Stochastic Nonlinear Model Predictive Control with Probabilistic Constraints

Transcription

1 14 American Control Conference (ACC June 4-6, 14. Portland, Oregon, USA Stochastic Nonlinear Model Predictive Control with Probabilistic Constraints Ali Mesbah 1, Stefan Streif 1,2, Rolf Findeisen 2, and Richard D. Braatz 1 Abstract Stochastic uncertainties are ubiquitous in complex dynamical systems and can lead to undesired variability of system outputs and, therefore, a notable degradation of closedloop performance. This paper investigates model predictive control of nonlinear dynamical systems subject to probabilistic parametric uncertainties. A nonlinear model predictive control framework is presented for control of the probability distribution of system states while ensuring the satisfaction of constraints with some desired probability levels. To obtain a computationally tractable formulation for real control applications, polynomial chaos expansions are utilized to propagate the probabilistic parametric uncertainties through the system model. The paper considers individual probabilistic constraints, which are converted explicitly into convex second-order cone constraints for a general class of probability distributions. An algorithm is presented for receding horizon implementation of the finite-horizon stochastic optimal control problem. The capability of the stochastic model predictive control approach in terms of shaping the probability distribution of system states and fulfilling state constraints in a stochastic setting is demonstrated for optimal control of polymorphic transformation in batch crystallization. I. INTRODUCTION Parametric uncertainties and exogenous disturbances are ubiquitous in complex dynamical systems and render the process stochastic. Deterministic robust model predictive control (MPC approaches have been investigated to address the receding horizon optimal control problem for uncertain systems (e.g., see [1], [2] and the citations therein. If the uncertainties are considered to be in bounded sets, robust MPC formulations enable analyzing the stability and performance of the system against worst-case perturbations. This has led to a number of applications, where providing guarantees on robust stability and performance is the main objective [1], [2]. However, the worst-case perturbations may have a vanishingly small probability of occurrence in real practice [3], so control design based on worst-case uncertainties can be too conservative and, therefore, may lead to infeasible control problems. In addition, it is seldom practical to specify precisely the uncertainty bounds for complex dynamical systems. If the actual uncertainty realizations are larger than those assumed a priori, the guarantees on robust stability and performance will be diminished; conversely, exceedingly Financial support is acknowledged from Novartis Pharma AG, and from the Research Center Dynamic Systems, Saxony-Anhalt, Germany. 1 Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. 2 Otto-von-Guericke University Magdeburg, Universitätsplatz 2, 398 Magdeburg, Germany. s: amesbah@mit.edu, stefan.streif@ovgu.de, rolf.findeisen@ovgu.edu, and braatz@mit.edu. large uncertainty bounds will lead to loss in the achievable closed-loop performance. In contrast to deterministic robust MPC, stochastic MPC (SMPC with probabilistic constraints (aka chance constraints takes a different approach to solve MPC problems under uncertainties. SMPC utilizes probabilistic descriptions of uncertainties and constraint violations and allows accounting for acceptable levels of risk during system operation. SMPC with probabilistic constraints is intended to control the predicted probability distributions of system states in some optimal manner over a finite prediction horizon, while ensuring the satisfaction of constraints with a desired probability level (e.g., see [4], [5], [6], [7], [8], [9], [], [11], [12]. Probabilistic constraints enable the SMPC formulation to directly incorporate the tradeoffs between the satisfaction of constraints and closed-loop performance and, therefore, alleviate the inherent conservatism of worst-case MPC. In addition, the probabilistic framework of SMPC allows the shaping of the probability distributions of system states (e.g., rather than merely optimizing their expected value and/or variance, which can have significant safety and economic implications for the operation of complex dynamical processes. SMPC formulations are particularly advantageous when the control cost functions are asymmetric and the highperformance system operation is realized in the vicinity of constraints in a stochastic setting [13]. The propagation of probabilistic