Delayed finite-dimensional observer-based control of 1-D parabolic PDEs

Designing delay-dependent functional observers for LTI systems with multiple known time-varying state delays and unknown time-varying input delays is studied. The input delays are arbitrary, but the state delays should be upper-bounded. In addition, two scenarios of slow-varying and fast-varying state delays are investigated. The results of the paper can also be considered as one of the first contributions considering unknown-input functional observer design for linear systems with multiple time-varying state delays. Based on the Lyapunov Krasovskii approach, delay-dependent sufficient conditions of the exponential stability of the observer in each scenario are established in terms of linear matrix inequalities. Because of using effective techniques, such as the descriptor transformation and an advanced weighted integral inequality, the proposed stability criteria can result in larger stability regions compared with the other papers that study functional observers for time-varying delay systems. Furthermore, to help with the design procedure, a genetic algorithm-based scheme is proposed to adjust a weighting matrix in the established linear matrix inequalities. Two numerical examples illustrate the design procedure and demonstrate the efficacy of the proposed observer in each scenario. Copyright © 2016 John Wiley & Sons, Ltd.

Functional electrical stimulation (FES) is commonly used to facilitate cycling tasks for people with lower-limb movement disorders. In this work, FES and motor controllers are designed to track a desired power and cadence, respectively, and a Lyapunov-based switched systems analysis is performed to guarantee uniformly ultimately bounded power tracking and global exponential cadence tracking for a switched, delayed, nonlinear, and uncertain FES-cycling system. A unique challenge in this problem is that there is an unknown time-varying input delay to produce force, and a different unknown time-varying residual input delay where force is still produced after stimulation is removed. These delays impact the dwell-time conditions that dictate stimulation timing, and if not properly accounted for can lead to undesired effects such as antagonistic muscles exerting force at the same time or potential instabilities. The proposed controllers were validated by experimental analysis of four participants with neurological conditions (NCs) and five able-bodied participants, and yielded average power and cadence tracking errors of 0.01 ± 0.09 W and −0.05 ± 0.65 revolutions per minute (RPM), respectively, for the able-bodied participants and 0.01 ± 1.11 W and −0.07 ± 1.17 RPM, respectively, for the participants with NCs.

Output Feedback Stabilization for a Wave-ODE Cascade System with the Time-Varying Input and Output Delay

This research paper deals with a novel delay-range-dependent approach for observer design for systems with time-varying delay in the state variables and the output measurement vector. Lyapunov Karasovskii (LK) functional, Jensen's inequality and linear matrix inequality(LMI) tools are employed to derive the stability conditions ensuring the state estimation error to asymptotically converge to the origin. Further, as particular cases of proposed design methodology, delay-range-dependent design conditions are separately deduced for systems with state delays or output delays. Numerical example is simulated to validate the efficiency of the derived results.

Robust backstepping control of uncertain nonlinear systems with unknown time-varying input delay

In this paper, a dynamic decoupling approach is proposed to improve the robust Takagi-Sugeno (T-S) fuzzy model-based control to cope with system uncertainty, input actuator nonlinearity, and input time delay. First, the basic dynamic decoupling concept is introduced by involving virtual input dynamics, such that the system uncertainty and control input are decoupled in each fuzzy rule. This leads to simplified linear matrix inequality (LMI) conditions. Next, the dynamic decoupling approach is extended to controlling uncertain systems with input actuator nonlinearity (e.g., saturation, quantization, dead-zone, etc.) or time-varying input delay. Due to the decoupling between uncertainty, actuator nonlinearity, and input delay, more relaxed stability conditions are obtained for the asymptotic stability and H ∞ control performance. Furthermore, the limit on the initial condition is removed when considering input saturation. Larger and faster time-varying state and input delays are allowed under fewer LMIs. Finally, to show the advantages of the developed control method, numerical simulations are carried out on an inverted pendulum (subject to either the saturation, quantization, or delay input), a delay mass-spring-damper system, and a delay truck-trailer system.

A robust controller is developed for uncertain, second-order nonlinear\nsystems subject to simultaneous unknown, time-varying state delays and known,\ntime-varying input delays in addition to additive, sufficiently smooth\ndisturbances. An integral term composed of previous control values facilitates\na delay-free open-loop error system and the development of the feedback control\nstructure. A stability analysis based on Lyapunov-Krasovskii (LK) functionals\nguarantees uniformly ultimately bounded tracking under the assumption that the\ndelays are bounded and slowly varying.\n

Motorized functional electrical stimulation (FES) cycling has been demonstrated to have numerous health benefits for individuals suffering from neurological disorders. FES-cycling is usually designed to track the desired trajectories in real time. However, there are input delays between the exertion of the stimulation and the corresponding muscle contraction that potentially destabilize the system and undermine training efforts. Meanwhile, muscle fatigue gives rise to a time-varying input delay and decreased force. Moreover, switching between FES and motor control can be chattering and destabilizing owing to the high frequency. This article constructs Lyapunov-Krasovskii functionals to analyze the stability and robustness of the nonlinear cycling system with time-varying input delay. A new average dwell time condition is then provided to ensure the input-to-state stability of the considered systems. Finally, numerical simulations are illustrated to verify the effectiveness of the developed controller.

