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Stochastic MPC of Systems with Additive Disturbance using Scenario Optimization*
Preprints, 5th IFAC Conference on Nonlinear Model Predictive Preprints, Preprints, 5th 5th IFAC IFAC Conference Conference on on Nonlinear Nonlinear Model Model Predictive Predictive Control Preprints, 5th IFAC Conference on Nonlinear Model Predictive Control Available online at www.sciencedirect.com Control September 17-20, 2015. Seville, Spain Control September September 17-20, 17-20, 2015. 2015. Seville, Seville, Spain Spain September 17-20, 2015. Seville, Spain
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Stochastic MPC of Systems with Additive Stochastic Stochastic MPC MPC of of Systems Systems with with Additive Additive Disturbance using Scenario Optimization Disturbance using Scenario Optimization Disturbance using Scenario Optimization ∗∗ Xiaohai Lin ∗∗ Daniel G¨ orges ∗∗ Sebastian Caba ∗∗ ∗∗ Steven Liu ∗∗ ∗ Daniel G¨ ∗ Sebastian Caba ∗∗ ∗∗ Steven Liu ∗∗ Xiaohai Lin o rges ∗ ∗ Xiaohai Lin Lin Daniel Daniel G¨ G¨ orges rges Sebastian Sebastian Caba Caba Steven Steven Liu Liu ∗∗ Xiaohai o ∗ ∗ Juniorprofessorship for Electromobility, Department of Electrical and ∗ Juniorprofessorship for Electromobility, Department of Electrical and ∗Computer Juniorprofessorship for Electromobility, Department Electrical Engineering, of Kaiserslautern, Kaiserslautern, Juniorprofessorship for University Electromobility, Department of of Electrical and and Computer Engineering, University of Kaiserslautern, Kaiserslautern, Computer University of Germany (e-mail: lin|[email protected]) Computer Engineering, Engineering, University of Kaiserslautern, Kaiserslautern, Kaiserslautern, Kaiserslautern, Germany (e-mail: lin|[email protected]) ∗∗ Germany (e-mail: lin|[email protected]) of Control Systems, Department of Electrical and Germany (e-mail: lin|[email protected]) ∗∗ Institute ∗∗ Institute of Control Systems, Department of Electrical and ∗∗ Institute of Control Control Systems,ofDepartment Department of Electrical Electrical and Computer Engineering, University Kaiserslautern, Kaiserslautern, Institute of Systems, of and Computer Engineering, University of Kaiserslautern, Kaiserslautern, Computer of Germany University (e-mail: caba|[email protected]) Computer Engineering, Engineering, University of Kaiserslautern, Kaiserslautern, Kaiserslautern, Kaiserslautern, Germany (e-mail: caba|[email protected]) Germany Germany (e-mail: (e-mail: caba|[email protected]) caba|[email protected])
Abstract: In this paper a stochastic model predictive control (SMPC) scheme for linear systems Abstract: In this paper a stochastic model predictive control (SMPC) scheme for linear systems Abstract: In paper model predictive control scheme for linear with additive disturbance is presented. The goal is to design(SMPC) a controller that the Abstract: In this this paper a a stochastic stochastic model predictive control (SMPC) scheme for minimizes linear systems systems with additive disturbance is presented. The goal is to design a controller that minimizes the with additive additive disturbance is presented. presented. The goal goal is to to design design controllerexponential that minimizes minimizes the expected valuedisturbance of an objective function while guaranteeing mean-square input-towith is The is aa controller that the expected value of an objective function while guaranteeing mean-square exponential input-toexpected value(MSE-ISS) of an an objective objective function while while guaranteeing mean-square exponential input-tostate stability and constraints on the states andmean-square inputs. The exponential SMPC is partitioned expected value of function guaranteeing input-tostate stability (MSE-ISS) and constraints on the states and inputs. The SMPC is partitioned state stability constraints on states and The SMPC partitioned into offline (MSE-ISS) computationand based on a bilinear (BMI) ensuring statean stability (MSE-ISS) and constraints on the thematrix statesinequality and inputs. inputs. The problem SMPC is isfor partitioned into an offline computation based on a bilinear matrix inequality (BMI) problem for ensuring into an offline computation based on a bilinear matrix inequality (BMI) problem for ISS, constraint satisfaction, and recursive feasibility and an online optimization based on a into an offline computation based on a bilinear matrix inequality (BMI) problem for ensuring ensuring ISS, constraint satisfaction, and recursive feasibility and an online optimization based on ISS, constraint constraintconstrained satisfaction, and recursive recursive feasibility and an anproblem online optimization optimization based on aaa quadratically quadratic programming (QCQP) for includingbased knowledge ISS, satisfaction, and feasibility and online on quadratically constrained quadratic programming (QCQP) problem for including knowledge quadratically constrained quadratic programming (QCQP) problem for including knowledge about the additive disturbance whileprogramming relaxing ISS (QCQP) to MSE-ISS. The for partition into knowledge an offline quadratically constrained quadratic problem including about the additive disturbance while relaxing ISS to MSE-ISS. The partition into an offline about disturbance while relaxing ISS partition an computation and online optimization stability,The feasibility, andinto performance about the the additive additive disturbance while allows relaxingaddressing ISS to to MSE-ISS. MSE-ISS. The partition into an offline offline computation and online optimization allows addressing stability, feasibility, and performance computation and online allows feasibility, and separately and therefore improving performance as well stability, as handling disturbances which may computation and online optimization optimization allows addressing addressing stability, feasibility, and performance performance separately and improving performance as well as which may separately and therefore therefore improving performance as The well effectiveness as handling handling disturbances disturbances which may not be independent identically distributed (i.i.d.). of the SMPCwhich scheme is separately and therefore improving performance as well as handling disturbances may not be independent identically distributed (i.i.d.). The effectiveness of the SMPC scheme is not be independent identically distributed (i.i.d.). The effectiveness of the SMPC scheme evaluated by simulations and assessed by comparison with a robust MPC scheme. not be independent identically distributed (i.i.d.). The effectiveness of the SMPC scheme is is evaluated by simulations and assessed by comparison with aa robust MPC scheme. evaluated by simulations and assessed by comparison with robust MPC scheme. evaluated by simulations and assessed by comparison with a robust MPC scheme. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Model predictive control, stochastic control, robust control, mean-square stability, Keywords: Model Model predictive predictive control, control, stochastic control, control, robust control, control, mean-square stability, stability, Keywords: input-to-state stability Keywords: Model predictive control, stochastic stochastic control, robust robust control, mean-square mean-square stability, input-to-state stability stability input-to-state input-to-state stability 1. INTRODUCTION putational demand is often high. As an alternative, tube1. INTRODUCTION INTRODUCTION putational demand is often high. As an alternative, tube1. putational demand is high. an tubebased model predictive approaches been proposed. 1. INTRODUCTION putational demand is often often high. As As have an alternative, alternative, tubebased model predictive approaches have been proposed. based model predictive approaches have been proposed. The control presented by Mayne et al. (2005) model scheme predictive approaches have been proposed. Model predictive control (MPC) has been developed since based presented by Mayne et (2005) Model predictive predictive control control (MPC) (MPC) has has been been developed developed since since The The control controlthescheme scheme presented by of Mayne et al. al. set (2005) Model guarantees exponential stability an “origin” and control scheme presented by Mayne et al. (2005) the latepredictive seventies and has (MPC) meanwhile a great impact Model control has gained been developed since The guarantees the exponential stability of an “origin” set and the late seventies and has meanwhile gained a great impact guarantees theviolation exponential stability of an an “origin” “origin” set only and the late seventies has meanwhile gained impact prevents the of the constraints. However, the exponential stability of set and on world and Badgwell, 2003). The guarantees the the late industrial seventies and and has (Qin meanwhile gained aa great great impact prevents the violation of the constraints. However, only on the industrial world (Qin and Badgwell, 2003). The prevents the violation of the constraints. However, only on industrial Badgwell, The the boundaries of the disturbance are takenHowever, into account. the violation of the constraints. only major MPC(Qin is theand ability to deal 2003). with input on the theadvantage industrial ofworld world (Qin and Badgwell, 2003). The prevents boundaries of the disturbance are taken into account. major advantage advantage of of MPC MPC is is the the ability ability to to deal deal with with input input the of taken account. major Thisboundaries can be conservative if furtherare knowledge the the boundaries of the the disturbance disturbance are taken into intoabout account. and state constraints (Mayne etability al., 2000). Thewith rationale major advantage of MPC is the to deal input the This can be conservative if further knowledge about the and state constraints (Mayne et al., 2000). The rationale This can be conservative if further knowledge about the and state constraints (Mayne et al., 2000). The rationale disturbance is available, e.g. the probability distribution. This can be conservative if further knowledge about the behind MPC is that at every time set rationale of future disturbance is available, e.g. the probability distribution. and state constraints (Mayne et al.,instant 2000). aThe behind MPC is that at every time instant a set of future disturbance is is available, available, e.g. e.g. the the probability probability distribution. distribution. behind at time instant set of control signalsis calculated minimizing behind MPC MPC is isthat that at every every by time instant aa an set objective of future future disturbance control signals signals is is calculated calculated by by minimizing minimizing an an objective objective Recently, stochastic MPC (SMPC) has been established, control stochastic MPC (SMPC) has been function for a finite horizon considering the current state. Recently, control signals is calculated by minimizing an objective Recently, stochastic MPC (SMPC) has Schildbach been established, established, e.g. by Calafiore and Fagiano (2013), et al. stochastic MPC (SMPC) has been established, function for for aa finite finite horizon horizon considering considering the the current current state. state. Recently, function by Calafiore and Fagiano (2013), Schildbach et al. Then thefor first element of the control sequence is applied function a finite horizon considering the current state. e.g. e.g. by by Cheng Calafiore and Fagiano (2013), Schildbach et al. (2014), et al. (2014), and Zhang et al. (2014). Most e.g. Calafiore and Fagiano (2013), Schildbach et al. Then the first element of the control sequence is applied Then first Cheng et al. (2014), and Zhang et al. (2014). Most to the the plant. Then the first element element of of the the control control sequence sequence is is applied applied (2014), (2014), Cheng et al. (2014), and Zhang et al. (2014). Most approaches assume that the disturbances are independent (2014), Cheng et al. (2014), and Zhang et al. (2014). Most to the plant. to approaches assume that the disturbances are independent to the the plant. plant. assume the are identically distributed Unfortunately, the disturapproaches assume that that(i.i.d.). the disturbances disturbances are independent independent The standard MPC formulation does not directly consider approaches identically distributed (i.i.d.). Unfortunately, the The standard MPC formulation does not directly consider identically distributed (i.i.d.). Unfortunately, the disturdisturThe MPC does directly bances are partly not i.i.d. in practical applications (Paridistributed (i.i.d.). Unfortunately, the disturmodel uncertainties and disturbances (Mu˜ noz de consider la Pe˜ na identically The standard standard MPC formulation formulation does not not directly consider bances are partly not i.i.d. in practical applications (Parimodel uncertainties and disturbances (Mu˜ n oz de la Pe˜ n a bances are partly2013; not i.i.d. i.i.d. in practical practical applications (Parimodel and (Mu˜ n n sio and are Glielmo, Di Cairano et al.,applications 2014). Bernardini partly not in (Pariet al., uncertainties 2005). Naturally, the feedback mechanism can model uncertainties and disturbances disturbances (Mu˜ noz oz de de la la Pe˜ Pe˜ naa bances sio and Glielmo, 2013; Di Cairano et al., 2014). Bernardini et al., 2005). Naturally, the feedback mechanism can sio and Glielmo, 2013; Di Cairano et al., 2014). Bernardini et the feedback mechanism andand Bemporad propose a SMPC schemeBernardini for linear Glielmo,(2012) 2013; Di Cairano et al., 2014). already provideNaturally, an inherent discussed can by sio et al., al., 2005). 