parameter uncertainties and exogenous disturbances through the system model and the reformulation of probabilistic constraints to computationally tractable expressions are key issues in SMPC for real applications. A common approach for probabilistic analysis and control of uncertain systems is to use samplingbased techniques, which solve convex chance constrained optimization problems (e.g., [14], [], [16], [17]. However, these approaches can be prohibitively expensive for realtime control due to the large number of samples required for uncertainty propagation. Additional complexity in SMPC arises from probabilistic constraints, which involve the computation of multivariate integrals. In [4], [5], [11], [18], the probabilistic constrains are transformed into deterministic expressions to facilitate solution of the optimal control problem. However, these analytic approximations are restrictive, as they hold only for certain classes of probability distributions and are limited to convex problems. The restriction with respect to special distributions is alleviated in sampling-based methods, where so-called scenarios are used to formulate an optimization problem that replaces the probabilistic constrained control /$ AACC 2413

2 problem [3], [19], [], [21], [22]. These methods can, however, result in conservative formulations unless special sampling techniques [9], which may require a prohibitive number of samples, are employed. Recently, [23] proposed an adaptive SMPC scheme that starts with a conservative probabilistic constrained formulation and, subsequently, adapts the constraints formulation online based on the experienced constraint violation frequencies. Guarantees of convergence to the desired level of constraints violation are established for a special class of linear systems. In this paper, the generalized polynomial chaos (PC theory is employed to devise a stochastic MPC formulation for a general class of nonlinear systems that are subject to parametric uncertainties. The PC theory [24], [], [26] provides computationally efficient spectral tools to replace or accelerate Monte Carlo-based uncertainty analysis methods. In the PC approach, the implicit mappings between the uncertain variables/parameters and the states (defined in terms of differential equations are replaced with explicit functions in the form of a series of orthogonal polynomials, whose statistical moments can be readily computed from the expansion coefficients. This approach is distinct from past works on the use of PC expansions for probabilistic simulation and controller synthesis of uncertain systems [], [11], [13], [27], [28], [29] in terms of the general framework proposed for stochastic nonlinear MPC with probabilistic constraints. The remainder of the paper is structured as follows. In Sec. II, the stochastic nonlinear optimal control problem is formulated to design the probability distributions of the system states, while satisfying probabilistic constraints on states. The PC expansion is introduced in Sec. III. An algorithm is presented in Sec. IV to facilitate the receding horizon implementation of the SMPC by updating the coefficients of PC expansions online at discrete time instants. To obtain a computationally tractable formulation, individual probabilistic constraints are converted explicitly into convex second-order cone constraints for a general class of probability distributions with known mean and covariance [30]. In Sec. V, the stochastic nonlinear MPC approach with probabilistic constraints is applied for optimal control of polymorphic transformation in batch crystallization, which is a process that most commonly occurs in the pharmaceutical industry. II. PROBLEM FORMULATION Consider an uncertain, discrete-time nonlinear system x t+1 = f(x t, u t, θ (1 where t is the discrete time, x t := [x 1 (t,, x nx (t] R nx denotes the system states, u t R nu denotes the control inputs, θ R n denotes the uncertain system parameters, and f : R nx R nu R n R nx denotes the nonlinear system dynamics. A probability triple (Ω, F, P is defined on the basis of sample space Ω, σ-algebra F, and probability measure P on (Ω, F. The parameter vector θ is composed of independent distributed random variables θ i, with known probability distribution functions (pdfs f θi, such that θ i L 2 (Ω, F, P i {1,, n}. Here, L 2 (Ω, F, P is the Hilbert space of all random variables θ i, whose L 2 -norm is finite. The expected value (first-order moment of a stochastic variable ψ : Ω R is denoted as E[ψ] := Ω ψdf ψ, where f ψ is the pdf of ψ over its support Ω. The variance (central second-order moment of a stochastic variable ψ is represented by Var[ψ] := E[(ψ E[ψ] 2 ]. The