Linear Time-invariant Single-input Single-output Research Articles

A robust output-feedback adaptive dynamic surface control for linear systems with input disturbance

In this paper, a robust output-feedback adaptive control is proposed for linear time-invariant (LTI) singleinput single-output (SISO) plants with unmeasurable input disturbance. Using dynamic surface control (DSC) technique, it is shown that the explosion of complexity problem in backstepping control can be eliminated. Furthermore, the proposed adaptive DSC scheme has the following merits: 1) by introducing an initialization technique, the L ∞ performance of system tracking error can be guaranteed even if the plant high-frequency gain is unknown and the input disturbance exists, and 2) the adaptive law is necessary only at the first design step, which significantly reduces the design procedure. It is proved that with the proposed scheme, all the closed-loop signals are semiglobally uniformly ultimately bounded. Simulation results are presented to demonstrate the effectiveness of the proposed scheme.

A robust adaptive dynamic surface control for linear systems

In this article, a robust output-feedback adaptive dynamic surface control (DSC) is proposed for linear time-invariant single-input single-output plants with unmodelled dynamics and unmeasurable output disturbance. With the proposed adaptive DSC scheme, the ‘explosion of terms’ problem inherent in backstepping control is eliminated and the adaptive law is necessary only at the first design step, which significantly reduces the design procedure. More importantly, it is proved that with an initialisation technique, the ℒ∞ performance of the tracking error can be guaranteed even with unmodelled dynamics, bounded output disturbance exists and the plant high-frequency gain is unknown.

Modified adaptive backstepping design method for linear systems

In this study, a modified adaptive backstepping control is proposed for linear time-invariant (LTI) single-input single-output (SISO) plants. By introducing arbitrary-order finite-time convergence exact differentiators, the explosion of complexity problem in backstepping control is eliminated and the tracking error converges to zero asymptotically. Furthermore, the adaptive laws are necessary only at the first design step, which significantly reduce the design procedure. It is proved that, with the proposed scheme, all the closed-loop signals are semi-globally uniformly ultimately bounded. Simulation results are presented to demonstrate the effectiveness of the proposed scheme.

When is the Achievable Discrete-Time Delay Margin Nonzero?

This technical note considers the delay margin problem for discrete-time finite dimensional linear time-invariant (FDLTI) single-input single-output (SISO) systems. This problem has been studied extensively in continuous time - it is well-known that, given a FDLTI plant and controller forming a strictly proper stable feedback connection, closed loop stability will be maintained for small delays; hence the achievable delay margin using FDLTI control is strictly greater than zero. In this work, we show that this is not the case for discrete-time systems; indeed, we show that a FDLTI SISO discrete-time plant has an achievable delay margin strictly greater than zero if and only if it has no negative real unstable poles.

An adaptive method for consistent estimation of real-valued non-minimum phase zeros in stable LTI systems

An adaptive algorithm, consisting of a recursive estimator for a finite impulse response model having two non-zero lags only, and an adaptive input are presented. The model is parametrized in terms of the first impulse response coefficient and the model zero. For linear time-invariant single-input single-output systems with real rational transfer functions possessing at least one real-valued non-minimum phase zero of multiplicity one, it is shown that the model zero converges to such a zero of the true system. In the case of multiple non-minimum phase zeros, the algorithm can be tailored to converge to a particular zero. The result is shown to hold for systems and noise spectra of arbitrary degree. The algorithm requires prior knowledge of the sign of the high frequency gain of the system as well as an interval to which the non-minimum phase zero of interest belongs.

Uncertainty and Disturbance Estimator–Based Control for Uncertain LTI-SISO Systems With State Delays

In this paper, a robust control strategy based on the uncertainty and disturbance estimator (UDE) is proposed for uncertain Linear Time Invariant-Single Input Single Output (LTI-SISO) systems with state delays. The knowledge of the bounds of uncertainties and disturbances is not needed during the design process although it is required for the stability analysis. Both the cases with known and unknown delays are considered. In the case of unknown delays, the terms involving the delays are treated as additional disturbances to the system. The robust stability of the closed-loop system is analyzed in detail, and a stability condition is proposed. Simulations are given to demonstrate the excellent tracking and disturbance rejection capabilities of the UDE-based control strategy.

