Optimal nonlinear model predictive control based on Bernstein polynomial approach Bhagyesh V. Patil§ and K. V. Ling† and J. M. Maciejowski‡ Abstract— In this paper, we compare the performance of global optimization procedures (cf. [11], [12]). The challenge Bernstein global optimization algorithm based nonlinear model involved in NMPC is two fold: predictive control (NMPC) with a power system stabilizer and (i) Can we complete the nonlinear iteration procedure until a linear model predictive control (MPC) for the excitation control of a single machine infinite bus power system. The control pre-specified convergence criterion is met so as to guarantee simulation studies with Bernstein algorithm based NMPC show the optimal solution of the optimization problem? improvement in the system damping and settling time when (ii) can we achieve (i) in a pre-specified sampling time limit? compared with respect to a power system stabilizer and linear Concerning these facts, our previous works have introduced MPC scheme. Further, the efficacy of the Bernstein algorithm is Bernstein global optimization procedures for NMPC appli- also compared with global optimization solver BMIBNB from YALMIP toolbox in terms of NMPC scheme and results are cations (cf. [13], [14]). Optimization procedures based on found to be satisfactory. this Bernstein form, also called Bernstein global optimization algorithms, have shown good promise to solve hard noncon- I. INTRODUCTION vex optimization problems. This procedure is based on the Over the past decades, model predictive control (MPC) Bernstein form of polynomials [15], and uses several nice has emerged as one of the prominent advanced control ‘geometrical’ properties associated with this Bernstein form. methodology for multivariable control. At the heart of MPC The current scope of the work is based on the sequential lies a system model, which predicts the future evolution of improvement of our previous works in [13], [14]. Specif- the system states. It generates control actions by iteratively ically, in this work we implement a hull pruning feature optimizing a performance criterion over a finite-time moving for the Bernstein global optimization algorithm to solve a window with reference to system constraints, and based nonconvex optimization problem at each NMPC iteration. on predictions of the system model [1], [2]. In practice, The hull pruning aids in discarding regions from solution MPC implementations utilizing linear models (a.k.a ‘linear search spaces that surely do not contain the global solution. MPC’) are preferred. This facilitates use of linear/convex As this feature provides the means to narrow the search programming techniques to exactly solve optimization prob- region for the optimization problem, we call it a narrowing lems at each sampling instant. However, some applications (‘hull pruning’) operator. The applicability of the Bernstein (like power systems) have significant nonlinear behaviour, algorithm with this hull pruning feature is demonstrated and for such applications, a linear MPC scheme may not by simulating a nonlinear model predictive control scheme yield desirable closed-loop performance [3], [4]. Hence, to for a classical single machine infinite bus (SMIB) power mitigate problems arising from the system nonlinearities, system [16]. SMIB has a strong nonlinear characteristics and many researchers have pursued MPC approach based on exhibits accurate representation of the synchronous generator nonlinear system models. This approach under large variation behaviour. The findings of our nonlinear MPC scheme based of dynamic behaviour of the system, may provide more on the Bernstein algorithm with hull pruning operator are satisfactory control than linear MPC. Model predictive con- compared with respect to well-established power system trol using nonlinear system models, usually called ‘nonlinear stabilizer (PSS) [16] and linear MPC. We also investigate our MPC’ (or NMPC), hence has attracted many researchers over findings with nonlinear MPC based on the YALMIP global the past decade [5], [6], [7], [8], [9], [10]. optimization solver BMIBNB [17]. We note that an NMPC formulation usually requires the In the rest of the paper, we first introduce a nonlinear solution of a nonlinear, usually nonconvex, optimization MPC formulation (Section 2). Next, we briefly describe the problem at each sampling instant. Therefore, NMPC requires Bernstein form and the hull pruning operator, followed by the presentation of the Bernstein global optimization algorithm §Bhagyesh V. Patil is with Cambridge Centre for Advanced Research and Education in Singapore (CARES), 50 Nanyang Ave, Singapore. (Section 3). We then report the simulation studies on a [email protected] nonlinear SMIB power system (Section 4). Finally, some †K. V. Ling is with the School of Electrical and Electronic Engi- concluding