints defining those sets are given by ( E x x e min E x A i w i := e (IV6 w i W i= which means that only a modification of the RHS of the state constraints is necessary, at no additional online computational effort This modification maes the precomputed inputs robustly feasible meaning that all state and input 643

5 constraints are guaranteed to be satified up to the prediction horizon for the projected inputs We do not need to compute the uncertainty set W explicitly, but we can instead evaluate the RHS of the modified state constraints IV6 offline To this end, we define B AB B B :=, E := I N E x (IV7 A N B AB B where denotes the Kronecer product, and rewrite the RHS of the modified constraints as e e p(ũ ũ e e p(ũ = min E B ũ ũ feasible (IV8 e N e p(ũ N ũ N This optimization problem can be written in tractable form as a mixed integer linear program (MILP min u,ũ u ũ u E B ũ u N ũ N (IV9a st M v = M ṽ (IV9b ( / 3 / (M / v = ( / 3 / (M / ṽ (IV9c M (δ + δ = (IV9d M ( δ + δ = (IV9e u satisfy (II8, (II9 (IV9f ũ satisfy (II8 (IV9g with min meaning row wise minimization Each row of (IV9a gives the amount by which the RHS value of one constraint for one stage in the prediction horizon needs to be tightened Conditions (IV9b and (IV9c define the projection Conditions (IV9d and (IV9e ensure that the switch is open at the end of every cycle Conditions (IV9f and (IV9g ensure input feasibility The maximal state deviations for the BC over two switching cycles are depicted in Figure 6 All computations for the constraint tightening can be done offline and need to be done only once Thus it does not matter that several computationally expensive MILP need to be solved Because of the hybrid nature of switched PWM systems, asymptotic stability is not possible for arbitrary reference values In [4], a method based on robust control invariant sets is proposed With this method comes a stronger notion of robust feasibility which does guarantee feasibility of all subsequent iterations as well as convergence to a control invariant terminal set This method can be readily used with our approach The proposed MPC based on the projection approach uses (A Inductor current (B Capacitor voltage 5 5 bounds on state uncertainty i L (A bounds on state uncertainty u C (V cycle, M = 5 uncertainty sets uncertainty sets samples, T sample = ms Fig 6 The uncertainty set W for the states of the BC, M = 5 The uncertainty is given for each sample in a horizon of two cycles Note that the uncertainty becomes small at the end of every cycle 5 5 i L /i L u C /u C Fig 7 Step response under MPC for the buc converter based on a QPrelaxation of the hybrid system model Depicted are the states and the inputs, ie the precomputed input by the controller (green dashed and the binary input that is actually applied to the system The constraint on i L is satisfied and the steady state error is very small the following finite horizon optimal control problem min J map (x, ũ = N J hyb (x, ũ x,ũ = = st hybrid dynamics (II6 (IV modified state constraints (IV6 switching constraints (II8 relaxed integrality constraints δ +, δ V EVALUATION The example of the BC (Figure 7 demonstrates that the problem of constraint violations as demonstrated in the step response of the MPC controller based on the averaged model in Figure 3 can be avoided by the MPC controller based on our projection approach In addition, the classical PWM model may lead to inaccuracies in the steady state behavior [5] The MPC controllers based on the hybrid model, both the hybrid and the relaxed one, are able achieve a better steady state RMS deviation from the reference The corresponding state trajectories are depicted in Figure 9 The steady state RMS deviation of the continuous time output voltage from the reference and the online computation times are summarized in Table II The optimization problems have been solved using Yalmip [] and Gurobi [] 643

6 i g(t L fg R fg L fi R fi s s u out /u out 5 95 (a averaged (b projected reference Vg C f VDC i L /i L 5 5 s s Fig 8 Single phase grid inverter circuit Buc Converter average hybrid QP-relaxation RMS deviation 6 V V 7 V comp time, N =, M = 5 6 ms 9 ms 34 ms AC-DC converter average hybrid QP-relaxation RMS deviation 45 V 98 V 85 V comp time, N = 4, M = 3 6 ms 533 ms 6 ms TABLE II MPC PERFORMANCE In order to demonstrate that our approach can be readily used with more complex PWM topologies, the proposed MPC approach has also been applied to a single phase grid inverter, which is used to connect a DC source or load to an AC grid The single phase grid inverter is a linear system operated by four switches (Figure 8, out