Unstable Linear Plants Research Articles

Stabilization of the Magnetic Levitation System

This paper contributes toward research on the control of the magnetic levitation plant, representing a typical nonlinear unstable system that can be controlled by various methods. This paper shows two various approaches to the solution of the controller design based on different closed loop requirements. Starting from a known unstable linear plant model—the first method is based on the two-step procedure. In the first step, the transfer function of the controlled system is modified to get a stable non-oscillatory system. In the next step, the required first-order dynamic is defined and a model-based PI controller is proposed. The closed loop time constant of this first-order model-based approach can then be used as a tuning parameter. The second set of methods is based on a simplified ultra-local linear approximation of the plant dynamics by the double-integrator plus dead-time (DIPDT) model. Similar to the first method, one possible solution is to stabilize the system by a PD controller combined with a low-pass filter. To eliminate the offset, the stabilized system is supplemented by a simple static feedforward, or by a controller proposed by means of an internal model control (IMC). Another possible approach is to apply for the DIPDT model directly a stabilizing PID controller. The considered solutions are compared to the magnetic levitation system, controlled via the MATLAB/Simulink environment. It is shown that, all three controllers, with integral action, yield much slower dynamics than the stabilizing PD control, which gives one motivation to look for alternative ways of steady-state error compensation, guaranteeing faster setpoint step responses.

The Value of Timing Information in Event-Triggered Control

We study event-triggered control for stabilization of unstable linear plants over rate-limited communication channels subject to unknown, bounded delay. On one hand, the timing of event triggering carries implicit information about the state of the plant. On the other hand, the delay in the communication channel causes information loss, as it makes the state information available at the controller out of date. Combining these two effects, we show a phase transition behavior in the transmission rate required for stabilization using a given event-triggering strategy. For small values of the delay, the timing information carried by the triggering events is substantial, and the system can be stabilized with any positive rate. When the delay exceeds a critical threshold, the timing information alone is not enough to achieve stabilization and the required rate grows. When the loss of information due to the communication delay perfectly compensates the implicit information carried by the triggering events, the delay equals the inverse of the entropy rate of the plant, and we obtain the same rate requirement prescribed by the data-rate theorem. When the delay is larger than this threshold, the required rate becomes larger than that required by the data-rate theorem. We also provide an explicit construction yielding a sufficient rate for stabilization, and generalize our results to vector systems. The results do not rely on any a priori probabilistic model of the delay or the initial conditions.

State-Secrecy Codes for Networked Linear Systems

In this paper, we study the problem of remote state estimation, in the presence of a passive eavesdropper. An authorized user estimates the state of an unstable linear plant, based on the packets received from a sensor, while the packets may also be intercepted by the eavesdropper. Our goal is to design a coding scheme at the sensor, which encodes the state information, in order to impair the eavesdropper's estimation performance while enabling the user to successfully decode the sent messages. We introduce a novel class of codes, termed State-Secrecy Codes, which use acknowledgment signals from the user and apply linear time-varying transformations to the current and previously received states. By exploiting the properties of the system's process noise, the channel physical model, and the dynamics, these codes manage to be fast, efficient, and suitable for real-time dynamical systems. We prove that under minimal conditions, State-Secrecy Codes achieve perfect secrecy, namely the eavesdropper's minimum mean square error (mmse) grows unbounded almost surely, whereas the user's estimation performance is optimal. These conditions only require that at least once, the user receives the corresponding packet while the eavesdropper fails to intercept it. Even one occurrence of this event renders the eavesdropper's mmse unbounded with asymptotically optimal rate of increase. State-Secrecy Codes are provided and studied for two cases, first, when direct state measurements are available and second, when we only have output measurements. The theoretical results are illustrated in simulations.

Control Over Gaussian Channels With and Without Source–Channel Separation

We consider the problem of controlling an unstable linear plant with Gaussian disturbances over an additive white Gaussian noise channel with an average transmit power constraint, where the signaling rate of communication may be different from the sampling rate of the underlying plant. Such a situation is quite common since sampling is done at a rate that captures the dynamics of the plant and that is often lower than the signaling rate of the communication channel. This rate mismatch offers the opportunity of improving the system performance by using coding over multiple channel uses to convey a single control action. In a traditional, separation-based approach to source and channel coding, the analog message is first quantized down to a few bits and then mapped to a channel codeword whose length is commensurate with the number of channel uses per sampled message. Applying the separation-based approach to control meets its challenges: first, the quantizer needs to be capable of zooming in and out to be able to track unbounded system disturbances, and second, the channel code must be capable of improving its estimates of the past transmissions exponentially with time, a characteristic known as anytime reliability. We implement a separated scheme by leveraging recently developed techniques for control over quantized-feedback channels and for efficient decoding of anytime-reliable codes. We further propose an alternative, namely, to perform analog joint source–channel coding, by this avoiding the digital domain altogether. For the case where the communication signaling rate is twice the sampling rate, we employ analog linear repetition as well as Shannon–Kotel’nikov maps to show a significant improvement in stability margins and linear-quadratic costs over separation-based schemes. We conclude that such analog coding performs better than separation, and can stabilize all moments as well as guarantee almost-sure stability.

Adaptive Stabilization of Discrete-Time Nonminimum Phase Systems

In this paper, we present a direct multirate adaptive control algorithm that ensures global stabilization of a class of (potentially unstable and invertible) linear time-invariant discrete-time plants of known order and relative degree one. An essential feature of our scheme is that no projections are needed, no appeal is made to the persistence of excitation arguments for the stability proof and it has a better transient performance than existing schemes. The implementation of the controller requires some prior information about its Markov parameters, namely, upper and lower bounds on the systems impulse response. It is directly applicable to plants with interlacing real poles and zeros, i.e., with Cauchy index equal to the plant order, provided the adaptation gain is restricted to be smaller than some value determined by the measure of relative primeness of the pole and zero polynomial. This class contains some practically interesting systems, for instance, resistor–inductor or resistance–capacitance circuits.

