Controlling a system with control and state constraints is one of the most important problems in control theory, but also one of the most challenging. Another important but just as demanding topic is robustness against uncertainties in a controlled system. One of the most successful approaches, both in theory and practice, to control constrained systems is model predictive control (MPC). The basic idea in MPC is to repeatedly solve optimization problems on-line to find an optimal input to the controlled system. In recent years, much effort has been spent to incorporate the robustness problem into this framework.The main part of the thesis revolves around minimax formulations of MPC for uncertain constrained linear discrete-time systems. A minimax strategy in MPC means that worst-case performance with respect to uncertainties is optimized. Unfortunately, many minimax MPC formulations yield intractable optimization problems with exponential complexity.Minimax algorithms for a number of uncertainty models are derived in the thesis. These include systems with bounded external additive disturbances, systems with uncertain gain, and systems described with linear fractional transformations. The central theme in the different algorithms is semidefinite relaxations. This means that the minimax problems are written as uncertain semidefinite programs, and then conservatively approximated using robust optimization theory. The result is an optimization problem with polynomial complexity.The use of semidefinite relaxations enables a framework that allows extensions of the basic algorithms, such as joint minimax control and estimation, and approx- imation of closed-loop minimax MPC using a convex programming framework. Additional topics include development of an efficient optimization algorithm to solve the resulting semidefinite programs and connections between deterministic minimax MPC and stochastic risk-sensitive control.The remaining part of the thesis is devoted to stability issues in MPC for continuous-time nonlinear unconstrained systems. While stability of MPC for un-constrained linear systems essentially is solved with the linear quadratic controller, no such simple solution exists in the nonlinear case. It is shown how tools from modern nonlinear control theory can be used to synthesize finite horizon MPC controllers with guaranteed stability, and more importantly, how some of the tech- nical assumptions in the literature can be dispensed with by using a slightly more complex controller.

Minimax approaches to robust model predictive control

In this thesis we study a number of nonlinear control problems motivated by their appearance in flight control. The results are presented in a general framework and can also be applied to other areas. The two main topics are backstepping and control allocation.Backstepping is a nonlinear control design method that provides an alternative to feedback linearization. Here, backstepping is used to derive robust linear control laws for two nonlinear systems, related to angle of attack control and flight path angle control, respectively. The resulting control laws require less modeling information than corresponding designs based on feedback linearization, and achieve global stability in cases where feedback linearization can only be performed locally. Further, a method for backstepping control of a rigid body is developed, based on a vector description of the dynamics. We also discuss how to augment an existing nonlinear controller to suppress constant input disturbances. Two methods, based on adaptive backstepping and nonlinear observer design, are proposed.Control allocation deals with actuator utilization for overactuated systems. In this thesis we pose the control allocation problem as a constrained least squares problem to account for actuator position and rate constraints. Efficient solvers based on active set methods are developed with similar complexity to existing, approximate, pseudoinverse methods. A method for dynamic control allocation is also proposed which enables a frequency dependent control distribution among the actuators to be designed. Further, the relationship between control allocation and linear quadratic control is investigated. It is shown that under certain circumstances, the two techniques give the same freedom in distributing the control effort among the actuators. An advantage of control allocation, however, is that since the actuator constraints are considered, the control capabilities of the actuator suite can be fully exploited.

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Backstepping and control allocation with applications to flight control

In many industrial robot applications it is a fact that the robot is programmed to do the same task repeatedly. By observing the control error in the different iterations of the same task it becomes clear that it is actually highly repetitive. Iterative Learning Control (ILC) allows to iteratively compensate for and, hence, remove this repetitive error.In the thesis different aspects of iterative learning control are covered. Although stability is the most important in practice the design aspect is also highlighted. Several design schemes for iterative learning control methods are presented, including first order as well as second order iterative learning control. An adaptive approach to iterative learning control is also discussed. Many of the suggested design methods are also given with stability and robustness results.The application, industrial robot control, that has been used as a testbed throughout the thesis is described. The description includes a general discussion on robot modeling and control as well as a specific discussion on the implementation of the functions needed in the commercial robot control software in order to make it possible to apply iterative learning control.The suggested iterative learning control design methods are all tested on the robot. Some practical aspects on the path following problem for industrial robots using iterative learning control are discussed. A potential solution to the path tracking problem using additional sensors is given, although it is not yet implemented on the robot.

Iterative Learning Control : Analysis, Design, and Experiments

Avoiding collisions is a crucial issue in most transportation systems as well as in many other applications. The task of a collision avoidance system is to track objects of potential collision risk ...

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Collision Avoidance Theory : with Application to Automotive Collision Mitigation

Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models. More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system. In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented. In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs. It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem.

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Linear models of nonlinear systems

Feedback control of systems modelled as single input, time invariant, linear and continuous time systems is the subject of this thesis. A growing interest in the design of feedback control systems that can cope with model uncertainties has been the motivation for the study. Some commonly used design approches for various kinds of systems are investigated with respect to robustness properties.The Smith controller, often propsed to controlling open loop stable plants with an input delay, state feedback control based on linear quadratic optimization techniques and output feedback control using a Kalman filter are considered.Also some new design strategies are given. A graphical design philosphy suitable for open loop stable plants with no modelled input delay, feedback stabilization of systems with a known number of unstable poles and modification of the Kalman filter based controller are proposed. The objective in common for these control strategies is to obtain control systems with good robustness properties.

Kjell Nordström

Papers

Uncertainty, Robustness and Sensitivity Reduction in the Design of Single Input Control Systems