of which two (s and s can be operated independently by PWM Using the parameters and the linear model described in [6], MPC controllers based on the averaged model, the hybrid model and the projection approach habe been implemented We operate the grid inverter such that power from the DC source is injected into the grid, which is achieved by tracing a sinusoidal reference current of 5 Hz with an amplitude of A The resulting RMS deviation of the continuous time output (grid current from the reference and computation times are reported in Table II In addition to achieving a better RMS deviation than MPC based on the averaged model, relaxation based MPC achieves superior computation times in comparison to hybrid MPC VI CONCLUSIONS This paper demonstrates a new predictive control approach for linear systems with PWM inputs In contrast to hybrid MPC, which solves mixed integer programs online, we use a projection function to generate feasible inputs from the solution of a relaxed optimization problem A robust MPC approach based on constraint tightening has been used to accommodate the additional uncertainty which is induced by the projection function This ensures robust feasibility and constraint satisfaction at no additional online computational effort In future, the approach could be adapted to accommodate less conservative techniques lie closed-loop robust MPC [9] The main benefit of our relaxation-and-projection approach is that only one single convex optimization problem as opposed to a mixed integer problem has to be solved Fig 9 Sampling times are indicated by o, the voltage u out is the system output (a Using the averaged model, model inaccuracies and the current ripple which is unnown to the controller lead to a significant steady state offset (b The projection based MPC achieves a better approximation of the system behavior through a more refined system model and reduces the RMS deviations significantly even in the presence of the uncertainty induced by the projection online, which is computationally much less demanding in both theory and practice We find a suboptimal solution in general, but as can be seen in the case of the buc converter, the performance degradation may be minimal while the necessary computation times are reduced significantly The choice of the projection function is the ey for good performance In order to build on this wor, it would be attractive to investigate the existence of performance bounds on the quality of the relaxed solution REFERENCES [] Daniel Axehill, Anders Hansson, and Lieven Vandenberghe, Relaxations applicable to mixed integer predictive control comparisons and efficient computations, Decision and Control, 7 46th IEEE Conference on, 7, pp [] Daniel Axehill, Lieven Vandenberghe, and Anders Hansson, On relaxations applicable to model predictive control for systems with binary s, PhD Thesis, Department of Electrical Engineering, Linöpings Universitet, 7 [3], Convex relaxations for mixed integer predictive control, Automatica 46 (, no 9, [4] Alberto Bemporad and Manfred Morari, Control of systems integrating logic, dynamics, and constraints, Automatica 35 (999, no 3, [5] C Fischer, S Mariethoz, and M Morari, Multisampled hybrid model predictive control for pulse-width modulated systems, 5th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC, December, pp [6] C Fischer, S Mariéthoz, and M Morari, A model predictive control approach to reducing low order harmonics in grid inverters with LCL filters, IEEE IECON, Industrial Electronics Conf, November 3 [7] Francesco Borrelli, Alberto Bemporad, Manfred Morari, Predictive control for linear and hybrid systems, available on mpcbereleyedu/mpc-course-material, Preprint, [8] T Geyer, G Papafotiou, and M Morari, Hybrid model predictive control of the step-down DC-DC converter, IEEE Transactions on Control Systems Technology 6 (8, no 6, 4 [9] Paul J Goulart, Eric C Kerrigan, and Jan M Maciejowsi, Optimization over state feedbac policies for robust control with constraints, Automatica 4 (6, no 4, [] Inc Gurobi Optimization, Gurobi optimizer reference manual, 3 [] J Löfberg, Yalmip : A toolbox for modeling and optimization in MATLAB, Proceedings of the CACSD Conference (Taipei, Taiwan, 4 [] Jan Maciejowsi, Predictive control with constraints, Prentice Hall, [3] G Papafotiou, T Geyer, and M Morari, Hybrid modelling and optimal control of switch-mode dc-dc converters, 4 IEEE Worshop on Computers in Power Electronics Proceedings, 4, pp [4] Arthur Richards and Jonathan How, Robust stable model predictive control with constraint tightening, American Control Conference, 6 Proceedings of,, 6, pp