2005). Naturally, therobustness feedback as mechanism can Bemporad (2012) propose aa SMPC scheme for linear already provide provide an an inherent inherent robustness robustness as as discussed discussed by by and Bemporad (2012) scheme for linear already systems with multiplicative disturbance may not and Bemporad (2012) propose propose a SMPC SMPC which scheme for be linear Mayne al. (2000). However, model inaccuracies already et provide an inherent robustness as discussedand by and systems with multiplicative disturbance which may be not Mayne et al. (2000). However, model inaccuracies and systems with multiplicative disturbance which may be not Mayne et al. (2000). However, model inaccuracies and i.i.d., guaranteeing mean-square exponential stability systems with multiplicative disturbance which may be not external perturbations may still lead to loss of performance Mayne et al. (2000). However, model inaccuracies and i.i.d., guaranteeing mean-square exponential stability and and external perturbations may still lead to loss of performance i.i.d., guaranteeing guaranteeing mean-square exponentialThese stability and external may to of satisfaction of statemean-square and input constraints. fruitful exponential stability and and evenperturbations stability. Therefore, robust strategy i.i.d., external perturbations may still still alead lead to loss losscontrol of performance performance of state and input constraints. These fruitful and even even stability. stability. Therefore, Therefore, aa robust robust control control strategy strategy satisfaction of These fruitful and results can unfortunately not beconstraints. applied for linear satisfaction of state state and and input input constraints. These systems fruitful is which explicitly copes with uncertainties and satisfaction anddesired even stability. Therefore, a robust control strategy results can unfortunately not be applied for linear systems is desired which explicitly copes with uncertainties and results can unfortunately not be applied for linear systems is desired which explicitly copes with uncertainties and with additive disturbance since a robust convergence to the can unfortunately not be applied for linear systems disturbances. is desired which explicitly copes with uncertainties and results with additive disturbance since aa robust convergence to disturbances. with additive disturbance since robust convergence to the the disturbances. origin cannot be enforced. Particularly, a key component, with additive disturbance since a robust convergence to the disturbances. cannot be enforced. Particularly, a key component, Several robust MPC laws based on the min-max policy origin origin cannot set, be enforced. enforced. Particularly,inaa another key component, component, the invariant must be constructed way. origin cannot be Particularly, key Several robust MPC laws based on the min-max policy Several robust MPC laws based on policy invariant set, must be constructed in another way. have been proposed, Scokaert andmin-max Mayne (1998) Several robust MPC e.g. lawsby based on the the min-max policy the the invariant invariant set, set, must must be be constructed constructed in in another another way. way. have been been proposed, proposed, e.g. e.g. by by Scokaert Scokaert and and Mayne Mayne (1998) (1998) the have In this paper a novel SMPC scheme for linear with and Bemporad et al. (2003). The intention is to minimize have been proposed, e.g. by Scokaert and Mayne (1998) In this paper a novel SMPC scheme for linear systems systems with and Bemporad et al. (2003). The intention is to minimize In this paper a novel SMPC scheme for linear systems with and Bemporad et The is minimize additive disturbance which may be for notlinear i.i.d. systems is presented. this paper a novel SMPC scheme with the the objective function. The control In and worst-case Bemporadvalue et al. al.of(2003). (2003). The intention intention is to to minimize disturbance which may be not i.i.d. is presented. the worst-case worst-case value value of of the the objective objective function. function. The The control control additive additive disturbance which may be not i.i.d. is presented. the The SMPC scheme which consists of be annot offline disturbance may i.i.d. computation is presented. action of these value schemes mayobjective be conservative com- additive the worst-case of the function.and Thethe control SMPC scheme consists of an computation action of of these these schemes may may be conservative conservative and the the com- The The an SMPC scheme consists of the an offline offline computation, computation action and onlinescheme optimization. With SMPC consists of an offline computation action of these schemes schemes may be be conservative and and the comcom- The This work has been partly supported by the Center for Commercial and an online optimization. With the offline computation, and an online optimization. With the offline computation, input-to-state stability (ISS), constraint satisfaction, and This work has been partly supported by the Center for Commercial and an online optimization. With the offline computation, This has partly by for Vehicle Technology (ZNT) atsupported the University ofCenter Kaiserslautern funded input-to-state stability (ISS), constraint satisfaction, and This work work has been been partly supported by the the Center for Commercial Commercial input-to-state stability (ISS), constraint satisfaction, and Vehicle Technology (ZNT) at the University of Kaiserslautern funded recursive feasibility are ensured. In the online optimizainput-to-state stability (ISS), constraint satisfaction, and Vehicle Technology (ZNT) at the the University of Rhineland-Palatinate. Kaiserslautern funded funded by Research Initiative of the Federal State of Vehicle Technology (ZNT) at University of Kaiserslautern recursive feasibility are ensured. In the online optimizaby Research Initiative of the Federal State of Rhineland-Palatinate. recursive feasibility are ensured. In the online optimizarecursive feasibility are ensured. In the online optimizaby Research Initiative of the Federal State of Rhineland-Palatinate. by Research Initiative of the Federal State of Rhineland-Palatinate. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 236 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 236 Peer review© of International Federation of Automatic Copyright © 2015 IFAC 236 Copyright ©under 2015 responsibility IFAC 236Control. 10.1016/j.ifacol.2015.11.289
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tion, knowledge about the disturbance is included using a multi-horizon scenario tree (Bernardini and Bemporad, 2012; Mu˜ noz de la Pe˜ na et al., 2005) and ISS is relaxed to mean-square exponential ISS (MSE-ISS). The partition into an offline computation and an online optimization allows addressing stability, feasibility and performance separately. This increases the flexibility in the online optimization and thus the performance and particularly allows including disturbances which are not i.i.d. The paper is organized as follows. The system model and a sufficient condition for MSE-ISS are introduced in Section 2. The offline computation in form of BMIs and the online optimization are then formulated in Section 3. The performance of the SMPC policy is compared with the performance of the robust MPC (RMPC) approach from Yang and Feng (2013) in Section 4 based on simulations. Finally conclusions and remarks on future work are given in Section 5. Throughout the paper the following notation is used. For a vector v, [v]j is the jth element. For a matrix T, [T]ij denotes the component of the ith row and jth column. E[k1 ,k2 ] stands for the conditional expectation with respect to a variable sequence {w(k1 ), ..., w(k2 )}. Ek corresponds to E[k,k] . The maximum and the minimum eigenvalues of the matrix A are indicated by λmax (A) and λmin (A). In denotes the identity matrix of the size n × n and Iin is the ith row. Pr(·) indicates the probability, diag(·) a blockdiagonal matrix. For a symmetric matrix E, the symbol E11 E12 E11 E12 is used and E = E22 is equivalent to ET E22 . 12
2. BACKGROUND
2.1 System Model The discrete-time linear system considered in the paper is x(k + 1) = Ax(k) + B1 u(k) + B2 w(k) (1) where k ∈ N0 is the discrete time, x ∈ Rn is the state vector, u ∈ Rm is the input vector, w ∈ W ⊂ Rmw is a random disturbance vector, and W = {w1 , w2 , . . . , ws } is a predefined finite set with s elements. The disturbance vector w is modeled as a discrete-time Markov chain with a transition probability matrix T ∈ Rs×s described by [T]ij = Pr[w(k) = wi |w(k − 1) = wj ] ∀i, j ∈ {1, 2, .., s}. (2) The probability distribution of w(k) is given by p(k) = Tp(k − 1), (3) where p(k−1) = [0, ..., 0, 1, 0, ..., 0]T ∈ Rs with [p(k−1)]j = 1 if w(k−1) = wj . Note that disturbance models like ARMA models or GARCH models (Bernardini and Bemporad, 2012) can be considered as well. 2.2 Mean-Square exponential Input-To-State Stability Lemma 1. Consider the discrete-time linear system x(k + 1) = Ax(k) + B2 w(k) (4) with wT w ≤ wTmax wmax . If there exist a Lyapunov function V = xT Px with P = PT > 0, a matrix L = LT > 0, and a constant γ such that (5) Ek (V (x(k + 1))) − V (x(k)) ≤ − xT (k)Lx(k)+ γ 2 Ek (wT (k)w(k)) ∀k ∈ N0 ,
237
237
then the discrete-time linear system (4) is mean-square exponentially input-to-state stable (MSE-ISS). Note that E[0,k] (V (x(k + 1))) stands for the conditional expectation of V (x(k + 1)) with respect to the variable sequence {w(0), ..., w(k)}. Proof. Consider E[0,k] (V (x(k + 1))) = E[0,k] (V (x(k)) + V (x(k + 1)) − V (x(k))) = E[0,k−1] (V (x(k)))+ E[0,k−1] (Ek (V (x(k + 1))) − V (x(k))) ≤ E[0,k−1] (−xT (k)Lx(k)+
γ 2 Ek (wT(k)w(k)))+E[0,k−1] (V (x(k)))
≤ E[0,k−1] (V (x(k))) + γ 2 wTmax wmax − λmin (L)E[0,k−1] (xT (k)x(k))
≤ E[0,k−1] (V (x(k))) + γ 2 wTmax wmax − λmin (L) E[0,k−1] (V (x(k))) λmax (P) λmin (L) E[0,k−1] (V (x(k))) + γ 2 wTmax wmax = 1− λmax (P) where the first inequality results from (5) and the second and third inequality from the min-max theorem. By induction the inequality E[0,k] (V (x(k+1))) ≤ak+1 V (x(0))+ (6) γ 2 (ak + ak−1 + ... + 1)wTmax wmax 1 − ak+1 2 T γ wmax wmax =ak+1 V (x(0)) + 1−a min (L) with a = 1 − λλmax (P) holds where the equality follows from the geometric progression. (5) can be written as (7) Ek (V (x(k + 1))) ≤V (x(k)) − xT (k)Lx(k)+
γ 2 Ek (wT (k)w(k)) ∀k ∈ N0 . According to the definition of the Lyapunov function, Ek (V (x(k + 1))) must be positive definite. Thereby the relations λmax (P) ≥ λmin (L) and a ∈ [0, 1) are imposed. Together with (6) this leads to 1 E[0,k] (xT (k+1)x(k+1)) ≤ E[0,k] (V (x(k + 1))) λmin (P) λmax (P) k+1 T ≤ a x (0)x(0)+ (8) λmin (P) 1 1 − ak+1 2 T γ wmax wmax , λmin (P) 1 − a where the inequalities follow from the min-max theorem. In accordance with Definition 1 by Zhu et al. (2013), (8) guarantees that the discrete-time linear system (4) is MSEISS. This completes the proof. Note that (5) enforces a decrease of the Lyapunov function for −xT (k)Lx(k) + γ 2 Ek (wT (k)w(k)) < 0 (9) and allows a possible increase of the Lyapunov function for −xT (k)Lx(k) + γ 2 Ek (wT (k)w(k)) ≥ 0. (10) This implies a convergence of the states to a region C defined by C = {x(k) | xT (k)Lx(k) ≤ γ 2 Ek (wT (k)w(k))}. (11)
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Note that the expected value of the Lyapunov function decreases for x outside of the set C. The parameter γ can be regarded as a kind of “robustness”, since a smaller γ leads to a smaller set C. 3. STOCHASTIC MPC In this section the SMPC design is addressed. Aiming at obtaining a less conservative control action in comparison to RMPC approaches, the available information about the disturbance is exploited. At the same time, MSEISS, recursive feasibility and constraint satisfaction are ensured. The SMPC design is partitioned into two parts. First, an invariant set for the discrete-time linear system (1) is computed offline considering the input vector u(k) = Kx(k). (12) This step ensures that a linear feedback control law (12) is available as a valid solution with respect to the desired properties above if the system is in the invariant set. Second, an online optimization problem is formulated in which the invariant set, the state and input limitations, and the MSE-ISS are expressed as constraints.
