PRAINCODERECI

4 6 8 10 12 14 16

0 100 200 300 400 500

Typical period T

(s)

A v er a g e p o w er (k W ) ^{Method 1} _{Method 2}

Latching No Control

Figure 3.7: Average absorbed power against typical wave period with the different control meth-ods, B_pto≈ 280 kN·s/m (B_pto≈ 95 kN·s/m for latching control), G = M + µ_∞, T_p= 8 s, H_s= 2m.

power when Tcis halved at Tp= 3s with the 0.1 kW difference at Tp= 8s. The integration of multi-step wave excitation prediction schemes in this sensitivity analysis, as done using an extended Kalman filter in [44], can also be considered.

3.5 Conclusion

In this chapter, model predictive control strategies were considered for implementation in a receding horizon fashion. A state space model of a generic point absorber, whose power take-off includes a linear damper and an active element, was formulated and used.

By considering a variational formulation of the optimal control problem with constraints only on the inputs, the solution was shown to be a bang-bang type.

A computationally inexpensive and globally convergent numerical scheme was devel-oped for solving the power maximization problem. A variation of the projected gradient method (PGM) was exploited and shown to converge in a small number of iterations un-der various wave conditions. Its performance has been compared to solving a directly collocated version of the problem using an interior point solver, IPOPT. It was shown that the PGM requires only a single state and costate evaluation at each iteration and is far less computationally demanding compared to a general NLP solver. For example, at each iteration IPOPT performs gradient and Hessian computations of the Lagrangian,

3.5 Conclusion 63

0 10 20 30 40 50

0 50 100 150 200

Prediction horizon (s)

A v er a g e p ow er (k W )

T

=1 s T

=0.5 s

Figure 3.8: Variations in average absorbed power against prediction horizon lengths for ‘Method 1’. JONSWAP wave with T_p= 8 s and Hs= 2 m used; G = B = 0.3 ∗ (M + µ∞).

as well as a number of line searches by solving a KKT system (see Section 3.3.5). On the other hand, the PGM method described in Section 3.3.3 requires only a single state and adjoint state evaluation; there is only marginal computational cost associated with the gradient computation or the input projection onto the feasible set . For this reason, to further reduce the cost incurred in the solution of the system and adjoint ODEs, we consider the use of exponential integrators in Chapter 5.

Time-domain simulations have also been used to evaluate the performance of the con-trollers developed in a receding horizon scheme. The active methods have been com-pared with the case of optimal command latching and the system with no control. The controllers developed were shown to widen the bandwidth of the wave energy converter and increase extracted power significantly. Unlike latching control, the active meth-ods widen the bandwidth of a WEC in both directions around the resonant frequency;

latching is effective only at frequencies lower than that of the buoy.

The PTO system optimization was shown to be important and highly dependent on the control scheme used. Although more energy could potentially be produced using larger actuation forces, physical constraints related to the maximum pressures and loads that actuation systems can withstand or deliver should be taken into account. In practice, the efficiency of the PTO should also be considered in the control problem and device optimization.

3.5 Conclusion 64

Similarly to other works in the literature, the availability of future excitation force infor-mation and radiation forces has been assumed; this is not true in practice. The synthesis of observers for the radiation force and their use is also the subject of this thesis and is discussed in Chapter 4.

65

Chapter 4

Robust Estimator Design for Bilinear Systems with Bounded Inputs

This chapter investigates low-order observer design for bilinear systems with input con-straints. A bilinear Luenberger-type observer with an H∞performance is formulated and the resulting synthesis problem is posed as a matrix inequality optimization for a linear parameter varying system. The resulting (nonconvex) bilinear matrix inequality prob-lem is then solved with an LMI-based algorithm to find low-order nominal and robust quadratically stable observers. The performance of these observers is compared with that of an extended Kalman fiter. In addition to alleviating the need to know the noise spectrum and its lower real-time computational burden, the H∞filter is shown to be ro-bust to model uncertainties. The online radiation force estimation problem for a wave energy converter with bilinear dynamics is considered as an example. A small order ra-diation subsystem is used to approximate the rara-diation process and the robustness of the observer to both model order and parameter uncertainties is investigated. Closed-loop receding horizon simulations are also used to assess the effectiveness of the observers under noisy measurements.

4.1 Introduction

Bilinear systems have dynamics that are linear in the input and linear in the state, but not jointly linear in both. These systems arise in a variety of modelling processes, includ-ing the discretization of certain partial differential equations with boundary actions or the bilinearization of some nonlinear systems, within various application areas such as plasma physics, quantum physics, biomedicine, metallurgy, economics, ecology and so on [70]. Motivated by estimation problems in wave energy conversion [71], in this

chap-4.1 Introduction 66

ter we investigate the design of observers for bilinear systems with input constraints. We consider observer design for such systems where physical actuator constraints give rise to convex input constraints [71]. Motivated by this problem, we will investigate the design of observers for bilinear systems with bounded inputs.

The literature on full-order H₂and H_∞ filters for linear systems is vast. The synthesis of full-order H2 filters is a well studied convex problem; the analysis and synthesis of H2 filters in the presence of polytopic uncertainty is discussed in [72, Ch. 9] and the literature therein. However, the filter performance degrades for a general uncertainty in the model or when the spectral density of the noise is not perfectly known — the optimal minimal error variance in the estimation error is no longer guaranteed. In comparison, H_∞ filters require no statistical knowledge of the input or disturbance, other than that its energy be bounded. In addition to being inherently robust to model uncertainty, the analysis and synthesis of H∞ filters also allows the explicit incorporation of various uncertainty types [73–75]. These cited works consider a system with structured and unstructured uncertainty and synthesize full-order H∞ filters with respect to the given uncertainty sets. This problem can be posed as a linear matrix inequality (LMI), which is a convex optimisation problem and so can be solved efficiently due to recent advances in the discipline [72]. As we review in the following, LMI based H_∞analysis and synthesis tools have been generalized to some nonlinear systems and bilinear systems. However, these standard methods produce observers that have the same order as the model. For high order systems, this may pose a high computational burden or even infeasibility for real-time implementation. In the following, we investigate the design of full-order and lower-order H∞filters for bilinear systems.

This chapter is organised as follows. In Section 4.2, we define the bilinear systems of interest and review existing literature on observer design for such systems. Stability and quadratic performance of LPV systems is reviewed in Section 4.3. In Section 4.4, we formulate the bilinear system with input constraints as a linear parameter varying (LPV) system and show that the lower-order H_∞filter synthesis can be posed as a bilinear ma-trix inequality (BMI) problem. Section 4.5 presents an LMI-based coordinate descent algorithm for solving the BMI locally. We also give the dynamics for an implementa-tion of an extended Kalman filter for comparison with the H∞ filter. In Section 4.7, we compare the two filters by examining the robust radiation force estimation problem for a wave energy converter (WEC) in [71].