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Publication# Robust control of constrained systems given an information structure

Résumé

We study finite horizon optimal control where the controller is subject to sensor-information constraints, that is, each input has access to a fixed subset of states at all times. In particular, we consider linear systems affected by exogenous disturbances with state and input constraints. We establish the class of sensor-information structures that allows for the formulation of this optimization problem as a convex program. In the literature, Quadratic Invariance (QI) is a well-established result that is applicable to the infinite horizon unconstrained case. We show that, despite state and inputs constraints being enforced, QI results can be naturally adapted to our problem. To this end, we highlight and exploit the connection between Youla parametrization and disturbance-feedback policies. Additionally, we provide graph-theoretic visual insight which is consistent with Partially Nested (PN) interpretations.

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Concepts associés (9)

Résolution de problème

vignette|Résolution d'un problème mathématique.
La résolution de problème est le processus d'identification puis de mise en œuvre d'une solution à un problème.
Méthodologie
Dans l'ind

Robustesse (ingénierie)

En ingénierie, la robustesse d'un système se définit comme la « stabilité de sa performance ».
On distingue trois types de systèmes :

- les systèmes non-performants, qui ne remplissent pas les fonct

Système

Un système est un ensemble d' interagissant entre eux selon certains principes ou règles. Par exemple une molécule, le système solaire, une ruche, une société humaine, un parti, une armée etc.
Un s

Publications associées (24)

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This thesis deals with dynamic optimization of batch processes, i.e. processes that are characterized by a finite time of operation and the frequent repetition of batches. The objective is to maximize the quantity of the desired product at final time while satisfying constraints during the batch (path constraints) and at final time (terminal constraints). The classical approach is to apply, in an open-loop fashion, input profiles that have been determined off-line using optimization. In practical applications, however, a conservative stand has to be taken since uncertainty is present, which may lead to constraint violations or non-optimal operation. Measurement-based optimization helps reduce this conservatism by using measurements to compensate for the effect of uncertainty. In order to achieve optimal operation, it has been proposed to track the necessary conditions of optimality (NCO). For the terminal cost optimization of batch processes, the NCO consist in four parts : Path constraints and path sensitivity conditions that have to be met during the batch, as well as terminal constraints and terminal sensitivity conditions that have to be met at final time. The intuitive approach is to handle the path conditions during the batch by using on-line measurements, and to implement the terminal conditions on a run-to-run basis since measurements of the terminal conditions are available off-line. However, run-to-run adaptation methods have two important disadvantages : i) Several runs are necessary to obtain optimal operation, and ii) within-run perturbations cannot be compensated. In this thesis, it is shown via a variational analysis of the NCO that meeting the constraint parts of the NCO is more important than meeting the sensitivity parts. Therefore, methods for steering the system toward the terminal constraints by using on-line measurements are examined in order to give priority to meeting the constraint-seeking conditions. This work proposes to track run-time references that lead to the constraints at final time. Thus, a two-time-scale methodology is proposed, where the task of meeting the active terminal constraints is addressed on-line using trajectory tracking, whilst pushing the sensitivities to zero is implemented on a run-to-run basis. Moreover, the use of Iterative Learning Control (ILC) to improve tracking is studied. Reference tracking is often accomplished with linear controllers of the PID type since they are widely accepted in industry and easy to implement. However, as batch processes operate over a wide region and are often highly nonlinear systems, tracking errors are inevitable. A possibility to reduce tracking errors is to use time-varying feedforward inputs. The feedforward inputs can be generated by using a process model, but the presence of model uncertainty can limit the performance. Instead of a process model, ILC uses error information from previous batches for computing the feedforward inputs iteratively. Since batch processes are often not reset to identical initial conditions, ILC schemes that provide a certain robustness to errors in initial conditions must be used, such as ILC with forgetting factor. In this work, a scheme that shifts the input of the previous run backward in time is proposed. The advantage of the input shift over the use of a forgetting factor is that, when the reference trajectory is constant and the system stable, the tracking error decreases with run time. In addition to a shift of the input, a shift of the error trajectory, which is known as anticipatory ILC, is also used. The foregoing methodological developments are illustrated by two case studies, a simulated semi-batch reactor and an experimental laboratory-scale batch distillation column. It is shown that meeting the terminal constraints is the most important optimality condition for the processes under study. Provided that selected composition measurements are available on-line, these measurements are used to steer the processes to the desired terminal constraints on product composition. In comparison to run-to-run optimization schemes, performance is improved from the first batch on, and within-run perturbations can be rejected.