leads to (18a)
0 Ξ T1 α1 Q α2 wT wj (B2 wj )T ≥ 0 (18b) j Q with Ξ 1 = AQ+B1 Y. The set Ω has the property that the states will remain in it if the initial state vector satisfies the condition x(0) ∈ Ω, expressed by Kothare et al. (96) as 1 xT (0) ≥ 0. (19) Q Next, let us consider componentwise peak bounds on the state and input vector, given respectively as x ∈ X = {x | |xi (k + l|k)| ≤ x ¯i , i = 1, 2, ..., n} (20a) u ∈ U = {u | |ui (k + l|k)| ≤ u ¯i , i = 1, 2, ..., m} (20b) with k, l ≥ 0. Other types of constraints will not be introduced for brevity. Substituting xi (k + l|k) by (1) and applying the triangle inequality, (20a) can be written as max |Iin (Ax(k + l − 1|k) + B1 u(k + l − 1|k)) + Iin B2 wj | l≥1
≤ max |Iin (Ax(k + l − 1|k) + B1 u(k + l − 1|k))| l≥1
3.1 Offline Computation
+
This subsection addresses the offline computation. The goal is to obtain the linear feedback control law (12). The resulting input vector will not lead to optimal performance and thus will not be employed in the online optimization. Nevertheless, a sufficient condition for the recursive feasibility is obtained which will be introduced later. The ellipsoid T
E2 ≤ 0
Ω = {x ∈ Rn |xT Q−1 x ≤ 1}
(13)
−1
with Q = Q = P is used in this paper as candidate invariant set. Assuming w(k)T w(k) ≤ wTmax wmax , due to the definition of invariance, it is required that xT (k)Q−1 x(k) ≤ 1 implies x(k + 1)Q−1 x(k + 1) ≤ 1. With the S-procedure the inequality (xT (k + 1)Q−1 x(k + 1)−1) − α1 (xT (k)Q−1 x(k) − 1)− α2 (wT (k)w(k) − β 2 ) ≤ 0 (14)
with wTmax wmax = β 2 , α1 > 0, and α2 > 0 is obtained.
Substituting (1) and (12), (14) is implied by the BMIs T −1 E1 Q E1 − α1 Q−1 ET1 Q−1 B2 wj 0 wTj (BT2 Q−1 B2 − α2 I)wj 0 ≤ 0 E2 (15) with j ∈ {1, 2, ..., s}, E1 = (A + B1 K) and E2 = −1 + α1 + α2 β 2 . Using the Schur complement, (15) can be transformed into α1 Q−1 0 0 ET1 α2 wTj wj 0 (B2 wj )T ≥ 0. (16) −E2 0 Q Using the congruence transformation diag(Q, I, I, I) and −1
K = YQ
(17) 238
max
j∈{1,...,s}
|Iin B2 wj | ≤ x ¯i ,
(21)
which means max |Iin (Ax(k + l − 1|k) + B1 u(k + l − 1|k))| l≥1
≤x ¯i −
max
j∈{1,...,s}
|Iin B2 wj |.
(22)
Using (12) and (17), from (22) the inequalities max |Iin (AQ + B1 Y)Q−1 x(k + l − 1|k)|2 l≥1
≤ max |Iin (AQ + B1 Y)Q−1 x|2 x∈Ω
1
1
≤ max (Iin (AQ + B1 Y)Q− 2 2 Q− 2 x 2 x∈Ω
1
≤ (Iin (AQ + B1 Y)Q− 2 2 (23) are obtained, where the second inequality follows from the Cauchy-Schwarz inequality. A sufficient condition for (22) is thus 1 (Iin (AQ + B1 Y)Q− 2 2 ≤(¯ xi −
max
j∈{1,...,s}
|Iin B2 wj |)2
(24)
and can be expressed as linear matrix inequalities (LMIs). Ξ T2 Q ≥ 0 ∀i ∈ {1, ..., n} (25) (¯ xi − Ξ3 )2 with Ξ 2 = IinΞ 1 and Ξ3 = maxj∈{1,...,s} |Iin B2 wj |.
By Kothare et al. (96) a sufficient condition for the fulfillment of (20b) is provided by the LMIs X Y ≥ 0 ∀i ∈ {1, ..., m} (26) Q ¯2i and X = XT . with Xii ≤ u
Finally, let us take the stability into account. Resorting to Definition 3.2 by Jiang and Wang (2001) we know that if the inequalities
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In conclusion, resorting to Lemma 2, the offline problem can be expressed as log det(Q) max
V (x(k + 1))−V (x(k)) ≤ −xT (k)Lx(k)+γ 2 wTj wj ∀k ∈ N0 (27) with j ∈ {1, ..., s} are satisfied, then ISS is guaranteed. In the online optimization, which will be introduced later, the inequalities (27) will be relaxed to ensure MSE-ISS. The inequalities (27) are implied by T −1 E1 Q E1 − Q−1 + L ET1 Q−1 B2 wj ≤ 0. wTj (BT2 Q−1 B2 − γ 2 I)wj (28)
Π ,Y,α1 ,α2 ,η Q,Π
s.t.
(35)
The inequalities (18b) are bilinear due to the term α1 Q, so that the optimization problem is extremely hard to solve, at least globally (L¨ofberg, 2004). From (18a) we know that 0 < α1 < 1. A linear search for α1 can be thus applied to treat the bilinearity. 3.2 Disturbance Model
1 2
where the variables Π = L > 0 and η = γ 2 are introduced to transform bilinear inequalities into LMIs.
By Bernardini and Bemporad (2012), Mu˜ noz de la Pe˜ na et al. (2005), Di Cairano et al. (2014) and the references therein an approach named multiple-horizon scenario tree T is proposed to model the disturbance. Assuming that the transition probability matrix T of the disturbance is available, the idea is to design a tree to describe the possible evaluation of the states. This construction scheme is applied in this paper. Note again that the stability and constraint satisfaction ensured by (35) are independent of the probability distribution p(k).
The following lemma summarizes how ISS and constraint satisfaction can be guaranteed with the linear feedback control law (12). Note again that the invariant set Ω has been introduced to enforce that the states and the inputs always stay in the feasible set. Lemma 2. If there exist positive definite symmetric matrices Q, Π , a general matrix Y, and positive scalars α1 , α2 and η such that the inequalities (18a), (18b), (19), (25), (26), and (29) hold, then the discrete-time linear system (1) is ISS and the constraints (20a) and (20b) are satisfied under the feedback control law (12).
3.3 Online Optimization This subsection addresses the online optimization, which consists in the quadratically constrained quadratic programming (QCQP) problem πi xTi Qx xi + πi uTi Qu ui + πi xTi Qs xi min U i∈S
A main point of Lemma 2 is that if the matrix inequalities above can be solved, then (12) provides a feasible solution for any k ∈ N0 in the online optimization introduced in the next subsection.
i∈T \(T1
Besides, if (12) is utilized, then ISS of the discrete-time linear system (1) can be guaranteed in the invariant set Ω by proper choices of L and γ, e.g. L = (1 − α1 )Q−1 (30) and √ γ = α2 . (31) As a result, if (15) is satisfied, then (28) holds. Similarly to (11), a maximum “origin” set Cmax = x | xT Lx ≤ γ 2 β 2 . (32) can be defined. Following (27), a convergence of the Lyapunov function V can be ensured for x ∈ / Cmax . By substituting L and γ in (32) with (30) and (31), the set can also be written as α2 β 2 T −1 . (33) Cmax = x | x Q x ≤ 1 − α1
According to (18a), it is easy to show that ≤ 1 and Cmax can be as large as the maximum invariant set Ωmax in correspondence with the initial states x(0). However, it is desired that the Lyapunov function V also decreases in some region in Ωmax . Therefore, a design parameter θ is introduced and (18a) is replaced by the sufficient condition 1 − α1 ≥ θα2 β 2 with θ > 1, (34) so that the relation Cmax ⊂ Ωmax is guaranteed. The influence will be illustrated later with a numerical example.
(34), (18b), (19), (25), (26), (29).
As shown by Boyd et al. (1998) the maximum volume of the ellipsoid can be achieved with the objective function max log det(Q). The offline problem (35) can be solved R using the toolbox Yalmip in Matlab (L¨ofberg, 2004).
