Direct optimization of dynamic systems described by differential‐algebraic equations

Abstract

This paper presents a method for the optimization of dynamic systems described by index-1 differentialalgebraic equations (DAE). The class of problems addressed include optimal control problems and parameter identification problems. Here, the controls are parameterized using piecewise constant inputs on a grid in the time interval of interest. In addition, the DAE are approximated using a Rosenbrock-Wanner (ROW) method. In this way the infinite-dimensional optimal control problem is transformed into a finite-dimensional nonlinear programming problem (NLP). The NLP is solved using a sequential quadratic programming (QP) technique that minimizes the L ∞ exact penalty function, using only strictly convex QP subproblems. This paper shows that the ROW method discretization of the DAE leads to (i) a relatively small NLP problem and (ii) an efficient technique for evaluating the function, constraints and gradients associated with the NLP problem. This paper also investigates a state mesh refinement technique that ensures a sufficiently accurate representation of the optimal state trajectory. Two nontrivial examples are used to illustrate the effectiveness of the proposed method.Here, we wish to find the control input u(t) ∈ R n u and the parameter vector p ∈ R n p that minimizes the scalar cost function J . The cost function is made up of the terminal penalty function, (t f , p), and the integrand L(y(t), u(t), p), which are both scalars. The initial time t i and the final time t f are both fixed. The system state vector, y(t) ∈ R n y , must satisfy the differential equations (2), where F(y(t), u(t), p) ∈ R n y . The matrix M ∈ R n y ×n y is constant in these systems. If M is singular then (2) represents a system of differential-algebraic equations (DAE), otherwise if M is nonsingular then (2) represents a system of ordinary differential equations (ODE).We assume that the DAE have differentiation index-1, i.e. rank[M F y (y, u, p)] = n y for all u, p, and y, where F y = *F/*y. Moreover, we assume that the ODE/DAE are 'stiff' [1]. Problems with higher index DAE can be treated via index reduction [1][2][3]. Also, note that optimal control problems with index-2 and higher index DAE are considered in [4,5].Equation (3) indicates that the state vector at the initial time is known. It is also assumed that these initial conditions are consistent, i.e. for a given u(t i ) and p there is aẏ( (4) indicates that the inequality constraints d j (y(t), u(t), p) 0, j = 1, 2, . . . , n d must be satisfied in the entire interval t i t t f . Equation (5) indicates that the state and parameters must satisfy the constraints j (y(t f ), p) = 0, j = 1, 2, . . . , n , at the final time. Here, it is assumed that the functions , L , F, d, and are at least twice continuously differentiable with respect to their arguments.Note that system (2) is autonomous; however, nonautonomous systems can be easily accommodated by adding a new state variable to represent the time. It should also be noted that minimum time problems and problems with unknown ini...

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