solution of (1 is assumed to exist and to be unique with probability one for any initial condition x 0 and any feasible input. Under the uncertainties in the system parameters, the solution trajectories of system (1 are stochastic processes. The goal of controller synthesis is to shape the probability distributions of x to possess desirable statistics. Assuming that the system states can be measured at all times, the finite-horizon SMPC problem with probabilistic constraints is stated as follows. Problem 1 (Finite-horizon stochastic MPC with probabilistic constraints: min u N J(x(k, u N subject to: x t+1 = f( x t, u t, θ, t = 0,..., N h( x t, u t, θ 0, t = 0,..., N Pr[g( x t 0] β, t = 0,..., N u t U, t = 0,..., N x 0 = x(k where u N := [u 0,..., u N ] denotes the decision variables (control policy, x t denotes the states predicted by the nonlinear system model, x(k denotes the measured states at discrete time instant k, h : R nx R nu R n R n h denotes the hard inequality constraints, Pr denotes probability, g : R nx R ng denotes the joint probabilistic constraints on the states, β (0, 1 R denotes the lower bound of the desired joint probability that the state constraints should satisfy under uncertainties, and U R nu denotes the convex compact set of input constraints. For the nonlinear system (1, Problem 1 considers joint probabilistic constraints on the states, hard constraints on the inputs, and general inequality constraints. In (2, the cost function J(x(t, u N can be defined in terms of some statistics or moments of the pdfs of the states x or, alternatively, in terms of some metrics (e.g., Wasserstein distance [31], [32] on the states pdfs when explicitly available. Using the latter in the cost function, the difference between the shapes of states pdfs and their desired pdfs can be quantified, which enables the shaping of the probability distributions of x. Sampling from the uncertain parameter distributions and, subsequently, simulating the system (1 can be used to derive the pdfs of states. To render the online solution of (2 computationally feasible, it is proposed to use PC expansions to propagate the parametric uncertainties through (1. Next, the main principles of PC expansions are introduced. III. POLYNOMIAL CHAOS EXPANSIONS Polynomial chaos expansions provide means for the approximation of stochastic variables with finite second- (2 2414

3 order moments [24], [33]. A stochastic variable ψ(θ L 2 (Ω, F, P has the L 2 -convergent expansion [26] ψ(θ = a k Φ αk (θ (3 k=0 where a k denotes the expansion coefficients and Φ αk (θ := n i=1 Φ α (θ i,k i denotes the multivariate polynomials with Φ αi,k being univariate polynomials in θ i of degree α i,k. The choice of the univariate polynomials Φ αi,k is made in accordance with the distribution of random variables θ i (e.g., Hermite polynomials are utilized for Gaussian random variables []. The polynomials should satisfy the orthogonality property Ω Φ α i Φ αj df θ = h 2 i δ ij, where δ ij is the Kronecker delta and h i is a constant term that corresponds to Ω Φ2 α i df θ. For practical reasons, the PC expansion (3 needs to be truncated. The total number L of terms in the truncated expansion depends on the number of uncertain parameters and the order m (i.e., n i=1 α i,k m α k of expansion (e.g., see [] L = (n + m!. (4 n!m! Hence, the truncated PC expansion takes the form L 1 ˆψ(θ := a k Φ αk (θ = a Λ(θ (5 k=0 with Λ(θ := [Φ α0,, Φ αl 1 ] and a := [a 0,, a L 1 ]. The PC expansion is convergent in the mean-square sense []. Probabilistic collocation methods are commonly used to compute the coefficients a for a nonlinear system (e.g., see [], [13], [34], [35]. These methods essentially consist of the least-squares estimation of the coefficients from a finite data set (i.e., a set of collocation points, requiring the residuals between the PC expansion and the nonlinear model predictions to be zero at the collocation points. The selection of the collocation points can greatly affect the accuracy of the PC approximation. A common choice for collocation points is the roots of the orthogonal polynomial of one degree higher than the order of the PC expansion [36]. Once the coefficients a in (5 are computed, the orthogonality property of multivariate polynomials can be exploited to efficiently compute the statistics of the stochastic variable ˆψ(θ. For instance, the first- and second-order moments of ˆψ(θ are defined by E[ ˆψ(θ] = a 0 (6 Var[ ˆψ(θ] L 1 = a 2 ke[φ 2 α k (θ] (7 k=1 where the terms E[Φ 2 α k (θ] k = 1,, L 1 are calculated only once and offline. These moments, along with the higher order moments [28], can be used to approximate the pdf of the stochastic variable ˆψ(θ. Next, PC