With the Schur complement and the congruence transformation diag{Q, I, I}, (28) can be formulated as Π Q)T Q 0 Ξ T1 (Π ηwT w (B w )T 0 j 2 j ≥ 0 ∀j ∈ {1, ..., s}, (29) j Q 0 I
α2 β 2 1−α1
239
i∈T \S
S)
s.t. x1 = x(k) xi = Axpre(i) + B1 upre(i) + B2 wpre(i) ∀i ∈ T \ T1 xi ∈ X ∀i ∈ T \ T1 ui ∈ U ∀i ∈ T \ S (Ax1 + B1 u1 + B2 wi ) ∈ X ∀i : succ(T1 , i) ∈ / T and pi (k) > 0
(36a) (36b) (36c) (36d) (36e)
(36f)
T
(Ax1 + B1 u1 + B2 wj ) P(Ax1 + B1 u1 + B2 wj ) ≤ 1 ∀j ∈ {1, 2, ..., s} : pj (k) > 0 (36g) s pj (k)(Ax1 + B1 u1 + B2 wj )TP(Ax1 + B1 u1 + B2 wj ) j=1
≤ xT1 Px1 − xT1 Lx1 + γ 2
s
pj (k)wTj wj ,
(36h)
j=1
where pj (k) is given in (3), and T , Ti , S, pre(i), succ(Ti , j), πi , Qx , Qu , and Qs are defined as in (Bernardini and Bemporad, 2012, Section III-A2). The expected value of a quadratic objective function is minimized over the input sequence U = {u1 , u2 , ..., unmax } considering a scenario tree T with nmax nodes (36a). The measured state x(k) is regarded in (36b) and the system dynamics in (36c). The constraints (36d) and (36e) ensure satisfaction of the state and input constraints. The constraints (36f) guarantee
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satisfaction of the state constraints (20a) at time k+1 even if some disturbance realizations are not included in the scenario tree T at time k. The constraints (36g) recursively enforce that the state remains in the invariant set Ωmax computed by (35). The constraint (36h) finally enforces MSE-ISS, resorting to Lemma 1.
Proof. The key point is to show that the linear feedback control law (12) with K = YQ−1 is always a feasible solution for the QCQP (36). Using the linear feedback control law (12) at time k = 0, the states remain in the invariant set Ωmax and the constraints on the inputs and the states are satisfied at time k = 1. Moreover, MSE-ISS is ensured by (36h) due to the satisfaction of (29) for any probability distribution. By induction it can be shown that at any time k ≥ 2 the control law (12) provides a feasible solution. Therefore, recursive feasibility of the QCQP (36) is guaranteed and the proof is completed. 4. SIMULATION STUDY In this section a simulation study is presented. Particularly a comparison between the proposed SMPC and the RMPC by Yang and Feng (2013) is provided to highlight the results in this paper. 1 0.1 ] , Consider a second-order system (1) with A = [ 0.5 1 1 0.5 B1 = [ 0 ] , B2 = [ 0 ], w ∈ W = {−0.06, 0.02, 0.1}, the ¯ = xx¯¯12 = [ 10 state constraints x 10 ], the input constraint u ¯ = 1, and wmax = 0.1. The weighting matrices are chosen 0 as Qx = Qs = [ 0.1 0 0.1 ] and Qu = 0.1.
In order to assess the performance of the SMPC scheme, 500 disturbance sequences are generated according to the 0.3385 0.3662 0.2717 transition probability matrix T = 0.3160 0.3471 0.3533 , 0.3455 0.2867 0.3750 each of them consisting of 40 discrete values from W. Correspondingly, 500 initial state vectors are sampled randomly from a proper region, so that the offline computation (35) and the RMPC by Yang and Feng (2013) remain feasible. Assuming that the preceding disturbance w(k−1) is measured, the current probability distribution p(k) can be calculated from (3). A comparison among the following control schemes is performed by running 500 simulations with the given disturbance sequences and initial states: (1) The RMPC developed by Yang and Feng (2013) based on the min-max policy guarantees ISS, recursive feasibility and input constraint satisfaction, however the state constraints can not be ensured. A linear feedback control law u(k) = Kx(k) is applied to minimize the objective function, where the gain K is optimized based on the current state x(k). Only the boundary of the disturbance is taken into account. In 240
Largest invariant set Ωmax Largest origin set C
15
max,1
10
10
5
5
x2
The recursive feasibility of the QCQP (36) when applying the input vector u(k) = u1 to the discrete-time linear system (1) with an initial state x(0) ∈ Ωmax at each time k ∈ N0 is guaranteed: Theorem 1. Consider the discrete-time linear system (1) with an initial state x(0) ∈ Ωmax . If the offline problem Π , Y, α1 , α2 , η, Q), (35) is feasible and admits a solution (Π then the QCQP (36) with the input vector u(k) = u1 and k ∈ N0 is recursively feasible.
15
Largest invariant set Ωmax Largest origin set C
max,2
x2
2015 IFAC NMPC 240 September 17-20, 2015. Seville, Spain
0
0
−5
−5
−10
−10
−10
−5
0
x1
5
10
−10
−5
0
x1
5
10
Fig. 1. Left: largest invariant set Ωmax and largest origin set Cmax,1 with (18a), Right: largest invariant set Ωmax and largest origin set Cmax,2 with (34) the example the state constraints are added and set to proper values so that a violation of the constraints does not occur. (2) The proposed SMPC makes use of the scenario tree to exploit the knowledge about the disturbance. Meanwhile, state and input constraints as well as MSEISS are ensured. Based on the offline computation, the crucial aspect of recursive feasibility of the online optimization is granted. The control law for the online procedure is not predefined which leads to more degrees of freedom than the first method. The number of the nodes nmax is set to 150 so that the maximum length of the paths in the scenario tree T and thus the maximum prediction horizon is 5. The size of the maximum origin set Cmax can influence the system behavior significantly. Therefore (18a) is replaced by (34). To demonstrate the effect, the design parameter θ = 2 is chosen and the results for the initial state vector x(0) = −3.5 are shown in Fig. 1, where 3.5 46.3269 −72.1136 −3 0.3224·10−3 , Q = −72.1136 160.5318 , L = 0.7178·10 −3 −3 0.3224·10 0.2071·10 Y = [ −2.7187 −2.1379 ], α1 = 0.99, α2 = 0.5, and γ = 0.7071 are obtained from (35). In the left diagram, the set Cmax,1 is as large as the invariant set Ωmax , while in the right diagram, Cmax,2 ⊂ Ωmax holds, which ensures a robust convergence to Cmax,2 in the ellipsoid Ωmax . (3) The prescient MPC (PMPC) relies on complete knowledge of the future disturbance sequence. Therefore the “absolute” minimum is to be expected. The resulting performance can thus serve as a benchmark. The prediction horizon is chosen as 5. To evaluate the performance of the control schemes, the cost function Ts {x(k)T Qx x(k) + u(k)T Qu u(k)} (37) J= k=1
is defined, where the duration of each simulation Ts is 39 time steps. To denote the cost of the control method i for the jth simulation, the notation Ji,j with i ∈ {RMPC, PMPC, SMPC} and j ∈ {1, ..., M } is introduced. The total number of simulations M is 500. The average relative
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improvement with respect to RMPC is defined as M 1 Ji,j ∆ = 1− with i ∈ {SMPC, PMPC} (38) M j=1 JRMPC,j
and summarized in Table 1. Table 1. Average relative improvements Control scheme SMPC PMPC
Average relative improvement ∆ 2.7 % 5.0 %
The performance of PMPC represents the maximal potential improvement and 2.7% 5.0% = 54% of it can be achieved by SMPC. There are two reasons for the improvement of SMPC with respect to RMPC. First, the inclusion of knowledge about the disturbance into the online optimization (36). Second, the partition into an offline computation (35) to ensure ISS and constraint satisfaction and an online optimization (36) to improve performance, yielding an increased flexibility. The improvements of SMPC and PMPC with respect to RMPC are expected to be even higher if a stronger disturbance perturbs the system since the utilization of the knowledge about the disturbance becomes more important and thus will result in more benefits. 5. CONCLUSIONS AND FUTURE WORK In this paper an SMPC scheme for linear systems with additive disturbance has been proposed. With an offline computation based on a BMI formulation, an invariant set and a linear feedback control law are determined to guarantee ISS and constraint satisfaction. In the online optimization based on a QCQP, the probability distribution of the disturbance is regarded using a multiple-horizon scenario tree and ISS is relaxed to MSE-ISS. Recursive feasibility of the online optimization is ensured using results from the offline computation. Benefits of SMPC in comparison with RMPC are shown by a simulation study. Future work will focus on alternative disturbance models to improve performance. Furthermore, implementation aspects will be studied. ACKNOWLEDGEMENTS The authors are grateful to the colleagues Professor Dr. Jun Hu, Sven Reimann, Sanad Al-Areqi and Felix Berkel for the helpful discussions and suggestions. REFERENCES Bemporad, A., Borrelli, F., and Morari, M. (2003). Minmax control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9), 1600–1606. Bernardini, D. and Bemporad, A. (2012). Stabilizing model predictive control of stochastic constrained linear systems. IEEE Transactions on Automatic Control, 57(6), 1468–1480. Boyd, S., Vandenberghe, L., and Wu, S. (1998). Determinant maximization with linear matrix inequality constraints. SIAM Journal on Matrix Analysis and Applications, 19(2), 499–533. 241
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Calafiore, G.C. and Fagiano, L. (2013). Robust model predictive control via scenario optimization. IEEE Transactions on Automatic Control, 58(1), 219–224. Cheng, Q., Cannon, M., Kouvaritakis, B., and Evans, M. (2014). Stochastic MPC for systems with both multiplicative and additive disturbances. In Proceedings of the 19th IFAC World Congress, 2291–2296. Di Cairano, S., Bernardini, D., Bemporad, A., and Kolmanovsky, I.V. (2014). Stochastic MPC with learning for driver-predictive vehicle control and its application to HEV energy management. IEEE Transactions on Control Systems Technology, 22(3), 1018–1031. Jiang, Z. and Wang, Y. (2001). Input-to-state stability for discrete-time nonlinear systems. Automatica, 37(6), 857–869. Kothare, M., Balakrishnan, V., and Morari, M. (96). Robust constrainted model predictive control using linear matrix inequalites. Automatica, 32(10), 1361–1379. L¨ofberg, J. (2004). YALMIP : A toolbox for modeling and optimization in MATLAB. In Proceedings of the 2004 IEEE International Symposium on Computer Aided Control Systems Design, 284–289. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O.M. (2000). Contrainted model predictive control: Stability and optimality. Automatica, 36, 789–814. Mayne, D.Q., Seron, M.M., and Rakovic, S.V. (2005). Robust model predictive control of constrained linear systems with bounded disturbances. Automatica, 41(2), 219–224. Mu˜ noz de la Pe˜ na, D., Bemporad, A., and Alamo, T. (2005). Stochastic programming applied to model predictive control. In 44th IEEE Conference on Decision and Control, and the European Control Conference, 1361–1366. Parisio, A. and Glielmo, L. (2013). Stochastic model predictive control for economic/environmental operation management of microgrids. In Proceedings of the 2013 European Control Conference, 2014–2019. Qin, S.J. and Badgwell, T.A. (2003). A survey of industrial model predictive control technology. Control Engineering Practice, 11(7), 733–764. Schildbach, G., Fagiano, L., Frei, C., and Morari, M. (2014). The scenario approach for stochastic model predictive control with bounds on closed-loop constraint violations. Automatica, 50(12), 3009–3018. Scokaert, P.O.M. and Mayne, D.Q. (1998). Min-max feedback model predictive control for constrained linear systems. IEEE Transactions on Automatic Control, 43(8), 1136–1142. Yang, W. and Feng, G. (2013). Robust model predictive control of discrete-time uncertain linear systems with persistent disturbances. In Proceedings of the 2013 IEEE International Conference on Cyber Technology in Automation, Control and Intelligent Systems, 58–63. Zhang, X., Grammatico, S., Margellos, K., Goulart, P., and Lygeros, J. (2014). Randomized nonlinear MPC for uncertain control-affne systems with bounded closedloop constraint violations. In Proceedings of the 19th IFAC World Congress, 1649–1654. Zhu, Q., Cui, J.R., and Hu, G. (2013). Mean-square exponential input-to-state stability of numerical solutions for stochastic control systems. Acta Automatica Sinica, 39(8), 1360–1365.
