Control Theory

Topics in this area

• Optimization based Control / Predictive Control
• Robust control / L1-optimal control
• Polynomial Control Systems
• Networked Control Systems
• Observer Design
• Differential-Algebraic Systems
• Adaptive lambda-tracking

Optimization based Control / Predictive Control

Model Predictive Control (MPC), often referred to as moving horizon control or receding horizon control, is one of the most successful and most popular advanced control methods. It is based on the repeated solution of a finite-horizon optimal control problem subject to a performance specification, constraints on states and inputs, and a system model. The reasons for the success of MPC are manifold. In many control problems it is desired to be optimal with respect to some performance specification. Typical examples include the maximization of profit or yield as well as the minimization of energy consumption or processing time. However, it is often hard or even impossible to find analytically a closed form solution to such a problem. Therefore, in MPC the given optimal control problem is solved repeatedly online based on the current measurement of the system states. In addition to its optimal performance MPC is one of the few control methods able to explicitly consider state and input constraints, where there is a high demand from industry for methods to deal with those. Constraints occur in a vast number of practical applications, such as the use of actuators, which are naturally limited, or the necessity to operate within safety bounds. Furthermore, the development of especially tailored numerical methods and steadily increasing power of today's computers seem promising for the future success of MPC. As of today, linear MPC theory can be considered as quite mature. However, many applications require the operation over a wide region instead of just a neighbourhood of an operating point. Here, linear models are not sufficient to guarantee a good performance, and one has to fall back on models with often complex nonlinear dynamics. Although quite a number of results in the field of nonlinear MPC (NMPC) have been published in the last 15 years, still many important problems remain unsolved. Thus, research at the IST focuses on at least partially solving these problems.

Basic MPC Loop

• Stability and robustness

  Optimization over a finite-horizon does not necessarily imply closed-loop stability. An important NMPC scheme, namely the so-called quasi-infinite horizon NMPC approach, which guarantees closed-loop stability in the nominal case, has been derived by members of the IST (H. Chen together with F. Allgöwer in 1998). One of the central questions of our current research in NMPC is how to develop and improve NMPC schemes which guarantee nominal and if possible robust stability of the closed-loop [Böhm et al., 2008; Böhm et al., 2009]. A new constraint tightened robust model predictive scheme for nonlinear systems is proposed in [Yu et al., 2010], where robust stability as well as recursive feasibility are guaranteed for the set which is feasible at the initial time instant.

• Efficient formulation of the optimization problem using LMIs

  Due to the nonlinear dynamics considered in standard NMPC, the underlying optimal control problem is often non-convex and may lead to suboptimal solutions. One of the key NMPC research fields at the IST is to develop approaches which approximate the original optimization problem with a convex one that is based on linear matrix inequalities. [Raff et al., 2004c; Böhm et al., 2009c; Böhm et al., 2009e; Reble et al., 2009]. Another field of research are linear parameter-varying (LPV) systems. MPC schemes with parameter-dependent control laws are developed such that an upper bound of the cost functions and the L2 gain from the disturbances to the performance outputs are minimized [Yu et al., 2009a, Yu et al. 2009b].

• Time-delay systems

  Many systems possess inherent time-delays in the states. Typical examples are transportation of material and propagation of data such as chemical or biological reactors, combustion engines, telemanipulation systems, or communication networks. However, there are still only few results on the control of nonlinear time-delay systems. At the IST, MPC schemes have been developed which guarantee stability of the closed loop [Raff et al., 2007a; Esfanjani et al., 2009].

• Distributed MPC

  In large-scale dynamical systems and networks of cooperating systems, it is often not possible or desirable to control the overall system with one centralized controller. Hence in recent years, the field of distributed MPC has gained significant attention, where each of the subsystems is locally controlled by an MPC controller and exchanges information about the predicted trajectories with its neighbors. At the IST, a distributed MPC scheme has been developed such that consistency within the overall system and convergence to the desired goal, like setpoint stabilization, consensus or synchronization, can be guaranteed [Müller et al., 2011].

• Output-feedback NMPC

  Most of the theoretical developments in the area of NMPC are based on the assumption that the full state is available for measurement. This assumption does not hold in the typical practical case. Results on the output feedback problem in NMPC with guaranteed stability are included in: [Findeisen et al., 2003b; Findeisen et al., 2003d; Findeisen and Allgöwer, 2005a; Böhm et al., 2008].

• Sampled-data systems

  Sampled-data NMPC refers to NMPC schemes in which the optimal control problem is only solved at discrete recalculation instants. Efficient implementations and theoretical questions concerning stability, robustness and compensation of delays have been investigated at the IST [Findeisen and Allgöwer, 2004d; Findeisen, 2006].