expansions are used to present a tractable formulation for Problem 1. IV. STOCHASTIC MODEL PREDICTIVE CONTROL The most computationally efficient way to solve the probabilistic control Problem 1 for online control applications is to recast (2 in terms of a deterministic optimal control problem. Theorem 1 (Distributionally robust probabilistic constraints [30]: Consider a single generic probabilistic constraint of the form Pr[l ν 0] 1 ɛ, ɛ (0, 1 (8 where l R n l denotes some random quantities and ν R n l denotes some variables. Denote with L the family of all distributions with known mean l and covariance Γ. For any ɛ (0, 1, the distributionally robust probabilistic constraint inf l L Pr[l ν 0] 1 ɛ (where l L denotes that the distribution of l belongs to the family L is equivalent to the convex second-order cone constraint κ ɛ Var[l ν] + E[l ν] 0, κ ɛ = (1 ɛ/ɛ where E[l ν] = l ν and Var[l ν] = ν Γν. Thm. 1 provides an explicit expression for the probabilistic constraint (8, which is enforced robustly with respect to an entire family of probability distributions on random quantities l. Assumption 1: In Problem 1, the probabilistic constraints on states are assumed to consist of merely individual constraints, each of which is a linear function of one state variable. Hence, the probabilistic constraints in (2 take the form Pr[c i x i (t + d i 0] β i, i I (9 where { x i (t}, i I {1,..., n x } is a subset of x t and β i (0, 1 denotes user-specified lower bounds for the individual probabilistic constraints (note that the choice of β i is independent of β in (2. By approximating { x i (t}, i I in terms of PC expansions (5 (i.e., x i (t ˆx i (t, (9 is rewritten as Pr[c i a t,iλ(θ + d i 0] β i, i I ( where a t,i denotes the coefficients of PC expansions corresponding to x i (t at different times. Thm. 1 can now be used to convert the individual probabilistic constraints ( into deterministic constraints c i (κ 1 βi Var [ˆx i (t ] + E [ˆx i (t ] + d i 0, i I (11 where E[ˆx i ] and Var[ˆx i ], i I are computed using (6 and (7, respectively. Using PC expansions to propagate the parametric uncertainties through the nonlinear model (1 and explicit expressions (11 to replace the probabilistic constraints, a deterministic formulation for the finite-horizon stochastic MPC Problem 1 is presented as follows. 24

4 Problem 2 (Deterministic surrogate for stochastic MPC with probabilistic constraints: min u N J(x(k, u N subject to: ˆx i (t = a t,i Λ(θ, t = 0,, N i = 1,..., n x h(ˆx nom i (t, u t, θ 0, t = 0,, N i = 1,..., n x c i (κ 1 βi Var [ˆx i (t ] + t = 0,, N E [ˆx i (t ] + d i 0, u t U, x 0 = x(k. i I t = 0,, N (12 In the optimal control problem (12, the coefficients of the PC expansions (i.e., a t,i depend on the control inputs. To solve (12 online, the coefficients a t,i, i = 1,..., n x are recomputed at each discrete time instant using the new state measurements x(k and the decision variables u N (see Algorithm 1. Note that the hard inequality constraints on states are enforced only at the nominal values of the states ˆx nom i (t. For online implementations, Problem 2 is embedded in a receding horizon algorithm. Algorithm 1. (Receding horizon implementation of PCbased MPC. Input: 1 Initial time step k 2 Feasible initial control inputs u k 3 Initial states x(k 4 Uncertainty description of parameter vector θ 5 Type and order of orthogonal polynomials 6 Number of collocation points s At discrete time instant k: 1 Use x 0 = x(k and u k to carry out s simulations of the system (1 using s samples drawn from the known pdfs of θ 2 Use simulation data from Step 1 to estimate the coefficients of the PC expansions along the prediction horizon 3 Compute the statistics of the PC-approximated states ˆx t 4 Solve the deterministic optimal control problem (12 to determine the optimal control policy. For each optimization iteration, this requires repeated simulation of the PC expansion and of the statistics as in Steps 1 to 3 5 Apply the first element u 0 to the system (start from Step 1 at each time instant k. Next, stochastic nonlinear model predictive control of polymorphic transformation in a batch crystallization system is investigated. V. STOCHASTIC CONTROL OF POLYMORPHIC TRANSFORMATIONS IN BATCH CRYSTALLIZATION Polymorphism, in which multiple crystal forms exist for a chemical compound, is of paramount importance in the spe- TABLE I KINETIC EXPRESSIONS FOR BATCH POLYMORPHIC CRYSTALLIZATION OF L-GLUTAMIC ACID [39] α-form crystal nucleation rate B α = k bα (S α 1µ α,3 { kgα(s α-form crystal growth/dissolution rate G α = α 1 gα if S α 1 k dα (S α 1 otherwise k gα = k gα,0 exp ( Egα 8.314(T +273 β-form crystal nucleation rate B β = k bβ,1 (S β 1µ α,3 + k bβ,2 (S β 1µ β,3 β-form crystal growth rate G β = k gβ,1 (S β 1 g β exp ( k gβ,2 S β 1 k gβ,1 = k gβ,0 exp ( E gβ 8.314(T +273 cialty chemical and pharmaceutical industries [37]. Polymorphic transformations lead to variations in physical properties (e.g., shape, solubility, and chemical reactivity of crystals, and, therefore, can be detrimental to their performance. The problem of controlling polymorphic transformations consists of ensuring consistent production of the desired polymorph in a stochastic environment [38]. The batch polymorphic crystallization of L-glutamic acid is investigated using the kinetic model developed in [39] for polymorphic transformations of metastable α-form and stable β-form L-glutamic acid crystals. Under operating conditions that promote nucleation and growth as the prevalent kinetic phenomena, the batch crystallization dynamics are described by a population balance equation [40] n i t + (G in i = B i δ(l L 0, i = α, β (13 L where n i is the crystal size distribution (CSD of the i- form crystals (#/m 4 with # representing the number of crystals, B i and G i are the nucleation (#/m 3 s and growth (m/s rates, respectively, L and L 0 are the characteristic size of crystals and nuclei, respectively (m, and δ( is a Dirac delta function. Equation (13 is coupled with the solute mass balance dc dt = 3000 ( ρα k vα G α µ α,2 + ρ β k v,β G β µ β,2 ρ s where C is the solute concentration (g/kg, ρ s is the solvent density (kg/m 3, ρ i is the density of the i-form crystals (kg/m 3, k vi is the volumetric shape factor (dimensionless, and µ i,2 is the second moment (#/m of the crystal size distribution defined by µ i,j = 0 L j n i dl, i = α, β j = 2, 3. The kinetic expressions for nucleation and growth/dissolution rates of the α- and β-form crystals are listed in Table I, where S i = C/C sat,i is the supersaturation, C sat,i (T is the saturation concentration (g/kg, and T is the solution temperature ( C. The initial conditions (i.e., seed properties, the values of physical properties, and the kinetic parameters and their respective 95% credible intervals are taken from [39]. The method of moments [40] is applied to the population balance equation (13, yielding a set of nonlinear differential algebraic equations to describe the system dynamics. The kinetic parameters g α, E gα, k bβ,2, k gβ,2, and E gβ (see Table I are considered to be Gaussian random variables, whose 2416

5 Solute Concentration (g/kg PCE collocation points PCE 40 collocation points PCE 60 collocation points Nonlinear model Time (s Fig. 1. Comparison between solute concentration profiles predicted by the nonlinear model and third-order PC expansions. statistics are characterized using the parameter estimation results in [39]. The crystallization control objective is to minimize the ratio of the nucleated crystal mass to the seed crystal mass of β-form crystals at the end of the batch ζ(t f = µ β,3(t f µ seeds β,3 µ seeds β,3 (14 where t f is the end time of the batch and µ seeds β,3 is the thirdmoment of the seed size distribution. The optimal control problem should prevent the nucleation and growth of α-form crystals and the dissolution of β-form crystals during polymorphic transformation. The control objective should be realized in a stochastic environment due to the kinetic parametric uncertainties. A stochastic nonlinear model predictive control approach is designed to solve a finite-horizon stochastic control problem at every discrete time instant min T (t subject to: E[ζ(t f ] + wvar[ζ(t f ] nonlinear model equations C(t f C max (t f Pr[C sat,β (t < C(t < C sat,α (t] 0.95 T min T (t T max ( where w is a weighting coefficient on the variance of the performance index, and T min and T max are the minimum and maximum temperatures, respectively. In (, the first inequality constraint ensures that the minimum crystallization yield required by economic considerations will be achieved; the probabilistic constraint guarantees that the solute concentration stays within the admissible region in the presence of parametric uncertainties (to avoid the formation and dissolution of α- and β-form crystals, respectively; and the hard input constraint ensures that the optimal temperature profile remains within the operating range of the crystallizer. The stochastic optimal control problem ( is converted into a deterministic optimal control problem using polynomial chaos expansions (see Problem 2. Third-order expansions of Hermite polynomials are utilized to describe the parametric uncertainties, resulting in PC expansions with 56 terms (see expression (4. Algorithm 1 facilitates the receding horizon implementation of the stochastic MPC, where 60 collocation points are used to