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Stochastic MPC of Systems with Additive Stochastic Stochastic MPC MPC of of Systems Systems with with Additive Additive Disturbance using Scenario Optimization Disturbance using Scenario Optimization Disturbance using Scenario Optimization ∗∗ Xiaohai Lin ∗∗ Daniel G¨ orges ∗∗ Sebastian Caba ∗∗ ∗∗ Steven Liu ∗∗ ∗ Daniel G¨ ∗ Sebastian Caba ∗∗ ∗∗ Steven Liu ∗∗ Xiaohai Lin o rges ∗ ∗ Xiaohai Lin Lin Daniel Daniel G¨ G¨ orges rges Sebastian Sebastian Caba Caba Steven Steven Liu Liu ∗∗ Xiaohai o ∗ ∗ Juniorprofessorship for Electromobility, Department of Electrical and ∗ Juniorprofessorship for Electromobility, Department of Electrical and ∗Computer Juniorprofessorship for Electromobility, Department Electrical Engineering, of Kaiserslautern, Kaiserslautern, Juniorprofessorship for University Electromobility, Department of of Electrical and and Computer Engineering, University of Kaiserslautern, Kaiserslautern, Computer University of Germany (e-mail: lin|[email protected]) Computer Engineering, Engineering, University of Kaiserslautern, Kaiserslautern, Kaiserslautern, Kaiserslautern, Germany (e-mail: lin|[email protected]) ∗∗ Germany (e-mail: lin|[email protected]) of Control Systems, Department of Electrical and Germany (e-mail: lin|[email protected]) ∗∗ Institute ∗∗ Institute of Control Systems, Department of Electrical and ∗∗ Institute of Control Control Systems,ofDepartment Department of Electrical Electrical and Computer Engineering, University Kaiserslautern, Kaiserslautern, Institute of Systems, of and Computer Engineering, University of Kaiserslautern, Kaiserslautern, Computer of Germany University (e-mail: caba|[email protected]) Computer Engineering, Engineering, University of Kaiserslautern, Kaiserslautern, Kaiserslautern, Kaiserslautern, Germany (e-mail: caba|[email protected]) Germany Germany (e-mail: (e-mail: caba|[email protected]) caba|[email protected])
Abstract: In this paper a stochastic model predictive control (SMPC) scheme for linear systems Abstract: In this paper a stochastic model predictive control (SMPC) scheme for linear systems Abstract: In paper model predictive control scheme for linear with additive disturbance is presented. The goal is to design(SMPC) a controller that the Abstract: In this this paper a a stochastic stochastic model predictive control (SMPC) scheme for minimizes linear systems systems with additive disturbance is presented. The goal is to design a controller that minimizes the with additive additive disturbance is presented. presented. The goal goal is to to design design controllerexponential that minimizes minimizes the expected valuedisturbance of an objective function while guaranteeing mean-square input-towith is The is aa controller that the expected value of an objective function while guaranteeing mean-square exponential input-toexpected value(MSE-ISS) of an an objective objective function while while guaranteeing mean-square exponential input-tostate stability and constraints on the states andmean-square inputs. The exponential SMPC is partitioned expected value of function guaranteeing input-tostate stability (MSE-ISS) and constraints on the states and inputs. The SMPC is partitioned state stability constraints on states and The SMPC partitioned into offline (MSE-ISS) computationand based on a bilinear (BMI) ensuring statean stability (MSE-ISS) and constraints on the thematrix statesinequality and inputs. inputs. The problem SMPC is isfor partitioned into an offline computation based on a bilinear matrix inequality (BMI) problem for ensuring into an offline computation based on a bilinear matrix inequality (BMI) problem for ISS, constraint satisfaction, and recursive feasibility and an online optimization based on a into an offline computation based on a bilinear matrix inequality (BMI) problem for ensuring ensuring ISS, constraint satisfaction, and recursive feasibility and an online optimization based on ISS, constraint constraintconstrained satisfaction, and recursive recursive feasibility and an anproblem online optimization optimization based on aaa quadratically quadratic programming (QCQP) for includingbased knowledge ISS, satisfaction, and feasibility and online on quadratically constrained quadratic programming (QCQP) problem for including knowledge quadratically constrained quadratic programming (QCQP) problem for including knowledge about the additive disturbance whileprogramming relaxing ISS (QCQP) to MSE-ISS. The for partition into knowledge an offline quadratically constrained quadratic problem including about the additive disturbance while relaxing ISS to MSE-ISS. The partition into an offline about disturbance while relaxing ISS partition an computation and online optimization stability,The feasibility, andinto performance about the the additive additive disturbance while allows relaxingaddressing ISS to to MSE-ISS. MSE-ISS. The partition into an offline offline computation and online optimization allows addressing stability, feasibility, and performance computation and online allows feasibility, and separately and therefore improving performance as well stability, as handling disturbances which may computation and online optimization optimization allows addressing addressing stability, feasibility, and performance performance separately and improving performance as well as which may separately and therefore therefore improving performance as The well effectiveness as handling handling disturbances disturbances which may not be independent identically distributed (i.i.d.). of the SMPCwhich scheme is separately and therefore improving performance as well as handling disturbances may not be independent identically distributed (i.i.d.). The effectiveness of the SMPC scheme is not be independent identically distributed (i.i.d.). The effectiveness of the SMPC scheme evaluated by simulations and assessed by comparison with a robust MPC scheme. not be independent identically distributed (i.i.d.). The effectiveness of the SMPC scheme is is evaluated by simulations and assessed by comparison with aa robust MPC scheme. evaluated by simulations and assessed by comparison with robust MPC scheme. evaluated by simulations and assessed by comparison with a robust MPC scheme. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Model predictive control, stochastic control, robust control, mean-square stability, Keywords: Model Model predictive predictive control, control, stochastic control, control, robust control, control, mean-square stability, stability, Keywords: input-to-state stability Keywords: Model predictive control, stochastic stochastic control, robust robust control, mean-square mean-square stability, input-to-state stability stability input-to-state input-to-state stability 1. INTRODUCTION putational demand is often high. As an alternative, tube1. INTRODUCTION INTRODUCTION putational demand is often high. As an alternative, tube1. putational demand is high. an tubebased model predictive approaches been proposed. 1. INTRODUCTION putational demand is often often high. As As have an alternative, alternative, tubebased model predictive approaches have been proposed. based model predictive approaches have been proposed. The control presented by Mayne et al. (2005) model scheme predictive approaches have been proposed. Model predictive control (MPC) has been developed since based presented by Mayne et (2005) Model predictive predictive control control (MPC) (MPC) has has been been developed developed since since The The control controlthescheme scheme presented by of Mayne et al. al. set (2005) Model guarantees exponential stability an “origin” and control scheme presented by Mayne et al. (2005) the latepredictive seventies and has (MPC) meanwhile a great impact Model control has gained been developed since The guarantees the exponential stability of an “origin” set and the late seventies and has meanwhile gained a great impact guarantees theviolation exponential stability of an an “origin” “origin” set only and the late seventies has meanwhile gained impact prevents the of the constraints. However, the exponential stability of set and on world and Badgwell, 2003). The guarantees the the late industrial seventies and and has (Qin meanwhile gained aa great great impact prevents the violation of the constraints. However, only on the industrial world (Qin and Badgwell, 2003). The prevents the violation of the constraints. However, only on industrial Badgwell, The the boundaries of the disturbance are takenHowever, into account. the violation of the constraints. only major MPC(Qin is theand ability to deal 2003). with input on the theadvantage industrial ofworld world (Qin and Badgwell, 2003). The prevents boundaries of the disturbance are taken into account. major advantage advantage of of MPC MPC is is the the ability ability to to deal deal with with input input the of taken account. major Thisboundaries can be conservative if furtherare knowledge the the boundaries of the the disturbance disturbance are taken into intoabout account. and state constraints (Mayne etability al., 2000). Thewith rationale major advantage of MPC is the to deal input the This can be conservative if further knowledge about the and state constraints (Mayne et al., 2000). The rationale This can be conservative if further knowledge about the and state constraints (Mayne et al., 2000). The rationale disturbance is available, e.g. the probability distribution. This can be conservative if further knowledge about the behind MPC is that at every time set rationale of future disturbance is available, e.g. the probability distribution. and state constraints (Mayne et al.,instant 2000). aThe behind MPC is that at every time instant a set of future disturbance is is available, available, e.g. e.g. the the probability probability distribution. distribution. behind at time instant set of control signalsis calculated minimizing behind MPC MPC is isthat that at every every by time instant aa an set objective of future future disturbance control signals signals is is calculated calculated by by minimizing minimizing an an objective objective Recently, stochastic MPC (SMPC) has been established, control stochastic MPC (SMPC) has been function for a finite horizon considering the current state. Recently, control signals is calculated by minimizing an objective Recently, stochastic MPC (SMPC) has Schildbach been established, established, e.g. by Calafiore and Fagiano (2013), et al. stochastic MPC (SMPC) has been established, function for for aa finite finite horizon horizon considering considering the the current current state. state. Recently, function by Calafiore and Fagiano (2013), Schildbach et al. Then thefor first element of the control sequence is applied function a finite horizon considering the current state. e.g. e.g. by by Cheng Calafiore and Fagiano (2013), Schildbach et al. (2014), et al. (2014), and Zhang et al. (2014). Most e.g. Calafiore and Fagiano (2013), Schildbach et al. Then the first element of the control sequence is applied Then first Cheng et al. (2014), and Zhang et al. (2014). Most to the the plant. Then the first element element of of the the control control sequence sequence is is applied applied (2014), (2014), Cheng et al. (2014), and Zhang et al. (2014). Most approaches assume that the disturbances are independent (2014), Cheng et al. (2014), and Zhang et al. (2014). Most to the plant. to approaches assume that the disturbances are independent to the the plant. plant. assume the are identically distributed Unfortunately, the disturapproaches assume that that(i.i.d.). the disturbances disturbances are independent independent The standard MPC formulation does not directly consider approaches identically distributed (i.i.d.). Unfortunately, the The standard MPC formulation does not directly consider identically distributed (i.i.d.). Unfortunately, the disturdisturThe MPC does directly bances are partly not i.i.d. in practical applications (Paridistributed (i.i.d.). Unfortunately, the disturmodel uncertainties and disturbances (Mu˜ noz de consider la Pe˜ na identically The standard standard MPC formulation formulation does not not directly consider bances are partly not i.i.d. in practical applications (Parimodel uncertainties and disturbances (Mu˜ n oz de la Pe˜ n a bances are partly2013; not i.i.d. i.i.d. in practical practical applications (Parimodel and (Mu˜ n n sio and are Glielmo, Di Cairano et al.,applications 2014). Bernardini partly not in (Pariet al., uncertainties 2005). Naturally, the feedback mechanism can model uncertainties and disturbances disturbances (Mu˜ noz oz de de la la Pe˜ Pe˜ naa bances sio and Glielmo, 2013; Di Cairano et al., 2014). Bernardini et al., 2005). Naturally, the feedback mechanism can sio and Glielmo, 2013; Di Cairano et al., 2014). Bernardini et the feedback mechanism andand Bemporad propose a SMPC schemeBernardini for linear Glielmo,(2012) 2013; Di Cairano et al., 2014). already provideNaturally, an inherent discussed can by sio et al., al., 2005). 2005). Naturally, therobustness feedback as mechanism can Bemporad (2012) propose aa SMPC scheme for linear already provide provide an an inherent inherent robustness robustness as as discussed discussed by by and Bemporad (2012) scheme for linear already systems with multiplicative disturbance may not and Bemporad (2012) propose propose a SMPC SMPC which scheme for be linear Mayne al. (2000). However, model inaccuracies already et provide an inherent robustness as discussedand by and systems with multiplicative disturbance which may be not Mayne et al. (2000). However, model inaccuracies and systems with multiplicative disturbance which may be not Mayne et al. (2000). However, model inaccuracies and i.i.d., guaranteeing mean-square exponential stability systems with multiplicative disturbance which may be not external perturbations may still lead to loss of performance Mayne et al. (2000). However, model inaccuracies and i.i.d., guaranteeing mean-square exponential stability and and external perturbations may still lead to loss of performance i.i.d., guaranteeing guaranteeing mean-square exponentialThese stability and external may to of satisfaction of statemean-square and input constraints. fruitful exponential stability and and evenperturbations stability. Therefore, robust strategy i.i.d., external perturbations may still still alead lead to loss losscontrol of performance performance of state and input constraints. These fruitful and even even stability. stability. Therefore, Therefore, aa robust robust control control strategy strategy satisfaction of These fruitful and results can unfortunately not beconstraints. applied for linear satisfaction of state state and and input input constraints. These systems fruitful is which explicitly copes with uncertainties and satisfaction anddesired even stability. Therefore, a robust control strategy results can unfortunately not be applied for linear systems is desired which explicitly copes with uncertainties and results can unfortunately not be applied for linear systems is desired which explicitly copes with uncertainties and with additive disturbance since a robust convergence to the can unfortunately not be applied for linear systems disturbances. is desired which explicitly copes with uncertainties and results with additive disturbance since aa robust convergence to disturbances. with additive disturbance since robust convergence to the the disturbances. origin cannot be enforced. Particularly, a key component, with additive disturbance since a robust convergence to the disturbances. cannot be enforced. Particularly, a key component, Several robust MPC laws based on the min-max policy origin origin cannot set, be enforced. enforced. Particularly,inaa another key component, component, the invariant must be constructed way. origin cannot be Particularly, key Several robust MPC laws based on the min-max policy Several robust MPC laws based on policy invariant set, must be constructed in another way. have been proposed, Scokaert andmin-max Mayne (1998) Several robust MPC e.g. lawsby based on the the min-max policy the the invariant invariant set, set, must must be be constructed constructed in in another another way. way. have been been proposed, proposed, e.g. e.g. by by Scokaert Scokaert and and Mayne Mayne (1998) (1998) the have In this paper a novel SMPC scheme for linear with and Bemporad et al. (2003). The intention is to minimize have been proposed, e.g. by Scokaert and Mayne (1998) In this paper a novel SMPC scheme for linear systems systems with and Bemporad et al. (2003). The intention is to minimize In this paper a novel SMPC scheme for linear systems with and Bemporad et The is minimize additive disturbance which may be for notlinear i.i.d. systems is presented. this paper a novel SMPC scheme with the the objective function. The control In and worst-case Bemporadvalue et al. al.of(2003). (2003). The intention intention is to to minimize disturbance which may be not i.i.d. is presented. the worst-case worst-case value value of of the the objective objective function. function. The The control control additive additive disturbance which may be not i.i.d. is presented. the The SMPC scheme which consists of be annot offline disturbance may i.i.d. computation is presented. action of these value schemes mayobjective be conservative com- additive the worst-case of the function.and Thethe control SMPC scheme consists of an computation action of of these these schemes may may be conservative conservative and the the com- The The an SMPC scheme consists of the an offline offline computation, computation action and onlinescheme optimization. With SMPC consists of an offline computation action of these schemes schemes may be be conservative and and the comcom- The This work has been partly supported by the Center for Commercial and an online optimization. With the offline computation, and an online optimization. With the offline computation, input-to-state stability (ISS), constraint satisfaction, and This work has been partly supported by the Center for Commercial and an online optimization. With the offline computation, This has partly by for Vehicle Technology (ZNT) atsupported the University ofCenter Kaiserslautern funded input-to-state stability (ISS), constraint satisfaction, and This work work has been been partly supported by the the Center for Commercial Commercial input-to-state stability (ISS), constraint satisfaction, and Vehicle Technology (ZNT) at the University of Kaiserslautern funded recursive feasibility are ensured. In the online optimizainput-to-state stability (ISS), constraint satisfaction, and Vehicle Technology (ZNT) at the the University of Rhineland-Palatinate. Kaiserslautern funded funded by Research Initiative of the Federal State of Vehicle Technology (ZNT) at University of Kaiserslautern recursive feasibility are ensured. In the online optimizaby Research Initiative of the Federal State of Rhineland-Palatinate. recursive feasibility are ensured. In the online optimizarecursive feasibility are ensured. In the online optimizaby Research Initiative of the Federal State of Rhineland-Palatinate. by Research Initiative of the Federal State of Rhineland-Palatinate. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright 2015 IFAC 236 Hosting by Elsevier Ltd. All rights reserved. Copyright 2015 IFAC 236 Peer review© of International Federation of Automatic Copyright © 2015 IFAC 236 Copyright ©under 2015 responsibility IFAC 236Control. 10.1016/j.ifacol.2015.11.289