Many real-world systems are intrinsically nonlinear. This thesis proposes various algorithms for designing control laws for input-affine single-input nonlinear systems. These algorithms, which are based on the concept of quotients used in nonlinear control design, can break down a single-input system into cascade of smaller subsystems of reduced dimension. These subsystems are well defined for feedback-linearizable systems. However, approximations are required to handle non-feedback-linearizable systems. The method proceeds iteratively and consists of two stages. During the forward stage, an equivalence relationship is defined to isolate the states that are not directly affected by the input, which reduces the dimension of the system. The resulting system is an input-affine single-input system controlled by a pseudo-input which represents a degree of freedom in the algorithm. The pseudo-input is a complementary state required to complete the diffeomorphism. This procedure is repeated (n − 1) times to give a one-dimensional system, where n is the dimension of the system. The backward stage begins with the one-dimensional system obtained at the end of the forward stage. It iteratively builds the control law required to stabilize the system. At every iteration, a desired profile of the pseudo-input is computed. In this next iteration, this desired profile is used to define an error that is driven asymptotically to zero using an appropriate control law. The quotient method is implemented through two algorithms, with and without diffeomorphism. The algorithm with diffeomorphism clearly depicts the dimension reduction at every iteration and provides a clear insight into the method. In this algorithm, a diffeomorphism is synthesized in order to obtain the normal form of the input vector field. The pseudo-input is the last coordinate of the new coordinate system. A normal projection is used to reduce the dimension of the system. For the algorithm to proceed without any approximation, it is essential that the last coordinate appears linearly in the projection of the transformed drift vector field. Necessary and sufficient conditions to achieve linearity in the last coordinate are given. Having the pseudo-input appearing linearly enables to represent the projected system as an input-affine system. Hence, the whole procedure can be repeated (n−1) times so as to obtain a one-dimensional system. In the second algorithm, a projection function based on the input vector field is defined that imitates both operators, the push forward operater and the normal projection operator of the previous algorithm. Due to the lack of an actual diffeomorphism, there is no apparent dimension reduction. Moreover, it is not directly possible to separate the drift vector field from the input vector field in the projected system. To overcome this obstacle, a bracket is defined that commutes with the projection function. This bracket provides the input vector field of the projected system. This enables the algorithm to proceed by repeating this procedure (n−1) times. As compared with the algorithm with diffeomorphism, the computational effort is reduced. The mathematical tools required to implement this algorithm are presented. A nice feature of these algorithms is the possibility to use the degrees of freedom to overcome singularities. This characteristic is demonstrated through a field-controlled DC motor. Furthermore, the algorithm also provides a way of approximating a non-feedback-linearizable system by a feedback-linearizable one. This has been demonstrated in the cases of the inverted pendulum and the acrobot. On the other hand, the algorithm without diffeomorphism has been demonstrated on the ball-on-a-wheel system. The quotient method can also be implemented whenever a simulation platform is available, that is when the differential equations for the system are not available in standard form. This is accomplished numerically by computing the required diffeomorphism based on the data available from the simulation platform. Two versions of the numerical algorithm are presented. One version leads to faster computations but uses approximation at various steps. The second version has better accuracy but requires considerably more computational time.

Optimization arises naturally when process performance needs improvement. This is often the case in industry because of competition – the product has to be proposed at the lowest possible cost. From the point of view of control, optimization consists in designing a control policy that best satisfies the chosen objectives. Most optimization schemes rely on a process model, which, however, is always an approximation of the real plant. Hence, the resulting optimal control policy is suboptimal for the real process. The fact that accurate models can be prohibitively expensive to build has triggered the development of a field of research known as Optimization under Uncertainty. One promising approach in this field proposes to draw a strong parallel between optimization under uncertainty and control. This approach, labeled NCO tracking, considers the Necessary Conditions of Optimality (NCO) of the optimization problem as the controlled outputs. The approach is still under development, and the present work is today's most recent contribution to this development. The problem of NCO tracking can be divided into several subproblems that have been studied separately in earlier works. Two main categories can be distinguished : (i) tracking the NCO associated with active constraints, and (ii) tracking the NCO associated with sensitivities. Research on the former category is mature. The latter problem is more difficult to solve since the sensitivity part of the NCO cannot be directly measured on the real process. The present work proposes a method to tackle these sensitivity problems based on the theory of Neighboring Extremals (NE). More precisely, NE control provides a way of calculating a first-order approximation to the sensitivity part of the NCO. This idea is developed for static and both nonsingular and singular dynamic optimization problems. The approach is illustrated via simulated examples: steady-state optimization of a continuous chemical reactor, optimal control of a semi-batch reactor, and optimal control of a steered car. Model Predictive Control (MPC) is a control scheme that can accommodate both process constraints and nonlinear process models. The repeated solution of a dynamic optimization problem provides an update of the control variables based on the current state, and therefore provides feedback. One of the major drawbacks of MPC lies in the expensive computations required to update the control policy, which often results in a low sampling frequency for the control loop. This limitation of the sampling frequency can be dramatic for fast systems and for systems exhibiting a strong dispersion between the predicted and the real state such as unstable systems. In the MPC framework, two main methods have been proposed to tackle these difficulties: (i) The use of a pre-stabilizing feedback operating in combination with the MPC scheme, and (ii) the use of robust MPC. The drawback of the former approach is that there exists no systematic way of designing such a feedback, nor is there any systematic way of analyzing the interaction between the MPC controller and this additional feedback. This work proposes to use the NE theory to design this additional feedback, and it provides a systematic way of analyzing the resulting control scheme. The approach is illustrated via the control of a simulated unstable continuous stirred-tank reactor and is applied successfully to two laboratory-scale set-ups, an inverted pendulum and a helicopter model called Toycopter. The stabilizing potential of NE control to handle fast and unstable systems is well illustrated. In the case of a strong dispersion between the state trajectories predicted by the model and the real process, robust MPC becomes infeasible. This problem can be addressed using robust MPC based on multiple input profiles, where the inherent feedback provided by MPC is explicitly taken into account, thereby increasing the size of the set of feasible inputs. The drawback of this scheme is its very high computational complexity. This work proposes to use the NE theory in the robust MPC framework as an efficient way of dealing with the feasibility issue, while limiting the computational complexity of the approach. The approach is illustrated via the control of a simulated unstable continuous stirred-tank reactor, and of an inverted pendulum.