Stabilization of a Class of Even Order Linear Time-Invariant Systems

This paper deals with the problem of stabilization of a class of linear time-invariant (LTI) even order controllable, single-input single-output (SISO) systems. The systems considered are open-loop unstable and in particular not stabilizable by static time-invariant output feedback. We propose a set of easily computable necessary and sufficient conditions in terms of scalar inequalities for the asymptotic stabilization of a class of n-dimensional even order LTI SISO systems using composite output feedback.

Frequency Response Estimation for Mechanical Systems Using Closed-Loop Step Response Data

In this paper, a simple scheme to estimate frequency transfer functions of linear time-invariant single-input single-output systems based on closed-loop step response data is investigated. The method uses only a single set of step response data of plants under closed-loop operating conditions. The input and output of a plant are filtered by a simple bandpass filter, and the frequency transfer function of the plant is obtained with the Discrete Fourier Transforms of the filtered input and output signals. The method is applied to a resonant mechanical system and its validity is illustrated through simulations and experiments.

RST Controller Design by Convex Optimization Using Frequency-Domain Data

AbstractA new robust fixed-order RST controller design approach by convex optimization is proposed in this paper. Only frequency-domain data are considered in the approach for linear time-invariant single-input single-output systems. It is shown that performance specifications given as the infinity norm of the weighted sensitivity functions can be represented as convex constraints in the Nyquist diagram. One of the advantage of the proposed method is that it can deal directly with multi-model uncertainty. Moreover, a solution to a real industrial problem is given showing the effectiveness of the method.

Fixed-order [formula omitted] controller design for nonparametric models by convex optimization

A new approach for robust fixed-order H ∞ controller design by convex optimization is proposed. Linear time-invariant single-input single-output systems represented by nonparametric models in the frequency domain are considered. It is shown that the H ∞ robust performance condition can be represented by a set of linear or convex constraints with respect to the parameters of a linearly parameterized controller in the Nyquist diagram. Systems with time-delay and with multimodel and frequency-domain uncertainty can be considered straightforwardly in the proposed approach. The proposed method is compared with the standard H ∞ control problem by a simulation example on an unstable system.

Adaptive control of linear singular time-invariant single-input single-output systems with external point delay

This paper is devoted to the robust adaptive control of solvable single-input single-output impulse-free linear time-invariant singular dynamic systems of known orders and unknown parameterizations subject to single external point delays. The control law is of pole-placement type and based on input/output measurements and parametrical estimation only. The parametrical estimation algorithm incorporates adaptation dead zones to prevent against potential instability caused by disturbances and unmodeled dynamics. The Weierstrass canonical form is investigated in detail concerning the properties of controllability and observability via testable conditions of the given arbitrary state-space realization of the same order. Robust closed-loop stability is proved under similar conditions to those used for (nonsingular) standard systems.

Representations of Linear Dual-Rate System Via Single SISO LTI Filter, Conventional Sampler and Block Sampler

In this brief, it is proved that a linear dual-rate system can be represented via a series cascade of: 1) a conventional expander, a single-input single-output (SISO) linear time-invariant (LTI) filter and a block decimator, or 2) a block expander, an SISO LTI filter and a conventional decimator. Hence, incompatible nonuniform filter banks could achieve perfect reconstruction via LTI filters, conventional samplers and block samplers without expanding the input-output dimension of a subsystem of linear dual-rate systems or converting the nonuniform filter banks to uniform filter banks. The main advantage of the proposed representations is to avoid complicated design of the circuit layout caused by connecting subsystems with large input-output dimension or a lot of subsystems together.

Fast Step-response Evaluation of Linear Continuous-time Systems With Time Delay in the Feedback Loop

The simulation of step, impulse and ramp responses of linear continuous-time time-invariant single-input single-output systems with a pure time delay in the feedback loop is tackled. Contrary to systems without time delay where a conversion from Laplace transform to z transform with a zero-order hold allows for an exact response, delay in the feedback loop requires either an unbounded number of states or approximations. This paper proposes an approximation which fulfills a set of criteria, including exactness of the response over a specified multiple of the delay and the conservation of the stability or lack of stability. The approach is illustrated with examples.