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tion, knowledge about the disturbance is included using a multi-horizon scenario tree (Bernardini and Bemporad, 2012; Mu˜ noz de la Pe˜ na et al., 2005) and ISS is relaxed to mean-square exponential ISS (MSE-ISS). The partition into an offline computation and an online optimization allows addressing stability, feasibility and performance separately. This increases the flexibility in the online optimization and thus the performance and particularly allows including disturbances which are not i.i.d. The paper is organized as follows. The system model and a sufficient condition for MSE-ISS are introduced in Section 2. The offline computation in form of BMIs and the online optimization are then formulated in Section 3. The performance of the SMPC policy is compared with the performance of the robust MPC (RMPC) approach from Yang and Feng (2013) in Section 4 based on simulations. Finally conclusions and remarks on future work are given in Section 5. Throughout the paper the following notation is used. For a vector v, [v]j is the jth element. For a matrix T, [T]ij denotes the component of the ith row and jth column. E[k1 ,k2 ] stands for the conditional expectation with respect to a variable sequence {w(k1 ), ..., w(k2 )}. Ek corresponds to E[k,k] . The maximum and the minimum eigenvalues of the matrix A are indicated by λmax (A) and λmin (A). In denotes the identity matrix of the size n × n and Iin is the ith row. Pr(·) indicates the probability, diag(·) a blockdiagonal matrix. For a symmetric matrix E, the symbol E11 E12 E11 E12 is used and E = E22 is equivalent to ET E22 . 12
2. BACKGROUND
2.1 System Model The discrete-time linear system considered in the paper is x(k + 1) = Ax(k) + B1 u(k) + B2 w(k) (1) where k ∈ N0 is the discrete time, x ∈ Rn is the state vector, u ∈ Rm is the input vector, w ∈ W ⊂ Rmw is a random disturbance vector, and W = {w1 , w2 , . . . , ws } is a predefined finite set with s elements. The disturbance vector w is modeled as a discrete-time Markov chain with a transition probability matrix T ∈ Rs×s described by [T]ij = Pr[w(k) = wi |w(k − 1) = wj ] ∀i, j ∈ {1, 2, .., s}. (2) The probability distribution of w(k) is given by p(k) = Tp(k − 1), (3) where p(k−1) = [0, ..., 0, 1, 0, ..., 0]T ∈ Rs with [p(k−1)]j = 1 if w(k−1) = wj . Note that disturbance models like ARMA models or GARCH models (Bernardini and Bemporad, 2012) can be considered as well. 2.2 Mean-Square exponential Input-To-State Stability Lemma 1. Consider the discrete-time linear system x(k + 1) = Ax(k) + B2 w(k) (4) with wT w ≤ wTmax wmax . If there exist a Lyapunov function V = xT Px with P = PT > 0, a matrix L = LT > 0, and a constant γ such that (5) Ek (V (x(k + 1))) − V (x(k)) ≤ − xT (k)Lx(k)+ γ 2 Ek (wT (k)w(k)) ∀k ∈ N0 ,
237
237
then the discrete-time linear system (4) is mean-square exponentially input-to-state stable (MSE-ISS). Note that E[0,k] (V (x(k + 1))) stands for the conditional expectation of V (x(k + 1)) with respect to the variable sequence {w(0), ..., w(k)}. Proof. Consider E[0,k] (V (x(k + 1))) = E[0,k] (V (x(k)) + V (x(k + 1)) − V (x(k))) = E[0,k−1] (V (x(k)))+ E[0,k−1] (Ek (V (x(k + 1))) − V (x(k))) ≤ E[0,k−1] (−xT (k)Lx(k)+
γ 2 Ek (wT(k)w(k)))+E[0,k−1] (V (x(k)))
≤ E[0,k−1] (V (x(k))) + γ 2 wTmax wmax − λmin (L)E[0,k−1] (xT (k)x(k))
≤ E[0,k−1] (V (x(k))) + γ 2 wTmax wmax − λmin (L) E[0,k−1] (V (x(k))) λmax (P) λmin (L) E[0,k−1] (V (x(k))) + γ 2 wTmax wmax = 1− λmax (P) where the first inequality results from (5) and the second and third inequality from the min-max theorem. By induction the inequality E[0,k] (V (x(k+1))) ≤ak+1 V (x(0))+ (6) γ 2 (ak + ak−1 + ... + 1)wTmax wmax 1 − ak+1 2 T γ wmax wmax =ak+1 V (x(0)) + 1−a min (L) with a = 1 − λλmax (P) holds where the equality follows from the geometric progression. (5) can be written as (7) Ek (V (x(k + 1))) ≤V (x(k)) − xT (k)Lx(k)+
γ 2 Ek (wT (k)w(k)) ∀k ∈ N0 . According to the definition of the Lyapunov function, Ek (V (x(k + 1))) must be positive definite. Thereby the relations λmax (P) ≥ λmin (L) and a ∈ [0, 1) are imposed. Together with (6) this leads to 1 E[0,k] (xT (k+1)x(k+1)) ≤ E[0,k] (V (x(k + 1))) λmin (P) λmax (P) k+1 T ≤ a x (0)x(0)+ (8) λmin (P) 1 1 − ak+1 2 T γ wmax wmax , λmin (P) 1 − a where the inequalities follow from the min-max theorem. In accordance with Definition 1 by Zhu et al. (2013), (8) guarantees that the discrete-time linear system (4) is MSEISS. This completes the proof. Note that (5) enforces a decrease of the Lyapunov function for −xT (k)Lx(k) + γ 2 Ek (wT (k)w(k)) < 0 (9) and allows a possible increase of the Lyapunov function for −xT (k)Lx(k) + γ 2 Ek (wT (k)w(k)) ≥ 0. (10) This implies a convergence of the states to a region C defined by C = {x(k) | xT (k)Lx(k) ≤ γ 2 Ek (wT (k)w(k))}. (11)
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Note that the expected value of the Lyapunov function decreases for x outside of the set C. The parameter γ can be regarded as a kind of “robustness”, since a smaller γ leads to a smaller set C. 3. STOCHASTIC MPC In this section the SMPC design is addressed. Aiming at obtaining a less conservative control action in comparison to RMPC approaches, the available information about the disturbance is exploited. At the same time, MSEISS, recursive feasibility and constraint satisfaction are ensured. The SMPC design is partitioned into two parts. First, an invariant set for the discrete-time linear system (1) is computed offline considering the input vector u(k) = Kx(k). (12) This step ensures that a linear feedback control law (12) is available as a valid solution with respect to the desired properties above if the system is in the invariant set. Second, an online optimization problem is formulated in which the invariant set, the state and input limitations, and the MSE-ISS are expressed as constraints.
leads to (18a)
0 Ξ T1 α1 Q α2 wT wj (B2 wj )T ≥ 0 (18b) j Q with Ξ 1 = AQ+B1 Y. The set Ω has the property that the states will remain in it if the initial state vector satisfies the condition x(0) ∈ Ω, expressed by Kothare et al. (96) as 1 xT (0) ≥ 0. (19) Q Next, let us consider componentwise peak bounds on the state and input vector, given respectively as x ∈ X = {x | |xi (k + l|k)| ≤ x ¯i , i = 1, 2, ..., n} (20a) u ∈ U = {u | |ui (k + l|k)| ≤ u ¯i , i = 1, 2, ..., m} (20b) with k, l ≥ 0. Other types of constraints will not be introduced for brevity. Substituting xi (k + l|k) by (1) and applying the triangle inequality, (20a) can be written as max |Iin (Ax(k + l − 1|k) + B1 u(k + l − 1|k)) + Iin B2 wj | l≥1
≤ max |Iin (Ax(k + l − 1|k) + B1 u(k + l − 1|k))| l≥1
3.1 Offline Computation
+
This subsection addresses the offline computation. The goal is to obtain the linear feedback control law (12). The resulting input vector will not lead to optimal performance and thus will not be employed in the online optimization. Nevertheless, a sufficient condition for the recursive feasibility is obtained which will be introduced later. The ellipsoid T
E2 ≤ 0
Ω = {x ∈ Rn |xT Q−1 x ≤ 1}
(13)
−1
with Q = Q = P is used in this paper as candidate invariant set. Assuming w(k)T w(k) ≤ wTmax wmax , due to the definition of invariance, it is required that xT (k)Q−1 x(k) ≤ 1 implies x(k + 1)Q−1 x(k + 1) ≤ 1. With the S-procedure the inequality (xT (k + 1)Q−1 x(k + 1)−1) − α1 (xT (k)Q−1 x(k) − 1)− α2 (wT (k)w(k) − β 2 ) ≤ 0 (14)
with wTmax wmax = β 2 , α1 > 0, and α2 > 0 is obtained.
Substituting (1) and (12), (14) is implied by the BMIs T −1 E1 Q E1 − α1 Q−1 ET1 Q−1 B2 wj 0 wTj (BT2 Q−1 B2 − α2 I)wj 0 ≤ 0 E2 (15) with j ∈ {1, 2, ..., s}, E1 = (A + B1 K) and E2 = −1 + α1 + α2 β 2 . Using the Schur complement, (15) can be transformed into α1 Q−1 0 0 ET1 α2 wTj wj 0 (B2 wj )T ≥ 0. (16) −E2 0 Q Using the congruence transformation diag(Q, I, I, I) and −1
K = YQ
(17) 238
max
j∈{1,...,s}
|Iin B2 wj | ≤ x ¯i ,
(21)
which means max |Iin (Ax(k + l − 1|k) + B1 u(k + l − 1|k))| l≥1
≤x ¯i −
max
j∈{1,...,s}
|Iin B2 wj |.
(22)
Using (12) and (17), from (22) the inequalities max |Iin (AQ + B1 Y)Q−1 x(k + l − 1|k)|2 l≥1
≤ max |Iin (AQ + B1 Y)Q−1 x|2 x∈Ω
1
1
≤ max (Iin (AQ + B1 Y)Q− 2 2 Q− 2 x 2 x∈Ω
1
≤ (Iin (AQ + B1 Y)Q− 2 2 (23) are obtained, where the second inequality follows from the Cauchy-Schwarz inequality. A sufficient condition for (22) is thus 1 (Iin (AQ + B1 Y)Q− 2 2 ≤(¯ xi −
max
j∈{1,...,s}
|Iin B2 wj |)2
(24)
and can be expressed as linear matrix inequalities (LMIs). Ξ T2 Q ≥ 0 ∀i ∈ {1, ..., n} (25) (¯ xi − Ξ3 )2 with Ξ 2 = IinΞ 1 and Ξ3 = maxj∈{1,...,s} |Iin B2 wj |.
By Kothare et al. (96) a sufficient condition for the fulfillment of (20b) is provided by the LMIs X Y ≥ 0 ∀i ∈ {1, ..., m} (26) Q ¯2i and X = XT . with Xii ≤ u
Finally, let us take the stability into account. Resorting to Definition 3.2 by Jiang and Wang (2001) we know that if the inequalities
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In conclusion, resorting to Lemma 2, the offline problem can be expressed as log det(Q) max
V (x(k + 1))−V (x(k)) ≤ −xT (k)Lx(k)+γ 2 wTj wj ∀k ∈ N0 (27) with j ∈ {1, ..., s} are satisfied, then ISS is guaranteed. In the online optimization, which will be introduced later, the inequalities (27) will be relaxed to ensure MSE-ISS. The inequalities (27) are implied by T −1 E1 Q E1 − Q−1 + L ET1 Q−1 B2 wj ≤ 0. wTj (BT2 Q−1 B2 − γ 2 I)wj (28)
Π ,Y,α1 ,α2 ,η Q,Π
s.t.
(35)
The inequalities (18b) are bilinear due to the term α1 Q, so that the optimization problem is extremely hard to solve, at least globally (L¨ofberg, 2004). From (18a) we know that 0 < α1 < 1. A linear search for α1 can be thus applied to treat the bilinearity. 3.2 Disturbance Model
1 2
where the variables Π = L > 0 and η = γ 2 are introduced to transform bilinear inequalities into LMIs.
By Bernardini and Bemporad (2012), Mu˜ noz de la Pe˜ na et al. (2005), Di Cairano et al. (2014) and the references therein an approach named multiple-horizon scenario tree T is proposed to model the disturbance. Assuming that the transition probability matrix T of the disturbance is available, the idea is to design a tree to describe the possible evaluation of the states. This construction scheme is applied in this paper. Note again that the stability and constraint satisfaction ensured by (35) are independent of the probability distribution p(k).
The following lemma summarizes how ISS and constraint satisfaction can be guaranteed with the linear feedback control law (12). Note again that the invariant set Ω has been introduced to enforce that the states and the inputs always stay in the feasible set. Lemma 2. If there exist positive definite symmetric matrices Q, Π , a general matrix Y, and positive scalars α1 , α2 and η such that the inequalities (18a), (18b), (19), (25), (26), and (29) hold, then the discrete-time linear system (1) is ISS and the constraints (20a) and (20b) are satisfied under the feedback control law (12).
