Sliding Mode Control in Discrete Time Linear Systems

Sain and Massey [1] have obtained necessary and sufficient conditions for the invertibility of continuous and discrete time linear systems and have also found bounds for what they termed the "inherent integration" (continuous time) or the "inherent delay" (discrete time) of an invertible linear system. In this note, we tighten these bounds by modifying an argument used to prove the Sain-Massey result.

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.

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.

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.

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.

This paper presents a novel interval observer design for uncertain locally Lipschitz continuous-time (CT) and discrete-time (DT) systems with noisy nonlinear observations that is input-to-state stable (ISS) and minimizes the L1-gain of the observer error system with respect to the uncertainties. Using mixed-monotone decompositions, the proposed observer is correct and positive by construction without the need for additional constraints/assumptions. This, in turn, allows us to directly leverage techniques for positive systems to design an ISS and L1-robust interval observer via mixed-integer (linear) programs instead of semi-definite programs with linear matrix inequalities. Further, our observer design offers additional degrees of freedom that may serve as a surrogate for coordinate transformations. Finally, we demonstrate the effectiveness of our proposed observer on some CT and DT systems.

It is shown how genetic algorithms can be applied for system identification of both continuous and discrete time systems. It is shown that they are effective in both domains and are able to directly identify physical parameters or poles and zeros. This can be useful because changing one physical parameter might affect every parameter of a system transfer function. The estimates of poles and zeros are then used to design a discrete time pole placement adaptive controller. Simulations for minimum and nonminimum phase systems and a system with unmodeled dynamics are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The question of power and asymptotic stability of infinite dimensional discrete-time state space systems is investigated. It is shown that every balanced realization is asymptotically stable. Conditions are given for balanced, input normal, or output normal realization to be asymptotically and/or power stable. >

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.

In this work we study the design and analysis for nonlinear discrete time active control system with boring charter. It is shown that most analysis result for continuous time nonlinear system can be extended to the discrete time case. In previous studies, a method of nonlinear Model Following Control System (MFCS) was proposed by Okubo (1985). In this study, the method of nonlinear MFCS will be extended to nonlinear discrete time system with boring charter. Nonlinear systems which are dealt in this study have the property of norm constraints ║ƒ (v (k))║≤α + β║v (k)║^γ , where α≥0, β≥0, 0≤γ≤1. When 0≤γ<1. It is easy to extend the method to discrete time systems. But in the case γ = 1 discrete time systems, the proof becomes difficult. In this case, a new criterion is proposed to ensure that internal states are stable. We expect that this method will provide a useful tool in areas related to stability analysis and design for nonlinear discrete time systems as well.

Robust optimization-based control of constrained linear discrete time systems with bounded disturbances

Using semi-tensor product and the vector form of multi-valued logical variables, the mix-valued logical system is expressed as a linear discrete time system and the mix-valued logical control system is expressed as a bilinear discrete time system with respect to the state and control variables. In this paper, the topological property of mix-valued logical network and the reachability and controllability of mix-valued logical control systems are discussed. From the algebraic form of mix-valued logical network, we obtain the number of the logical system. The formula we obtained is based on the rank of structure matrix. Also, the necessary and sufficient conditions are obtained for testing the reachability and controllability.

We address the problems of computing operator norms of matrices induced by given norms on the argument and the image space. It is known that aside of a fistful of “solvable cases”, most notably, the case when both given norms are Euclidean, computing operator norm of a matrix is NP-hard. We specify rather general families of norms on the argument and the images space (“ellitopic” and “co-ellitopic”, respectively) allowing for reasonably tight computationally efficient upper-bounding of the associated operator norms. We extend these results to bounding “robust operator norm of uncertain matrix with box uncertainty”, that is, the maximum of operator norms of matrices representable as a linear combination, with coefficients of magnitude <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≤</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>, of a collection of given matrices. Finally, we consider some applications of norm bounding, in particular, (1) computationally efficient synthesis of affine non-anticipative finite-horizon control of discrete time linear dynamical systems under bounds on the peak-to-peak gains, (2) signal recovery with uncertainties in sensing matrix, and (3) identification of parameters of time invariant discrete time linear dynamical systems via noisy observations of states and inputs on a given time horizon, in the case of “uncertain-but-bounded” noise varying in a box.

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

SUMMARY The LQ+ problem, i.e. the finite-horizon linear quadratic optimal control problem with nonnegative state constraints, is studied for positive linear systems in continuous time and in discrete time. Necessary and sufficient optimality conditions are obtained by using the maximum principle. These conditions lead to a computational method for the solution of the LQ+ problem by means of a corresponding Hamiltonian system. In addition, the necessary and sufficient conditions are proved for the LQ+-optimal control to be given by the standard LQ-optimal state feedback law. Then sufficient conditions are established for the positivity of the LQ-optimal closed-loop system. In particular, such conditions are obtained for the problem of minimal energy control with penalization of the final state. Moreover, a positivity criterion for the LQ-optimal closed-loop system is derived for positive discrete-time systems with a positively invertible (dynamics) generator. The main results are illustrated by numerical examples. Copyright q 2010 John Wiley & Sons, Ltd.