|Sažetak (engleski)|| |
This thesis is concerned with the control law synthesis for nonlinear dynamical systems that guarantees that the effect of the external and/or internal uncertainty of the system is kept under permissible level and ensures the stability of the closed-loop system. As a measure of the uncertainty effect, the L2-gain of the system is considered. The problem belongs to the field of robust optimisation, i.e. a class of mathematical problems in which minimisation and maximisation of the same performance criterion are simultaneously carried out – minimax optimisation. As it is well known, the L2-gain optimal control problem requires solving a Hamilton- Jacobi-Isaacs (HJI) equation. In the nonlinear case HJI partial differential equation is difficult or impossible to solve, and can not have a global analytical solution even in very simple cases, for an example in some scalar systems. Therefore, many approximation methods have been developed to solve the so-called sub-optimal problem in which locally L2-gain is less than or equal to some prescribed number, see for example [1, 2, 3, 4, 5, 6] to name a few. From the differential game  point of view, L2-gain optimal control can be considered as a problem in which the control vector is “player” that minimises the optimality criterion, while the uncertainty vector is “player” that maximises the optimality criterion. Therefore, problem is also known as minimax optimal control . In order to avoid solving the corresponding HJI equation for L2-gain optimisation, in this thesis we propose an algorithm for direct minimisation of the performance criterion with respect to the control input, with simultaneous maximisation of the same performance criterion with respect to the uncertainty. The proposed algorithm uses: combination of subgradient and Newton’s method for optimisation; recursive calculation of derivatives; Adams method for the time discretisation and automatic differentiation, in order to achieve numerical robustness, stability and fast convergence. The version of the algorithm which gives the best efficiency and accuracy is determined by numerical simulations. The stability conditions are established by Lyapunov direct method. The developed algorithm is experimentally compared with the conventional approaches on the electrohydraulic systems. This thesis is organized in seven chapters, as follows: Chapter 1: Introduction. This chapter presents the relevance of the research. It reviews the literature related to the topic of the thesis and explains the research scope. The first chapter is divided into five sections which includes: motivation, overview of the previous research, research goals and hypotheses, methodology and plan of the research within this thesis, contributions and outline of the thesis. Chapter 2: Mathematical preliminaries and notation. The fundamental mathematical background, results and notation that are critical to the development of later chapters are collected and summarised in the Chapter 2. This chapter gives some basic concepts such as definitions and properties of the gradient of a vector, Jacobian matrix, Hessian matrix, vector and function norms, vectorisation of a matrix, Kronecker product, convexity and etc. Chapter 3: L2 control of nonlinear dynamical systems. This chapter is mainly designed to recall the main concepts and results from the L2-gain control theory in order to understand the contributions of this thesis. It is mostly based on the references that deal with: analysis of nonlinear control systems [9, 10, 11], H∞ control and related minimax problems, [8, 12, 13, 14, 15], differential games theory [7, 16]. Chapter 4: Synthesis of the algorithm for the minimax optimal control. This chapter is concerned with derivation of the algorithm for L2-gain optimisation of input-affine nonlinear systems. Chapter presents a more detailed version of the results published in authors previous works [17, 18, 19, 20]. In contrast to [1, 2, 3, 4, 5, 6], the aim is to avoid solving the HJI equation. The feedback control and the uncertainty variables are formulated as a linear combinations of approximation functions. It is well known that the problem of L2-gain optimisation is equivalent to the minimax fractional optimisation problem. Therefore, in order to solve this problem, we propose the algorithm which uses relationship between two-player zero-sum differential game with fractional cost function and two-player zero-sum differential game with the cost function including a parameter. Since the control, uncertainty and state variables are coupled via system dynamics in an input-affine form, the Newton-type optimisation algorithm with recursive calculation of the performance criterion gradients and Hessians with respect to the weights of the approximation functions is proposed. The methodology of recursive computation is based on the results of previous studies published in [17, 19, 21]. The algorithm presented in this thesis is an extension toward second order partial derivatives calculation. This recursive computation requires time discretisation of the overall problem, and for this purpose multistep Adams method is used. By using the Newton’s method the faster convergence to a solution