Francisco Javier Bejarano

Mini-max integral sliding-mode control for multimodel linear uncertain systems

An original linear time-varying system with matched and unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external unmatched bounded disturbances. Such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system containing only matched uncertainties leads to a linear multi-model system with known unmatched bounded disturbances and unknown matched disturbances as well. Each model from a given finite set is characterized by a quadratic performance index. The developed minimax integral sliding mode control strategy gives an optimal minimax linear quadratic (LQ)-control with additional integral sliding mode term. The design of this controller is reduced to a solution of an equivalent mini-max LQ problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex. The additional integral sliding mode controller part completely dismisses the influence of matched uncertainties from the initial time instant. Two numerical examples illustrate this study.

Switched Observers for Switched Linear Systems With Unknown Inputs

Two state observers are designed for some classes of switched linear systems with unknown inputs. The design of the proposed observers assumes that all switching subsystems fulfill a property of “strong detectability” that allows to implement suitable reduced-order unknown-input switched observers. The synthesis of the observers is based on the feasibility of a certain system of LMIs. Two main schemes are presented. For the case when the set of possible unknown-input distribution matrices are linearly dependent, an observer is suggested that guarantees the asymptotic state reconstruction without imposing any slow-switching dwell-time constraint about the sequence of the switching times. For the general case, the existence of a minimal average dwell-time for every switching sequence is assumed. By appropriate Lyapunov analysis, the convergence of the state estimate is proven to be exponential in both cases. Simulation results confirm the predicted performance.

High order sliding mode observer for linear systems with unbounded unknown inputs

A global observer is designed for strongly detectable systems with unbounded unknown inputs. The design of the observer is based on three steps. First, the system is extended taking the unknown inputs (and possibly some of their derivatives) as a new state; then, using a global high-order sliding mode differentiator, a new output of the system is generated in order to fulfil, what we will call, the Hautus condition, which finally allows decomposing the system, in new coordinates, into two subsystems; the first one being unaffected directly by the unknown inputs, and the state vector of the second subsystem is obtained directly from the original system output. Such decomposition permits designing of a Luenberger observer for the first subsystem, which satisfies the Hautus condition, i.e. all the outputs have relative degree one w.r.t. the unknown inputs. This procedure enables one to estimate the state and the unknown inputs using the least number of differentiations possible. Simulations are given in order to show the effectiveness of the proposed observer.

Robust Control With Exact Uncertainties Compensation: With or Without Chattering?

The problem of robust exact output control for linear systems with smooth bounded matched unknown inputs is considered. The higher order sliding mode observers provide both theoretically exact observation and unknown input identification. In this paper, a methodology is proposed to select the most adequate output control strategy for matched perturbations compensation. The aim of this paper is to investigate the possibility for exact uncertainties compensation using signals identified by high order sliding mode observers. Towards this aim, we modify the hierarchical super-twisting observer in order to have the best observation and identification accuracy possible. Then, two controllers are compared. The first one is an integral sliding mode controller based on the observed values of the state variables. The other strategy is based on the direct compensation of matched perturbations using their identified values. The performance of both controllers is estimated in terms of the deterministic noise upper bounds, sampling step and execution time. Based on these estimations, the designer may select the proper controller for the system. Experimental results are given for an inverted rotary pendulum system.

Unknown Input and State Estimation for Unobservable Systems

The concept of strong detectability and its relation with the concept of invariant zeros is reviewed. For strongly detectable systems (which includes the strongly observable systems), it is proposed a hierarchical design of a robust observer whose trajectories converge to those of the original state vector. Furthermore, it is shown that neither left invertibility is a sufficient condition nor strong detectability is a necessary condition to estimate the unknown inputs. It is shown that the necessary and sufficient condition for estimating the unknown inputs is that the set of the invariant zeros that do not belong to the set of unobservable modes be within the interior of the left half plane of the complex space. This shows that the unknown inputs could be estimated even if it is impossible to estimate the entire state vector of the system. Two numerical examples illustrate the effectiveness of the proposed estimation schemes.

Output integral sliding mode control based on algebraic hierarchical observer

The problem of the realization of integral sliding mode controllers based only on output information is discussed. The implementation of an output integral sliding mode controller ensures insensitivity of the state trajectory with respect to the matched uncertainties from the initial time moment. In the case when the number of inputs is more than or equal to the number of outputs, the closed loop system, describing the output integral sliding mode dynamics, is shown to lose observability. For the case when the number of inputs is less than the number of outputs, a hierarchical sliding mode observer is proposed. The realization of the proposed observer requires a filtration to obtain the equivalent output injections. Assigning the first order low-pass filter parameter small enough (during this filter realization), the convergence time and the observation error can be made arbitrarily small. The results obtained are illustrated by simulations.

State exact reconstruction for switched linear systems via a super-twisting algorithm

This article discusses the problem of state reconstruction synthesis for switched linear systems. Based only on the continuous output information, an observer is proposed ensuring the reconstruction of the entire state (continuous and discrete) in finite time. For the observer design an exact sliding mode differentiator is used, which allows the finite time convergence of the observer trajectories to the actual trajectories. The design scheme includes both cases: zero control input and nonzero control input. Simulations illustrate the effectiveness of the proposed observer.

Output Integral Sliding Mode for Min-Max Optimization of Multi-Plant Linear Uncertain Systems

We consider the application of a min-max optimal control based on the LQ-index for a set of systems where only the output information is available. Here every system is affected by matched uncertainties, and we propose to use an output integral sliding mode to compensate the matched uncertainties right after the beginning of the process. For the case when the extended system is free of invariant zeros, a hierarchical sliding mode observer is applied. The error of realization of the proposed control algorithm is estimated in terms of the sampling step and actuator time constant. An example illustrates the suggested method of design.

Hierarchical second-order sliding-mode observer for linear time invariant systems with unknown inputs

The problem of observability for systems with unknown inputs is revised. The sufficient and necessary conditions are used for the design of an observer for linear systems with bounded unknown inputs. To realize the observation of the state, a second-order sliding-mode observer is suggested to be applied. Such an observer provides a robust estimate of the state vector in a finite time, without filtration. The design is based on the concept of the hierarchical output injection maintaining zero value for output tracking error at each level of the hierarchy. The equivalent control is used to identify the unknown inputs. A numerical example illustrates the effectiveness of the suggested technique.

Observability of linear systems with commensurate delays and unknown inputs

This paper investigates the observability analysis for linear time systems whose outputs are affected by unknown inputs. Three different definitions of observability are proposed. By introducing the Smith form and comparing the invariant factors, a sufficient condition is deduced for each proposed definition observability . Several examples are given for the purpose of highlighting the effectiveness of the proposed approach.