• Consideration of structural properties

  The use of a prediction model in NMPC allows the explicit consideration of structural system properties, such as observability or identifiability of parameters. In [Böhm et al., 2009a] an approach has been developed which ensures that the considered system is observable along optimal trajectories, thus fulfilling a basic requirement for state estimation.

• Applications

  The NMPC research at the IST is not only theoretically motivated but also has the goal to make predictive control more attractive for practical applications. The methods developed at the IST have been successfully tested in simulation in the fields of magnetic spacecraft control [Böhm et al., 2009b], chemical batch reactor processes [Nagy et al., 2005c; Nagy, et al. 2004c] turbocharged Diesel engines [Herceg et al., 2006], and vehicle dynamics [Böhm et al., 2009a].

Robust Control / L1-Optimal Control

Norm-based optimization is used to address robustness and performance issues of a controlled system. In particular the work at the IST currently focuses on l1-optimal control, multi-objective synthesis, and linear parameter-varying (LPV) control problems. It is one goal of our investigations to find computationally efficient formulations for optimal and robust controllers for these problems.

The l₁-paradigm addresses persistent bounded disturbances and time domain performance specifications as disturbance rejection, overshoot, bounded magnitude and bounded slope or actuator saturation. In contrast to quadratic performance objectives l₁controller design can be used directly to achieve time-domain performance issues. Currently we extend l₁-optimal control strategies to linear descriptor systems. As an application example an l₁-optimal control strategy has been successfully applied to improve the performance of an atomic force microscopy.

Multi-objective synthesis problems arise when a controller has to achieve different types of performance goals. These design processes come with a high numerical burden and we are especially interested in complexity reduction and efficient algorithms for controller design.

Modelling and identification often results in uncertain system parameters and/or uncertain system dynamics which leads to our interest in the development of new robust controller design methods. One of our goals is to extend multi-objective synthesis to systems with parametric and/or dynamic model uncertainties. Another interest is the design of controllers for linear parameter-varying (LPV) systems since these models also fit in the robust controller synthesis framework. We investigate both, time-invariant and time-varying parameter-dependent controllers. In particular, we have developed a novel structure to synthesize gain-scheduled controllers for linear parameter-varying systems.Moreover we are interested in LMI relaxations for robust analysis and synthesis with low conservatism.

Polynomial Control Systems

Analysis & Design Strategy

Nonlinear control design methods has made major advances in the last decade. However, other then in linear control design methods, where the control engineer can revert to a powerful arsenal of well-developed software tools, there is a lack of computational tools for nonlinear analysis and design strategies. This drawback becomes more and more relevant for complex control systems, which usually appear in real world applications. One reason for this gap between theory and computation lies in the fact that many nonlinear analysis and design strategies lead to compuationally intractable solutions, i.e., computationally ill-posed, or are of heuristic nature. The objective of this research project is to find computationally tractable solutions for the analysis and design of polynomial control systems. Computationally tractable solutions are solutions in terms of relations which can be efficiently and reliably solved on a computer. Polynomial control systems are in-between linear and nonlinear systems and many practical relevant problems can be formulated and approximated by polynomial control systems or can be transformed into polynomial control systems. The applied computational machinery is semidefinite programming, a theoretically well-founded efficient and reliable optimization "technology". One idea is to use advanced Lyapunov stability theory (dissipation inequalities) and the so-called sum-of-squares decomposition (semidefinite programming) to obtain systems analysis and design tools for polynomial control systems.

Networked Control Systems

Observer Design

In many if not even most real life processes it is not possible to measure all process variables of interest. But exactly the knowledge of these system states is often needed for supervision, fault detection and control tasks. This is especially true for the control of nonlinear systems since most of current controller design techniques require the knowledge of the full system state for their implementation. Thus, reliable and robust techniques for the reconstruction of the system states (or system variables) are needed. Commonly this task is performed utilizing a dynamical system called an observer. This observer computes the system states from the knowledge of the inputs and outputs of the system to be observed. Our group focuses in this area on the following topics: Finite Time Observers, High Gain Observers, Observers for Nonlinear Time Delay Systems

Differential-Algebraic Systems

Differential-algebraic systems (DAEs), sometimes also referred to as singular, semistate or descriptor systems, describe a broad class of systems that are of great interest both theoretically and practically. For example, models of chemical processes or mechanical systems with holonomic and non-holonomic constraints typically consist of DAEs. Available results for these systems generally apply to the restricted case of index-one DAEs, but in practice these systems often have an index greater than one. We have developed various approaches to strugle with the problem of higher index DAEs based on LMI methods and predictive control.

Adaptive lambda-tracking