On the use of input-output feedback linearization techniques for the control of nonminimum-phase systems

The objective of the thesis is to study the possibility of using input-output feedback linearization techniques for controlling nonlinear nonminimum-phase systems. Two methods are developed. The first one is based on an approximate input-output feedback linearization, where a part of the internal dynamics is neglected, while the second method focuses on the stabilization of the internal dynamics. The inverse of a nonminimum-phase system being unstable, standard input-output feedback linearization is not effective to control such systems. In this work, a control scheme is developed, based on an approximate input-output feedback linearization method, where the observability normal form is used in conjunction with input-output feedback linearization. The system is feedback linearized upon neglecting a part of the system dynamics, with the neglected part being considered as a perturbation. Stability analysis is provided based on the vanishing perturbation theory. However, this approximate input-output feedback linearization is only effective for very small values of the perturbation. In the general case, the internal dynamics cannot be crushed and need to be stabilized. On the other hand, predictive control is an effective approach for tackling problems with nonlinear dynamics, especially when analytical computation of the control law is difficult. Therefore, a cascade-control scheme that combines input-output feedback linearization and predictive control is proposed. Therein, inputoutput feedback linearization forms the inner loop that compensates the nonlinearities in the input-output behavior, and predictive control forms the outer loop that is used to stabilize the internal dynamics. With this scheme, predictive control is implemented at a re-optimization rate determined by the internal dynamics rather than the system dynamics, which is particularly advantageous when internal dynamics are slower than the input-output behavior of the controlled system. Exponential stability of the cascade-control scheme is provided using singular perturbation theory. Finally, both the approximate input-output feedback linearization and the cascade-control scheme are implemented successfully, on a polar pendulum 'pendubot' that is available at the Laboratoire d'Automatique of EPFL. The pendubot exhibits all the properties that suit the control methodologies mentioned above. From the approximate input-output feedback linearization point of view, the pendubot is a nonlinear system, not input-state feedback linearizable. Also, the pendubot is nonminimum phase, which prevents the use of standard input-output feedback linearization. From the cascade control point of view, although the pendubot has fast dynamics, the input-output feedback linearization separates the input-output system behavior from the internal dynamics, thus leading to a two-time-scale systems: fast input-output behavior, which is controlled using a linear controller, and slow reduced internal dynamics, which are stabilized using model predictive control. Therefore, the cascade-control scheme is effective, and model predictive control can be implemented at a low frequency compared to the input-output behavior.

Neighboring extremals in optimization and control

Sébastien Gros

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.

EPFL2007

Neighboring extremals in optimization and control

Sébastien Gros

EPFL2007