Numerical conditioning

In order to analyse the numerical conditioning of the square-root and the normal Ric- cati method, we applied them to the waste water treatment problem forκinit= 0.01 and decreasing factorτ= 0.9 and for a fixedκ= 0.01. The top panels in Figure 4.16 show the condition number of a matrix involved in the measurement update for a state with an actively constrained component. For the normal Riccati method the condition

number of the following matrix is depicted  Rk Inik  + DkΣkDTk  (4.25)

which needs to be inverted (see Eq.(4.13)), or in practice is factorized and its factors are applied. For the square-root method, the condition number of

  TkDTk Tk  Vk Inik   , (4.26)

is depicted. Which is the matrix that is factorized in the measurement update of the square-root method (see Eq.(4.14)). The bottom panels in Figure 4.16 show the loga- rithmic growth of the condition number of the square-root barrier Hessian Mk. In case

of a decreasing barrier parameter, the effect is slightly amplified. For a decreasingκ the algorithm converges in 10 iterations while for a fixedκconvergence is achieved in 14 iterations. It can be seen from Figure 4.16 that the condition number for the typcial matrix in the square-root method is the square of the one in the normal method, as expected. This motivates the use of the square-root Riccati method in the context of (primal barrier) interior point methods.

The same problems of numerical conditions are also observed in the method of weight- ing for constrained least-squares (see [22, 77]). In this case it is recommended to use row and column permutations in the factorization process [22]. For most problems it is sufficient to initially order the rows with potentially large terms first as done in the proposed row-reordering for the measurement update. Although Van Loan [187] shows on certain contrived examples that this is not always sufficient, we have not experienced any problems in our implementations.

4.5. Conclusions

In this chapter Riccati based solutions for constrained MHE using a primal barrier interior-point method were presented. It was demonstrated how the barrier terms en- ter in the measurement update step of the recursion and constraints can be interpreted as perfect measurements. The square-root algorithm exploits both the structure and the symmetry in the problems and employs structured QR methods at the core. Sev- eral types of constraints were considered: mixed or separate, general or bound con- straints. A hot-starting strategy was presented which showed improved performance in the first iterations. The use of square-root Riccati methods within an interior-point method could be strongly motivated by the fact that the condition number of the matri- ces involved in the factorization typically grow logarithmically. A C implementation

4.5. Conclusions 2 4 6 8 10 12 14 100 101 102 103 104 iteration Square−root update Normal update 2 4 6 8 10 100 101 102 103 104 105 iteration Square−root update Normal update 2 4 6 8 10 12 14 100 101 102 103 104 iteration

Square−root of barrier hessian Inverse of barrier parameter

2 4 6 8 10 100 101 102 103 104 iteration

Square−root of barrier hessian Inverse of barrier parameter

Figure 4.16. Logarithmic growth of the numerical conditioning in function of the iteration num-

ber for a state with actively constrained components. Top row: Evolution condition number of a matrix involved in the measurement update of the square-root (solid) and normal Riccati method (dashed), for a fixed barrier parameter (left) and a decreasing barrier parameter (right).

Bottom row: Inverse barrier parameter (dashed) and evolution of the condition number of the

square-root of the barrier hessian (solid), for a fixed barri er parameter (left) and a decreasing barrier parameter (right).

demonstrated the performance of the square-root algorithm. It was shown by nu- merical examples that the method can be run in the milisecond range on a standard computer for moderate horizons.

CHAPTER

5

Active-set methods for MHE

5.1. Introduction

Albuquerque and Biegler [3] first proposed a structure-exploiting algorithm for MHE which scales linearly in the horizon length. They applied a type of condensing in which the controls and multipliers are eliminated and the reduced QP is formulated and solved in the state space. By construction this approach should have a strong

5.1. Introduction

relation with Riccati recursion, although the proposed algorithm does not explicitly employ a Riccati recursion. Riccati based methods for MHE problems have been proposed before, e.g. by Tenny et al [173] and Jorgensen et al [108]. Typically, normal Riccati (Kalman filter) recursions are proposed to solve the KKT system. A structure-exploiting interior-point MHE method using square-root was presented in [90].

Active-set methods for quadratic programming can be classified into primal feasible or dual feasible methods. In primal feasible methods, a Phase I calculation to find an feasible initial point is typically required. Subsequently, constraints are added or deleted in order to reduce the objective while maintaining primal feasiblity. Upon convergence, dual feasibility is obtained. Dual methods, on the other hand start with a dual feasible point, which usually can be computed cheaply (no Phase I), and main- tain dual feasibility during subsequent QP iterations. QPOPT [140] is a primal fea- sible active-set method which uses the null-space approach. It maintains a dense Cholesky factorization of the reduced Hessian.QPKWIK [161] is a dual feasible active- set method based on the famous method of Goldfarb and Idnani [76]. BothQPOPT and QPKWIK are dense methods and both require a positive definite Hessian. Therefore, they are not directly suitable for MHE problems.

A primal Schur complement active-set method, calledSchurQP, was presented by Gill et al [70]. This method applies the Schur complement for every change in the working set which can be addition or removal of one constraint. A dual Schur complement active-set method, calledQPSchur, was presented by Bartlett et al [10, 11]. It allows structure exploitation in the KKT system matrices and is applied to MPC problems. The method, however, still requires a positive definite Hessian and proceeds by adding or deleting one constraint at a time as usual in active-set methods.

Axehill and Hansson [7, 8] have presented a dual active-set method for MPC which uses gradient projection. In their algorithm the two step procedure, comprising a Cauchy point calculation and a Newton direction computation, is directly applied to the dual MPC problem. Inequalities are discarded in the Cauchy point calculation and (forward) normal Riccati recursions are used in the projected Newton step. Their method permits multiple active-set changes per iteration, requires few QP iterations and is shown to perform very well on several linear and hybrid MPC examples. In this chapter, we present a Schur complement active-set method tailored to MHE problems. The method inherits properties from both primal and dual methods. The algorithm is motivated by two observations: (1) the unconstrained MHE problem, i.e. discarding inequalities, can be solved very efficiently using Riccati recursions and (2) in MHE only a small number of inequality constraints is expected to be active at the solution. Therefore, the method uses the unconstrained MHE solution as the starting point and solves a number of QPs in the reduced space of working set constraints until the active set is determined. The method allows multiple active-set changes per itera-

tion and typically converges in a few iterations. For the unconstrained MHE solution, a Riccati based method using square-roots is proposed which allows fast computa- tion of the QP matrices. The underlying QPs are non-negativity constrained quadratic problems for which a gradient projection method using projected Newton steps is presented. Modified Cholesky factorizations are suggested for solving the QP KKT systems and subsequent changes are exploited by Cholesky downdates and updates at the level of outer and inner active set iterations. The method bears resemblances with the method for MPC by Axehill et al [7, 8], but differs in the following: (1) it is specifically designed for MHE, (2) it uses the unconstrained solution, hence empty initial working set, as a starting point, which can be motivated for MHE, (3) it pro- ceeds by adding sets of constraints per iteration and constraints are only removed upon convergence, inactive constraints are automatically assigned multipliers equal to zero (4) by using Cholesky factorizations for the solution of underlying KKT systems and updating factorizations at both the lower and higher level (inner and outer active set iterations), the efficiency is maximized.

5.2. Overview of active-set methods for quadratic programming