3.3 Online Optimization This subsection addresses the online optimization, which consists in the quadratically constrained quadratic programming (QCQP) problem πi xTi Qx xi + πi uTi Qu ui + πi xTi Qs xi min U i∈S
A main point of Lemma 2 is that if the matrix inequalities above can be solved, then (12) provides a feasible solution for any k ∈ N0 in the online optimization introduced in the next subsection.
i∈T \(T1
Besides, if (12) is utilized, then ISS of the discrete-time linear system (1) can be guaranteed in the invariant set Ω by proper choices of L and γ, e.g. L = (1 − α1 )Q−1 (30) and √ γ = α2 . (31) As a result, if (15) is satisfied, then (28) holds. Similarly to (11), a maximum “origin” set Cmax = x | xT Lx ≤ γ 2 β 2 . (32) can be defined. Following (27), a convergence of the Lyapunov function V can be ensured for x ∈ / Cmax . By substituting L and γ in (32) with (30) and (31), the set can also be written as α2 β 2 T −1 . (33) Cmax = x | x Q x ≤ 1 − α1
According to (18a), it is easy to show that ≤ 1 and Cmax can be as large as the maximum invariant set Ωmax in correspondence with the initial states x(0). However, it is desired that the Lyapunov function V also decreases in some region in Ωmax . Therefore, a design parameter θ is introduced and (18a) is replaced by the sufficient condition 1 − α1 ≥ θα2 β 2 with θ > 1, (34) so that the relation Cmax ⊂ Ωmax is guaranteed. The influence will be illustrated later with a numerical example.
(34), (18b), (19), (25), (26), (29).
As shown by Boyd et al. (1998) the maximum volume of the ellipsoid can be achieved with the objective function max log det(Q). The offline problem (35) can be solved R using the toolbox Yalmip in Matlab (L¨ofberg, 2004).
With the Schur complement and the congruence transformation diag{Q, I, I}, (28) can be formulated as Π Q)T Q 0 Ξ T1 (Π ηwT w (B w )T 0 j 2 j ≥ 0 ∀j ∈ {1, ..., s}, (29) j Q 0 I
α2 β 2 1−α1
239
i∈T \S
S)
s.t. x1 = x(k) xi = Axpre(i) + B1 upre(i) + B2 wpre(i) ∀i ∈ T \ T1 xi ∈ X ∀i ∈ T \ T1 ui ∈ U ∀i ∈ T \ S (Ax1 + B1 u1 + B2 wi ) ∈ X ∀i : succ(T1 , i) ∈ / T and pi (k) > 0
(36a) (36b) (36c) (36d) (36e)
(36f)
T
(Ax1 + B1 u1 + B2 wj ) P(Ax1 + B1 u1 + B2 wj ) ≤ 1 ∀j ∈ {1, 2, ..., s} : pj (k) > 0 (36g) s pj (k)(Ax1 + B1 u1 + B2 wj )TP(Ax1 + B1 u1 + B2 wj ) j=1
≤ xT1 Px1 − xT1 Lx1 + γ 2
s
pj (k)wTj wj ,
(36h)
j=1
where pj (k) is given in (3), and T , Ti , S, pre(i), succ(Ti , j), πi , Qx , Qu , and Qs are defined as in (Bernardini and Bemporad, 2012, Section III-A2). The expected value of a quadratic objective function is minimized over the input sequence U = {u1 , u2 , ..., unmax } considering a scenario tree T with nmax nodes (36a). The measured state x(k) is regarded in (36b) and the system dynamics in (36c). The constraints (36d) and (36e) ensure satisfaction of the state and input constraints. The constraints (36f) guarantee
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satisfaction of the state constraints (20a) at time k+1 even if some disturbance realizations are not included in the scenario tree T at time k. The constraints (36g) recursively enforce that the state remains in the invariant set Ωmax computed by (35). The constraint (36h) finally enforces MSE-ISS, resorting to Lemma 1.
Proof. The key point is to show that the linear feedback control law (12) with K = YQ−1 is always a feasible solution for the QCQP (36). Using the linear feedback control law (12) at time k = 0, the states remain in the invariant set Ωmax and the constraints on the inputs and the states are satisfied at time k = 1. Moreover, MSE-ISS is ensured by (36h) due to the satisfaction of (29) for any probability distribution. By induction it can be shown that at any time k ≥ 2 the control law (12) provides a feasible solution. Therefore, recursive feasibility of the QCQP (36) is guaranteed and the proof is completed. 4. SIMULATION STUDY In this section a simulation study is presented. Particularly a comparison between the proposed SMPC and the RMPC by Yang and Feng (2013) is provided to highlight the results in this paper. 1 0.1 ] , Consider a second-order system (1) with A = [ 0.5 1 1 0.5 B1 = [ 0 ] , B2 = [ 0 ], w ∈ W = {−0.06, 0.02, 0.1}, the ¯ = xx¯¯12 = [ 10 state constraints x 10 ], the input constraint u ¯ = 1, and wmax = 0.1. The weighting matrices are chosen 0 as Qx = Qs = [ 0.1 0 0.1 ] and Qu = 0.1.
In order to assess the performance of the SMPC scheme, 500 disturbance sequences are generated according to the 0.3385 0.3662 0.2717 transition probability matrix T = 0.3160 0.3471 0.3533 , 0.3455 0.2867 0.3750 each of them consisting of 40 discrete values from W. Correspondingly, 500 initial state vectors are sampled randomly from a proper region, so that the offline computation (35) and the RMPC by Yang and Feng (2013) remain feasible. Assuming that the preceding disturbance w(k−1) is measured, the current probability distribution p(k) can be calculated from (3). A comparison among the following control schemes is performed by running 500 simulations with the given disturbance sequences and initial states: (1) The RMPC developed by Yang and Feng (2013) based on the min-max policy guarantees ISS, recursive feasibility and input constraint satisfaction, however the state constraints can not be ensured. A linear feedback control law u(k) = Kx(k) is applied to minimize the objective function, where the gain K is optimized based on the current state x(k). Only the boundary of the disturbance is taken into account. In 240
Largest invariant set Ωmax Largest origin set C
15
max,1
10
10
5
5
x2
The recursive feasibility of the QCQP (36) when applying the input vector u(k) = u1 to the discrete-time linear system (1) with an initial state x(0) ∈ Ωmax at each time k ∈ N0 is guaranteed: Theorem 1. Consider the discrete-time linear system (1) with an initial state x(0) ∈ Ωmax . If the offline problem Π , Y, α1 , α2 , η, Q), (35) is feasible and admits a solution (Π then the QCQP (36) with the input vector u(k) = u1 and k ∈ N0 is recursively feasible.
15
Largest invariant set Ωmax Largest origin set C
max,2
x2
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0
0
−5
−5
−10
−10
−10
−5
0
x1
5
10
−10
−5
0
x1
5
10
Fig. 1. Left: largest invariant set Ωmax and largest origin set Cmax,1 with (18a), Right: largest invariant set Ωmax and largest origin set Cmax,2 with (34) the example the state constraints are added and set to proper values so that a violation of the constraints does not occur. (2) The proposed SMPC makes use of the scenario tree to exploit the knowledge about the disturbance. Meanwhile, state and input constraints as well as MSEISS are ensured. Based on the offline computation, the crucial aspect of recursive feasibility of the online optimization is granted. The control law for the online procedure is not predefined which leads to more degrees of freedom than the first method. The number of the nodes nmax is set to 150 so that the maximum length of the paths in the scenario tree T and thus the maximum prediction horizon is 5. The size of the maximum origin set Cmax can influence the system behavior significantly. Therefore (18a) is replaced by (34). To demonstrate the effect, the design parameter θ = 2 is chosen and the results for the initial state vector x(0) = −3.5 are shown in Fig. 1, where 3.5 46.3269 −72.1136 −3 0.3224·10−3 , Q = −72.1136 160.5318 , L = 0.7178·10 −3 −3 0.3224·10 0.2071·10 Y = [ −2.7187 −2.1379 ], α1 = 0.99, α2 = 0.5, and γ = 0.7071 are obtained from (35). In the left diagram, the set Cmax,1 is as large as the invariant set Ωmax , while in the right diagram, Cmax,2 ⊂ Ωmax holds, which ensures a robust convergence to Cmax,2 in the ellipsoid Ωmax . (3) The prescient MPC (PMPC) relies on complete knowledge of the future disturbance sequence. Therefore the “absolute” minimum is to be expected. The resulting performance can thus serve as a benchmark. The prediction horizon is chosen as 5. To evaluate the performance of the control schemes, the cost function Ts {x(k)T Qx x(k) + u(k)T Qu u(k)} (37) J= k=1
is defined, where the duration of each simulation Ts is 39 time steps. To denote the cost of the control method i for the jth simulation, the notation Ji,j with i ∈ {RMPC, PMPC, SMPC} and j ∈ {1, ..., M } is introduced. The total number of simulations M is 500. The average relative
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improvement with respect to RMPC is defined as M 1 Ji,j ∆ = 1− with i ∈ {SMPC, PMPC} (38) M j=1 JRMPC,j
and summarized in Table 1. Table 1. Average relative improvements Control scheme SMPC PMPC
Average relative improvement ∆ 2.7 % 5.0 %
The performance of PMPC represents the maximal potential improvement and 2.7% 5.0% = 54% of it can be achieved by SMPC. There are two reasons for the improvement of SMPC with respect to RMPC. First, the inclusion of knowledge about the disturbance into the online optimization (36). Second, the partition into an offline computation (35) to ensure ISS and constraint satisfaction and an online optimization (36) to improve performance, yielding an increased flexibility. The improvements of SMPC and PMPC with respect to RMPC are expected to be even higher if a stronger disturbance perturbs the system since the utilization of the knowledge about the disturbance becomes more important and thus will result in more benefits. 5. CONCLUSIONS AND FUTURE WORK In this paper an SMPC scheme for linear systems with additive disturbance has been proposed. With an offline computation based on a BMI formulation, an invariant set and a linear feedback control law are determined to guarantee ISS and constraint satisfaction. In the online optimization based on a QCQP, the probability distribution of the disturbance is regarded using a multiple-horizon scenario tree and ISS is relaxed to MSE-ISS. Recursive feasibility of the online optimization is ensured using results from the offline computation. Benefits of SMPC in comparison with RMPC are shown by a simulation study. Future work will focus on alternative disturbance models to improve performance. Furthermore, implementation aspects will be studied. ACKNOWLEDGEMENTS The authors are grateful to the colleagues Professor Dr. Jun Hu, Sven Reimann, Sanad Al-Areqi and Felix Berkel for the helpful discussions and suggestions. REFERENCES Bemporad, A., Borrelli, F., and Morari, M. (2003). Minmax control of constrained uncertain discrete-time linear systems. IEEE Transactions on Automatic Control, 48(9), 1600–1606. Bernardini, D. and Bemporad, A. (2012). Stabilizing model predictive control of stochastic constrained linear systems. IEEE Transactions on Automatic Control, 57(6), 1468–1480. Boyd, S., Vandenberghe, L., and Wu, S. (1998). Determinant maximization with linear matrix inequality constraints. SIAM Journal on Matrix Analysis and Applications, 19(2), 499–533. 241
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