CORE Metadata, citation and similar papers at core.ac.uk Provided by Apollo Nonlinear model predictive control based on Bernstein global optimization with application to a nonlinear CSTR Bhagyesh V. Patil§ and Jan Maciejowski† and K. V. Ling‡ Abstract— We present a model predictive control based model predictive control using nonlinear process models, tracking problem for nonlinear systems based on global usually called ‘nonlinear MPC’ ( or NMPC), has attracted optimization. Specifically, we introduce a ‘Bernstein global many researchers over the past decade [6], [7], [8]. optimization’ procedure and demonstrate its applicability to the aforementioned control problem. This Bernstein global An NMPC formulation requires the solution of a (usu- optimization procedure is applied to predictive control of a ally nonconvex) nonlinear optimization problem at each nonlinear CSTR system. Its strength and benefits are compared sampling instant. As such, NMPC is a challenging field, with those of a sub-optimal procedure, as implemented in and is dependent on good global optimization procedures. MATLAB using fmincon function, and two well established Motivated by this, in the present work we introduce one global optimization procedures, BARON and BMIBNB. such global optimization procedure for NMPC applications. I. INTRODUCTION This procedure is based on the well-known Bernstein form of polynomials [9], and uses several nice properties associated Model predictive control (MPC), also known as moving with this Bernstein form. Optimization procedures based on horizon control or receding horizon control, is an advanced this Bernstein form, also called Bernstein global optimization control scheme for multivariable control systems. Typically, algorithms, have shown good promise to solve hard non- MPC derives a control signal by optimizing a pre-defined convex NLP and MINLP problems (see, for instance, [10], performance criterion repeatedly over a finite-time moving [11], [12]). They are therefore very promising for NMPC horizon within system constraints, and based on a dynamic applications. model of the system to be controlled [1], [2]. MPC has been In this work, we present one such Bernstein global applied predominantly in the process industries, especially optimization algorithm to solve a nonlinear optimization refining and petrochemicals [3]. An excellent survey of problem at each NMPC iteration. Specifically, we use the industrial applications of MPC can be found in [4], and nonlinear system model for the predictions, followed by references therein. the formulation of a nonlinear programming (NLP) problem In practice, the majority of MPC applications employ based on these predictions. Then, the Bernstein global opti- linear models derived from system identification proce- mization algorithm is used as a tool to solve this nonlinear dures, combined with linear inequality constraints. The MPC optimization problem in terms of the control inputs (u’s) as scheme in such instances is also known as ‘linear MPC’. decision variables. The overall approach is demonstrated on Linear MPC is widely preferred, due to its simplicity and a simulation study for predictive control of a nonlinear CSTR the applicability of convex optimization algorithms (see [5] system, and the findings are compared with those of a sub- for instance). However, some processes may have either optimal procedure implemented in MATLAB using fmincon semi-batch characteristics, a large operating regime, or other function, and two well established global optimization pro- source of significantly nonlinear behaviour. Approaches such cedures, BARON and BMIBNB. as gain scheduling and switching between multiple linear The rest of the paper is organized as follows. In Section models based on the operating region are possible approaches II, we introduce a nonlinear MPC formulation. In Section for such processes. On the other hand, use of a nonlinear III, we briefly describe the Bernstein form, followed by the process model may come with attractive benefits, such as presentation of the Bernstein global optimization algorithm. higher product quality, tighter regulation of process param- In Section IV, we report the simulation studies on a nonlinear eters, and the possibility of operating the process (with a CSTR system with the Bernstein global optimization algo- good control authority) in different operating regimes. Hence, rithm and compare with MATLAB fmincon function, global optimization procedures, BARON and BMIBNB. Finally, in §Bhagyesh V. Patil is with Cambrdige Centre for Advanced Research in Energy Efficiency in Singapore (CARES), 50 Nanyang Ave, Singapore. Section V, we present some concluding remarks. [email protected] † Jan Maciejowski is with Department of Engineering, University