The stabilization of nonlinear stochastic systems with input and output delays is investigated in this article. First, an observer that includes input delay information is designed to estimate the state information after eliminating the effect of input delay. Second, a predictive scheme with delay information is utilized to predict the system state affected by output delay, and an event‐triggered predictive controller is established to eliminate the effects of input and output delays simultaneously. Subsequently, the convergence of the prediction error is analyzed by utilizing a recursive method, and a sufficient condition to ensure the global exponential stability is given by constructing a new closed‐loop system. Finally, numerical simulations of two nonlinear stochastic systems with different delay sizes are utilized to verify the impact of the event‐triggered predictive control in addressing the stabilization of nonlinear stochastic systems with time delays.

This paper considers the design problems of a delay-dependent robust and non-fragile guaranteed cost controller for singular systems with parameter uncertainties and time-varying delays in state and control input. The designed controller, under the possibility of feedback gain variations, can guarantee that a closed-loop system is regular, impulse-free, stable, an upper bound of guaranteed cost function, and non-fragility in spite of parameter uncertainties, time-varying delays, and controller fragility. The existence condition of the controller, the controller’s design method, the upper bound of guaranteed cost function, and the measure of non-fragility in the controller are proposed using the linear matrix inequality (LMI) technique. Finally, numerical examples are given to illustrate the effectiveness and less conservatism of the proposed design method.

The output feedback stabilization of polytopic-type uncertain discrete systems with interval-like time-varying state and input delays is studied. Based on a new bounding inequality technique, combining a parameter-dependent Lyapunov functional, a stability criterion is firstly presented in terms of a set of simple convex feasibility tests. Then, the output feedback stabilization conditions are formulated in the form of non-convex matrix inequalities, of which a feasible solution can be obtained by solving an LMI-based minimization problem. The newly proposed inequality lies in the partitioning idea of the varying interval and shows its more tightness over some existing bounding techniques. No free weighting matrix is involved. Two illustrative examples are finally given to verify the advantage and effectiveness of the proposed method.

The dissipative analysis and control synthesis for uncertain singular systems with interval time-varying state and input delays are investigated in this paper. The parametric uncertainty in this paper is assumed to be norm bounded. A sufficient condition, in the form of linear matrix inequality (LMI) with nonlinear constraints, on the existence of robust dissipative controller is established based on the Lyapunov stability theory. In order to solve the nonlinear problem, Cone Complementarity Linearization (CCL) is offered.

In this paper, we propose continuous-time and sampled-data output feedback controllers for nonlinear multi-input multi-output systems with time-varying measurement and input delays, with no restriction on the bound or serious limitations on the growth of the nonlinearities. A state prediction is generated by chains of saturated high-gain observers with switching error-correction terms and the state prediction is used to stabilize the system with saturated controls. The observers reconstruct the unmeasurable states at different delayed time-instants, which partition the maximal variation interval of the time-varying delays. These delayed time instant depend both on the magnitude of the delays and the growth rate of the nonlinearities. We also design sampled-data stabilizers as zero-order discretization of a hybrid modification (with continuous-time states and discrete-time control and innovations) of the continuous-time stabilizers.

In this paper, a novel online tube model predictive control (Tube-MPC) method is proposed for flexible hypersonic vehicle with time-varying input delay. The polytopic linear parameter varying (LPV) model with time-varying input delay is established and further transformed into a polytopic LPV delay-free model by input delay model transformation (IDMT). Since the flexible states cannot be measured accurately due to vehicle body deformation, the polytopic Luenberger observer is developed. To reduce control conservatism, substitute observations are used to replace the actually observed states such that the new reduced control error invariant set (CEI) is generated. On the basis, the Tube-MPC controller is designed, where the offline robust feedback control law and the online nominal control law are devised via linear matrix inequality (LMI) separately. For figuring out the invariant set of nominal control, an offline-online synthesized method is proposed by combining interpolation-based offline computation (IOFC) with substitute observation-based online computation (SOOC). In IOFC, the sequences of observation error invariant set (OEI) and CEI are computed offline iteratively and further interpolated so as to reduce online computation burden. The online judgement and update mechanism is proposed in SOOC. When the observed states enter the expanded invariant set constructed by the interpolated OEI and CEI, the substitute observation-based control error invariant set (SOCEI) will be calculated online and accordingly the nominal control law can also be updated. The simulation results verify the proposed control method’s effectiveness for flexible hypersonic vehicle with time-varying input delay.

Autonomous systems often use feedback from an imaging system for guidance, navigation, and control. Such feedback requires an image processing method that yields an inherent delay in state measurements. For some camera-to-hand applications, the image-based sensing and control computations may be computed by a more capable processor and then be transmitted to a sensorless vehicle, resulting in a delay in control execution. To account for these delays, an approach is developed in this paper that applies a transformation, to separate the image coordinates into spatial and time domain components. This transformation, along with a novel integral error system, allows for the development of a robust controller that compensates for uncertain time-varying state delays and known time-varying input delays for nonlinear image dynamics. A Lyapunov-based stability analysis is used to prove that the developed controller enables a sensorless autonomous vehicle to yield uniformly ultimately bounded (UUB) tracking of a desired image trajectory.