Calculation of All H∞ Robust Stabilizing Gains for SISO LTI Systems

Robust stabilization of continuous time single-input single-output (SISO) linear time invariant (LTI) systems with multiplicative uncertainties is considered in this paper. In particular, it has been shown that all gains that robustly stabilize a given uncertain SISO LTI system can be found by utilizing a generalization of the Nyquist theorem. The method proposed involves calculation of roots of two real polynomials and does not require any search or gridding over a parameter, and as a result offers computational advantages over existing methods in literature.

Minimum variance in fast, slow and dual‐rate control loops

AbstractIn certain industrial applications, the control updating rate is faster than the output sampling rate by a certain factor, which leads to dual‐rate (DR) control problems. Generally speaking, a DR controller performs better than a slow single rate (SSR) controller but worse than a fast single rate (FSR) controller in the sense of minimum variance control. This conjecture is theoretically justified in this paper for a continuous linear time‐invariant (LTI) single‐input single‐output (SISO) system. The optimal FSR, DR and SSR controllers are designed under the same performance criterion: variance of the fast sampled output. The discretization of continuous stochastic disturbance models is investigated preserving certain basic statistical properties. A linear matrix inequality (LMI) approach is developed to calculate the optimal controllers for DR and SSR loops. The theoretical results are illustrated by two simulation examples. Copyright © 2005 John Wiley & Sons, Ltd.

Nonlinear PI Compensators That Achieve High Performance

In this paper, linear time-invariant single-input single-output (SISO) systems that are stabilizable by a (linear) proportional and integral (PI) compensator are considered. For such systems a five-parameter nonlinear PI compensator is proposed. The parameters of the proposed compensator are tuned by solving an optimization problem. The optimization problem always has a solution. Additionally, a general non-linear PI compensator is proposed and is approximated by easy-to-compute compensators, for instance, a six-parameter nonlinear compensator. The parameters of the approximate compensators are tuned to satisfy an optimality condition. The superiority of the proposed nonlinear PI compensators over the linear PI compensator is discussed and is demonstrated for a feedback system.

A new two-step iterative method for optimal reduction of linear SISO systems

A new two-step iterative procedure is proposed for the optimal reduced-order modeling of linear time-invariant single-input single-output (SISO) systems. The performance index of optimal reduction is taken to be a quadratic function of the error between the time responses of the original and reduced models. At each iteration cycle, the numerator dynamics is first determined by solving a set of linear equations, and the denominator polynomial is then determined by a gradient-based search technique. The main features of the proposed procedure are that it searches the Routh stability parameters rather than the denominator polynomial coefficients of the reduced model, and computes the performance index and its gradients by a computationally efficient parametric algorithm. As a consequence, the need of stability monitoring in the step of searching optimal denominator polynomial for the reduced model is avoided, and the gradient vector evaluated exactly and efficiently for a gradient-based parameter search. Moreover, the constraint of zero steady-state response error can be easily handled.

Design and optimal tuning of nonlinear PI compensators

In this paper, linear time-invariant single-input single-output (SISO) systems that are stabilizable by linear proportional and integral (PI) compensators are considered. For such systems, a five-parameter nonlinear PI compensator is proposed. The parameters of the proposed compensator are tuned by solving an optimization problem. The optimization problem always has a solution.

Controller design for maximizing the size of a step disturbance in non-minimum phase, unstable uncertain systems

Considered in this paper is the problem of maximization of tolerable disturbances in the presence of hard time-domain constraints on system states and the input for a class of linear time-invariant single-input single-output uncertain systems. In particular, a methodology proposed by the authors is extended to include non-minimum phase and/or unstable systems. An inner-outer factorization separates a nominal minimum phase transfer function from the uncertain plant set thus facilitating the generation of the required frequency domain design region. In dealing with unstable systems, an intermediate prestabilization step may become necessary.

Recursive state and parameter estimation of SISO singular systems

In this paper, recursive algorithms are presented for the online state and parameter estimation of a linear time invariant single-input single-output (SISO) discrete-time singular system. The model considered is in the canonical observable form. The approach is based on the generalised Kalman filter and can be developed in two steps. First, the parameters are estimated by recursive least squares method. These parameters are then used to estimate the state by the generalised Kalman filter in the second step. The results are illustrated by a numerical example.