compute the coefficients of the PC expansions at the discrete time instants (i.e., every 0 s. The SMPC is implemented in Matlab based on the sequential optimization strategy, in which a stiff ODE integrator is exploited in combination with the optimization subroutine fmincon. Fig. 1 shows the solute concentration profiles predicted by the nonlinear model and PC expansions, whose coefficients are determined using different numbers of collocation points. Using and 40 collocation points resulted in inadequate estimation of the coefficients of the PC expansions and poor approximation of the solute concentration profile. On the other hand, when 60 points are used to estimate a in (5, the PC expansions provide accurate approximations of the system dynamics. To evaluate the performance of the control approach, Monte Carlo simulations of the closed-loop system are performed using 0 random parameter vectors generated from the Gaussian pdfs of the kinetic parameters (the parameters are used for simulating the process model to which the optimal control inputs are applied. The performance of the SMPC approach is compared to that of a nonlinear model predictive control (NMPC approach with the same performance index (14. Fig. 2 shows the histograms of the performance index based on 0 closed-loop simulations of control approaches. The NMPC and SMPC approaches result in comparable mean performance indexes of and 29.71, respectively. However, the SMPC approach leads to a significant reduction (i.e., 26% in the variance of the performance index in the presence of parametric uncertainties, as compared to the NMPC approach, due to the probabilistic formulation of the optimal control problem ( that enables Frequency Frequency (t f 5 (a Nonlinear model predictive control (t f (b Stochastic nonlinear model predictive control Fig. 2. Histograms of the performance index ζ(t f based on 0 closedloop simulations. 2417

6 Solute Concentration (g/kg Solute Concentration (g/kg Temperature ( C 30 (a Nonlinear model predictive control Temperature ( C (b Stochastic nonlinear model predictive control Fig. 3. Solute concentration trajectories for 0 closed-loop simulations. The insets show a magnification of the α-crystals solubility curve to illustrate constraint violations. shaping the distribution of the performance index (in terms of the first- and second-order moments in a stochastic environment. Fig. 3 depicts the solute concentration trajectories, which should be constrained between the solubility curves of α- and β-form crystals. Fig. 3a indicates that in the NMPC approach the solute concentration may violate its upper bound (the solubility curve of α-form crystals with a likelihood of 39% in the presence of parametric uncertainties. As shown in the inset figure, the constraint violation occurs in the initial phase of the batch process, where the abrupt dissolution of the α-form seeds upon introduction into the crystallizer leads to rapid supersaturation generation that is subsequently consumed by the growth of the β-form crystals. On the other hand, the SMPC approach ensures that the solute concentration remains within its admissible region in an uncertain setting by inclusion of the probabilistic constraint in (. Fig. 3b shows that the probabilistic constraint is fulfilled in 99% of closed-loop simulations, which is greater than the pre-specified probability of 95%. The simulation results of batch polymorphic crystallization clearly indicate the capability of the SMPC approach with probabilistic constraints to control the pdfs (in terms of their moments of system states, while guaranteeing the fulfillment of state constraints in the presence of stochastic uncertainties. VI. CONCLUSIONS AND FUTURE WORK A framework for stochastic MPC of nonlinear systems in the presence of parametric uncertainties is presented. The key challenges in stochastic MPC of real applications are the propagation of parametric uncertainties through the system model and the reformulation of probabilistic constraints to computationally tractable expressions. To obtain a deterministic surrogate for the stochastic MPC problem, polynomial chaos expansions are used to propagate the parametric uncertainties and the individual probabilistic constraints are transformed into convex second-order cone constraints that are valid for a general class of probability distributions with known mean and covariance. The application of the stochastic control approach to batch polymorphic crystallization reveals its ability in shaping the probability distribution of system states, as well as ensuring the fulfillment of state constraints in a stochastic setting. 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