remarks are given in Section 5. neering, Nanyang Technological University, 50 Nanyang Ave, Singapore. [email protected] II. NMPC CONTROLLER FORMULATION ‡J. M. Maciejowski is with Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom. We consider a class of continuous-time systems described [email protected] by the following nonlinear model This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore, under its Campus for Research Excellence and Technological Enterprise (CREATE) programme. x˙ = f (x;u); x(t0) = x0 (1) n m where x 2 R and u 2 R denote the vectors of states and control inputs, respectively. In practice, the continuous-time model (1) used for predictions is discretized with a sampling time Dt, such as Euler’s method Constraint handling c(xk, uk) <= 0 * Optimization solver u 0 x xk+1 = xk + Dt: f (xk;uk) (2) B(xk, uk) (BBB algorithm Plant in Section III) where k denotes sampling instant. Use of Lemma 1 The nonlinear optimization problem in the NMPC formu- Nonlinear optimization problem Measurement Unit lation at each sampling instant k is stated by (3)-(6). The (Equations 3-6) control objective is to maintain equilibrium point (origin), by minimizing the cost (3) subjected to discretized nonlinear Prediction xk (Equation 2) predictive model (2), and fulfilling constraints of the form (5). Fig. 1. Schematic of Bernstein algorithm based NMPC. N−1 min L (xk;uk) (3) u ∑ k k=0 subject to xk+1 = xk + Dt: f (xk;uk) (4) brevity, we only present notions about the univariate Bern- stein form (see [18] for the multivariate case). c(x ;u ) ≤ 0 (5) k k We can write a generic polynomial of degree l as for k = 0;1;:::;N − 1 (6) l i where N(> 1) denotes the prediction horizon and c(xk;uk) p(x) = ∑ aix ; ai 2 R (9) are the nonlinear constraints arising due to safety and oper- i=0 ational requirements of the control system. It may be noted where x is the variable, and fa0;a1;:::;alg are the coeffi- that, c(xk;uk) subsume the constraints on the state and input cients of the power basis Bp given by the following set of of the following form: monomials 2 l min max Bp = f1;x;x ;:::;x g: (10) xk ≤ xk ≤ xk : (7) min max uk ≤ uk ≤ uk : (8) We assume that p(x) is defined over a real bounded and x = [ ; ] ∗ ∗ closed interval 0 1 . The unit interval is not a restriction, We assume at the equilibrium point xk;uk , the cost (3) ∗ ∗ since any nonempty compact interval can be mapped affinely should be zero, i.e. L (xk;uk) = 0. onto it. The NMPC algorithm at each sampling instant k involves Now the polynomial p can be expressed into the Bernstein the following steps: polynomial form of the same degree [15]: (a) Measure the state xk of the system (we assume in the l above NMPC formulation all states are available for l p(x) = ∑ biBi (x) (11) measurement). i=0 (b) Solve the nonlinear optimization problem (3)-(6) with l where B (x) are the Bernstein basis polynomials and bi are initial state xk. Denote the obtained optimal control i sequence as u∗; u∗;:::; u∗ . the Bernstein coefficients: 0 1 N−1 (c) Implement the first step of the optimal control sequence, l l i l−i ∗ Bi(x) = x (1 − x) : (12) u0 to the system (1) to obtain a new updated state until i the next sampling instant. (d) Repeat from (a). i i The overall scheme of the Bernstein algorithm based j bi = ∑ a j; i = 0;:::;l: (13) NMPC is depicted in Fig. 1. The Bernstein algorithm can j=0 l be applied when the system model, cost function and the j constraints are polynomials. In many other applications the Equation (11) is referred as the Bernstein form of (9) and problem can be approximated (as closely as desired) by satisfies the following range enclosure property [15]: polynomials (such as our example in Section IV). III. BERNSTEIN GLOBAL OPTIMIZATION APPROACH p(x) ⊆ B(x) := [min bi; max bi]: (14) In the NMPC formulation (Section II), we solve a nonlin- where p(x) denote the range of p on a given interval x. ear optimization problem at each sampling instant to derive a control law for the nonlinear system (1). [5] and [7] Remark 1: Equation (14) says that the minimum and max- report some optimization approaches to achieve this goal. imum coefficients of bi provide lower and upper bounds for This section briefly introduces one such (global) optimization the range of p. This forms the Bernstein range enclosure, approach based on the Bernstein form of polynomials. For defined by B(x). Further, this Bernstein range enclosure 2 can successively be sharpened by the continuous domain Example: Consider a constraint x2 = x1, x1 2 [1;2] and subdivision procedure (see, for instance [19]). x2 2 [0;2:5]. First perform constraint inversion and interval The following properties follow immediately from the arithmetic operations for x1 Bernstein range enclosure (14). 0 1 x1 = (x2) 2 1 Lemma 1: Let B(x) be the Bernstein range enclosure = [0;2:5] 2 for a polynomial p(x) on a given box x.