than in the case of the conjugate gradient algorithm in [17, 19, 21] is achieved. While in our previous works sub-optimal problem is considered, in this thesis, an algorithm which finds optimal L2-gain solution is developed. We show that the solution corresponds to a particular value of single parameter which can be found by subgradient method. Jacobian and Hessian matrices of the nonlinear dynamics with respect to the state vector are calculated using automatic differentiation (AD). Application of AD provides significant reduction of the algorithm computational time and improved accuracy than numerical differentiation applied in . We develop an algorithm such that system dynamics are not included in the performance criterion as equality constraints, but rather included directly through derivation of gradients and Hessians that appear in the Newton-type minimax optimal computation of the approximation functions weights. Hence, in comparison with the common nonlinear programmingbased algorithms that use penalty function method or method of Lagrange multipliers in order to get unconstrained reformulations, the proposed algorithm has no high-dimensional sparse structure. Since the performance criterion is nondifferentiable due to minmax operator, the subgradient method is used. The subgradient method is known to be simple algorithm because of its low computational cost. The Newton’s method with line search strategy satisfying the Wolfe conditions is employed. A line search method satisfying the Wolfe conditions is introduce in order to globalize the Newton’s method, i.e. to insure the convergence from any starting point. In order to recursively calculate the first and the second order derivatives that appear in the Newton’s method, the time discretisation of system dynamics based on multistep Adams method is carried out. It is shown that Adams method can be transformed into causal state-space form and this fact, in comparison with popular Runge-Kutta method, significantly simplifies the calculation of derivatives. More specifically, the k-th order Runge-Kutta method requires the k computations in a sampling time, while Adams method requires only one computation. Discrete-time results converge towards continuous-time by decreasing the step size and by increasing the order of Adams method. In this way we are able to calculate the gradients and Hessians of the performance criterion with respect to the approximation functions weights. A solution to this problem is obtained by performing some elementary chain rule arithmetic, which results in recursive adjustment of the constant approximation functions weights. Therefore, the proposed algorithm provides the minimax optimal computation of the constants, so that we have continuous-time nonlinear controller which guarantee the best performance in the presence of the continuous-time uncertainty. Furthermore, Chapter 4 is also concerned with L2-gain optimisation of input-affine nonlinear systems controlled by analytic fuzzy logic system. Unlike the conventional fuzzy-based strategies, the nonconventional analytic fuzzy control method does not require an explicit fuzzy rule base. The presented approach is based on the fuzzy basis function expansion as an approximate realisation of the control and uncertainty variables. Instead of using conventional linguistic fuzzy IF-THEN rules, we use an analytical determination of the positions of the centres of the output fuzzy sets suggested in [22, 23]. In this thesis we prove that the fuzzy system without rule base has universal approximation property. Chapter 5: Stability analysis of the proposed control law with application to Euler- Lagrange systems. The subject of this chapter is the Lyapunov-based L2 stability analysis of nonlinear Euler-Lagrange systems in closed-loop with proposed form of the control law. Since, in the general case of nonlinear dynamical systems the construction of Lyapunov function is not easy, in this chapter we consider the control law with certain properties that make construction of Lyapunov function much easier, and thus the determination of L2 stability conditions. Chapter 6: Controller synthesis of the electrohydraulic systems. The main objective of this chapter is to apply an approach of the L2-gain performance criterion gradients and Hessians calculation, which is presented in Chapter 4, for controller synthesis of electro-hydraulic servo-systems (EHSS), both in simulations and experiments. Regarding the controlled output variable, three major types of EHSS have been investigated: the rotational motion control system, the linear motion control system and the force control system. Classical approaches, like PID regulators and linear static state feedback controllers for hydraulic drives, do not give satisfactory performance. Due to the existing limitations of such approaches various feedback linearisation, adaptive, backstepping and intelligent control algorithms have emerged. However, to the knowledge of the author, the problem of a nonlinear L2-gain optimal control of hydraulic actuator has not been investigated yet. Chapter 7: Conclusion. This chapter summarises the main contributions of the dissertation and presents several recommendations for future research.