of Cam- II. PROBLEM FORMULATION FOR NMPC bridge, Cambridge CB2 1PZ, United Kingdom. [email protected] ‡K. V. Ling is with the School of Electrical and Electronic Engi- We consider a class of continuous-time systems described neering, Nanyang Technological University, 50 Nanyang Ave, Singapore. [email protected] by the following nonlinear model This work was supported by the Singapore National Research Foundation (NRF) under its Campus for Research Excellence and Technological Enter- x˙ = f (x;u); x(0) = x0 (1) prise (CREATE) programme, and Cambridge Centre for Advanced Research in Energy Efficiency in Singapore (CARES). y = g(x;u) (2) n m where x 2 R and u 2 R denote the vectors of states and III. BERNSTEIN GLOBAL OPTIMIZATION ALGORITHM p control inputs, respectively; y 2 R is the controlled output. This section briefly presents some notions about the Bern- The state of the system and the control input applied at stein form. Due to space limitation, a simple univariate sampling instant k are denoted by x(k) and u(k), respectively. Bernstein form is introduced. A comprehensive background The system is subject to the state and input constraints of and mathematical treatment for a multivariate case can be the following form: found in [12]. x(k) 2 X; 8 k ≥ 0 (3) We can write a univariate l-degree polynomial p over an x u(k) 2 U; 8 k ≥ 0 (4) interval in the form l X n U m X U i where ⊆ R and ⊆ R . In simplest form, and are p(x) = ∑ aix ; ai 2 R : (7) given by bound constraints of the form: i=0 n Now the polynomial p can be expanded into the Bernstein X := fx 2 R j xmin ≤ x ≤ xmaxg: m polynomials of the same degree as below [9] U := fu 2 R j umin ≤ u ≤ umaxg: l In the present work, we consider the design of an NMPC l p(x) = ∑ bi (x)Bi (x) (8) controller for (1) to track a desired reference xs, while i=0 fulfilling constraints of the form (3)-(4). Further, dropping where Bl(x) are the Bernstein basis polynomials and b (x) the index k for simplicity, the general form of NMPC control i i are the Bernstein coefficients give as below law can be derived at each sampling instant k by the solution of the following NLP problem. l l i 1−i Bi(x) = x (1 − x) : (9) N−1 i 2 2 min k xi − xi;s k + k Dui k (5) u ∑ Q R i i=0 i i subject to (1); (3); and (4) for i = 0;1;:::;N − 1 j bi (x) = a j; i = 0;:::;l: (10) ∑ l where xi;s denotes the set-point (reference) at instant i; j=0 n×n m×m j Q 2 R and R 2 R denote positive definite, symmetric weighting matrices; Dui = ui − ui−1 denotes the control Equation (8) is referred as the Bernstein form of a polyno- increment, N(≥ 1) denotes the prediction horizon. mial and obeys the following property: At the outset, the nonlinear model (1) is used for predic- Theorem 1: (Range enclosure property) Let p be a poly- tions based on the initial state x0. The predicted control input nomial of degree l, and let p(x) denote the range of p on a profile is denoted by ui, i = 0;2;:::;N − 1 Then, assuming given interval x. Then, that the optimization problem has a feasible solution, an p(x) ⊆ B(x) := [min (b (x)); max (b (x))]: (11) optimizer (in this work, we use Bernstein global optimization i i Proof: See [13]. algorithm) computes an optimal control sequence based on the NMPC optimization problem formulated in (5), defined Remark 2: The above theorem says that the minimum and as 2 ∗ 3 maximum coefficients of b (x) provide lower and upper u0 i ∗ bounds for the range of p. This forms the Bernstein range 6 u1 7 6 7: (6) 6 . 7 enclosure, defined by B(x) in equation (11). Figure 1 shows 4 . 5 for a univariate polynomial p, its Bernstein coefficients u∗ N−1 (b0;b1;:::;b5). The minimum (b0;b4) and maximum (b1) ∗ Only the first step of this optimal control sequence, u0 is Bernstein coefficients encloses the range of p. Further, this applied to the system (1) to obtain a new updated state. Bernstein range enclosure can successively be sharpened Then the whole process is repeated, with x0 obtained from by the continuous domain subdivision procedure. Figure 2 the latest measurements, until the state is steered to its illustrates this fact. desired reference. We now present the global optimization algorithm based on the above Bernstein form. This algorithm uses the Remark 1: The cost function in the optimization problem Bernstein range enclosing property, followed by a domain (5) depends nonlinearly on the state and input variables. subdivision, to correctly locate the global solution (global Hence, the optimization problem turns out to be a NLP. minimum and global minimizers) for a given NLP problem. We note that, in some instances, such as a nonlinear CSTR (see Section IV, Equations (12)-(13)), the nonlinearity Algorithm Bernstein: [ye; pe;U]=BBBC(N;aI;x;ep;ex;ezero) appears only in terms of the state variables.