L₁-Robust Interval Observer Design for Uncertain Nonlinear Dynamical Systems

This paper proposes a novel unified interval-valued observer synthesis approach for locally Lipschitz nonlinear continuous-time (CT) and discrete-time (DT) systems with nonlinear observations. A key feature of our proposed observer, which is derived using mixed-monotone decompositions, is that it is correct by construction (i.e., the true state trajectory of the system is framed by the states of the observer) without the need for imposing additional constraints and assumptions such as global Lipschitz continuity or contraction, as is done in existing approaches in the literature. Furthermore, we derive sufficient conditions for designing stabilizing observer gains in the form of Linear Matrix Inequalities (LMIs). Finally, we compare the performance of our observer design with some benchmark CT and DT observers in the literature.

In this paper, sliding mode observer design principles based on the equivalent control approach are discussed for a linear time invariant system both in continuous and discrete time. For the continuous case, the observer is designed using a recursive procedure; however, the observer is eventually expressed as a replica of the original system with an additional auxiliary input with a certain nested structure. A direct discrete time counterpart of the sliding mode realization of a reduced order asymptotic observer using the discrete time equivalent control is also developed. Simulation of a linearized truck-trailer during a manoeuvre illustrates the approach. Results show the effectiveness and the finite time convergence characteristics of the proposed discrete and continuous time sliding mode observers.

Sliding Mode Control in Discrete Time Linear Systems

In our previous paper 4) -12), we have studied “Theory of Invariance” for nonlinear continuous-time systems with their inputs appearing linearly. The aim of this paper is to derive necessary and sufficient conditions that inputs never effect outputs (so called conditions of “Invariance”) for nonlinear discrete-time and continuous-time systems without the assumption of linearity in inputs. Applying these conditions to the decoupling problem, we obtained necessary and sufficient conditions which yield decoupling control laws for both the continuous-time systems and the discrete-time systems. The conditions of decoupling given by van der Schaft 13) and Grizzle 14) via geometric approach are different in forms from ours. But, it seems difficult to derive control laws from their conditions because the integration of partial deferential equations is needed there.

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In this paper, a decentralized controller is designed for cross coupled MIMO Twin rotor aerodynamic system (TRAS). The decentralized controller design is done by decoupling the MIMO system into two decoupled, a pitch and a yaw system. Then 2-DOF controller is designed using control technique in continuous and discrete time. The performance of these controllers are tested through extensive simulations in MatLAB and results are compared between discrete and continuous time systems.

The stability of an inverted pendulum is a textbook example of control. The easiest case is to put the pendulum on a cart and apply feedback force control on it to keep the upright position stable. This paper compares the no delay case (feed‐in‐time control: an anticipatory effect) and the delay differential equation approach. Then we study both continuous and discrete time systems. The main aim of the work is to investigate the behaviour of such systems at the stability boundaries by using numerical simulation. The principal points of interest are how continuous time systems differ from discrete time system at a bifurcation point and how time delay or an anticipatory feed‐in‐time control acts on its behaviour. Other exciting questions are how sampling delay can be taken into consideration and is bifurcation a robust phenomenon.

On the functional observers for linear descriptor systems

This paper considers the design of a variable structure observer for discrete-time systems. The output of the system is assumed to be multidimensional, which extends existing work in which unidimensional outputs are assumed. The system is affected by both matched and unmatched disturbances. An ultimately bounded stable variable structure observer is proposed. The main novelty of the approach is that the design of the observer is written as an optimization problem with linear constraints. The objective is the minimization of the ultimate boundedness of the response. Using Linear Matrix Inequalities, the degrees of freedom of the observer are constructively exploited, which provides advantages when compared with existing frameworks.

In this paper, a step by step algorithm is given to design functional unknown input observer for nonlinear discrete time systems under the Polytopic Takagi-Sugeno framework (also known as multiple models systems). Applying the Lyapunov theory and the ℒ 2 attenuation, Linear Matrix Inequalities (LMI)s conditions are deduced which are solved for feasibility to obtain observer design matrices. To reject the effect of unknown input, classical approach of decoupling the unknown input for the linear case is used. A solver based solution is proposed. The novelty of the proposed approach consists in solving simultaneously both structural constraints and LMIs, which ensure a mean for the efficient design of the gains of the observers. To illustrate the proposed theoretical results, an application example of model reference tracking control applied to an electro-mechanical model of a motor with a time varying parameter is discussed.

The model-following servo controller is useful for the nonlinear robot arm. We analyzed unit step response of this controller and cleared transient characteristics in both cases of continuous and discrete time systems. We used linear quadratic optimal control law with integral action and a criterion function considering the square of control error and weighted controlling input. The overshoot is always 8.1% regardless of continuous time, discrete time and the weight for controlling input. A function of the weight and the sampling time determines the speed of responses.

Steady-state, harmonic response and moments of linear systems with periodic jumps

<abstract><p>Neural networks (NNs) with values in multidimensional domains have lately attracted the attention of researchers. Thus, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and their generalization, Clifford-valued neural networks (ClVNNs) have been proposed in the last few years, and different dynamic properties were studied for them. On the other hand, time scale calculus has been proposed in order to jointly study the properties of continuous time and discrete time systems, or any hybrid combination between the two, and was also successfully applied to the domain of NNs. Finally, in real implementations of NNs, time delays occur inevitably. Taking all these facts into account, this paper discusses ClVNNs defined on time scales with leakage, time-varying delays, and infinite distributed delays, a type of delays which have been relatively rarely present in the existing literature. A state feedback control scheme and a generalization of the Halanay inequality for time scales are used in order to obtain sufficient conditions expressed as algebraic inequalities and as linear matrix inequalities (LMIs), using two general Lyapunov-like functions, for the exponential synchronization of the proposed model. Two numerical examples are given in order to illustrate the theoretical results.</p></abstract>

This paper combines a Smith Predictor with an approach for graphically determining all Proportional-Integral-Derivative (PID) controllers in either the Continuous-Time (CT) or Discrete-Time (DT) domains that meet Gain Margin (GM) and Phase Margin (PM) specifications. Smith Predictors (SP) are often used when designing controllers for systems that exhibit "relatively" large delays that may cause the system's stability and/or performance to deteriorate. The PID controller gains, namely Proportional gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> ), Integral gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ) and Derivative gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> ), will be determined graphically using only the frequency response of the system's components, i.e., plant with delay and SP structure. The inclusion of a SP along with a PID controller can significantly improve stability margins and performance when compared to relying solely on a PID controller. The improvement can be observed even if there is a mismatch between the actual process and its corresponding SP model. By using the delta operator, the procedure outlined applies to both CT and DT systems, hence, the unified approach designation. Additionally, this procedure will ensure meeting predesign requirements in terms of gain and phase margins. The stability boundaries of the PID controller gains will be determined graphically where within the boundaries, nominal stability is guaranteed and stability margins requirements are met.

The contributions of this paper are threefold. We first explore the differences in uncertainty representations and control designs between continuous time systems and discrete time systems, particularly the high frequency interpretation does not work for discrete time systems. It is shown that uncertainties appeared in sampled data systems may be highly structured, however they do not fall into any category which is currently available in the literature. Secondly, we define a measure on these uncertainties which confines a neighborhood to the nominal model in terms of frequency content, and then we propose a control design method which guarantees the stability of the closed loop system in the face of uncertainties. Last, we consider a generalization of a classical continuous time approach in representing the multiplicative uncertainty to sampled data systems by using a slightly modified sampler and present a method to calculate a frequency bound on the uncertainty in sampled data systems.