Absolute stability analysis of non‐linear active disturbance rejection control for single‐input–single‐output systems via the circle criterion method

This study focuses on the stability analysis of non-linear active disturbance rejection control (ADRC) for single-input–single-output systems. Firstly, a non-linear ADRC system for a linear plant is transformed into a Lurie system. Secondly, two extended circle criteria are obtained, and two numerical examples are presented to illustrate the absolute stability analysis, including both stable and unstable linear plants. Thirdly, local asymptotic stability of a non-linear ADRC system for a non-linear plant is also performed through linearisation by Taylor expansion. Finally, a comparison with the existed processing methods is further made, including the describing function method and time domain stability analysis method. It can be concluded that the circle criterion method is more convenient and practical for its frequency domain and graphical interpretation. The circle criterion method can also be extended to the stability analysis of a control system which applies linear ADRC to a plant with one non-linear term.

Dynamic source–channel coding for estimation and control over binary symmetric channels

The authors study causal dynamic source-channel codes for mean-square stabilisation of a linear plant with random initial state, over binary symmetric channels (BSCs). A binary expansion source coding scheme and two types of channel coding schemes are investigated: (i) dynamic repetition encoding under sequential majority logic decoding, and (ii) dynamic Fountain encoding under sequential BP decoding. The coding schemes are first analysed for an open-loop system, where the goal is to transmit a real-valued source over BSCs with the objective of minimising end-to-end mean-square error distortion. They derive upper- and lower bounds on the achievable distortion which is a function of time. Based on these distortion bounds, they show that the dynamic Fountain codes are able to stabilise (in mean-square sense) an unstable linear plant over BSCs.

Adaptive Controller for Linear Plant with Parametric Uncertainties, Input Delay And Unknown Disturbance

We present a new stabilization approach for unstable linear plants with long input delay, unknown parameters and disturbance. The predictor-based algorithm providing plant identification and stabilization is proposed. Also the extension is considered with estimation and cancellation of an unknown disturbance. A numerical examples are given to illustrate the efficiency of our adaptive controller.

Adaptive Delta Modulation in Networked Controlled Systems With bounded Disturbances

This technical note investigates the closed-loop properties of the differential coding scheme known as Delta Modulation (Δ - M) when used in feedback loops within the context of linear systems controlled through a communication network. We propose a new adaptive scheme with variable quantization step Δ, by defining an adaptation law exclusively in terms of information available at both the transmitter and receiver. With this approach, global asymptotic stability of the networked control system is achieved for a class of controllable (possibly unstable) linear plants. Moreover, thanks to the globally defined switching policy, this architecture enjoys a disturbance rejection property that allows the system to recover from any finite-time unbounded disturbance or communication loss.

Constructive nonlinear anti-windup design for exponentially unstable linear plants

In this paper we give a constructive method for anti-windup design for general linear saturated plants with exponentially unstable modes. The constructive solution is independent of the controller dynamics so that the size of the (necessarily bounded) operating region in the exponentially unstable directions of the plant state space is large. Desirable properties of the closed-loop are formally proved and shown to induce a very desirable behavior on a MIMO example with two exponentially unstable modes.

Adaptive stabilization of input-saturated plants with known unstable poles

This paper considers the problem of controlling partially uncertain, possibly unstable and non-minimum phase linear plants submitted to a control saturation constraint. It seeks both plant l∞-stabilization and output tracking. It proposes a direct adaptive controller and shows that if the closed loop poles are placed in a specific region, then the desired stability and tracking objectives are achieved.

Global Stabilization of a Linear Plant by Means of a Pulse-Frequency Modulated Signal

A system consisting of an unstable linear plant and of a pulse-frequency modulator, as a controller, is considered. Algebraic inequalities, which guarantee the boundedness of all the solutions, are obtained.

Stabilizability of Unstable Linear Plants Under Bounded Control

Considered in this paper is the problem of stabilizing a linear unstable plant under bounded control. It is shown that a linear strictly unstable plant can never be globally stabilized under bounded control, and a linear unstable plant can never be globally stabilized under bounded linear feedback control. Hence, any stabilizing controller {be it linear or nonlinear) for a strictly unstable plant can only be locally stabilizing. Consequently, it is important to estimate the domain of attraction achievable with a locally stabilizing control under a prespecified control bound. Since, it is difficult to construct the domain of attraction, we use the domain within which the control is not saturated to approximate it. The relation between these two domains are characterized and computable expressions for the biggest stability ball and the biggest stability polytope that lie inside the domain of nonsaturating control are given.

The stability of constrained receding horizon control

An infinite horizon controller that allows incorporation of input and state constraints in a receding horizon feedback strategy is developed. For both stable and unstable linear plants, feasibility of the contraints guarantees nominal closed-loop stability for all choices of the tuning parameters in the control law. The constraints' feasibility can be checked efficiently with a linear program. It is always possible to remove state constraints in the early portion of the infinite horizon to make them feasible. The controller's implementation requires only the solution of finite-dimensional quadratic programs. >

Model-Predictive Control of Chemical Processes

Abstract This paper discusses model-predictive control, a scheme in which an open-loop performance objective is optimized over a finite moving time horizon. Model-predictive control is shown to provide performance superior to conventional feedback control for nonminimum phase systems or systems with input constraints when future set points are known. Stabilizing unstable linear plants and controlling nonlinear plants with multiple steady states are also discussed.

Optimal control of unstable linear plants with inaccessible states

A theory is presented for the optimal design of linear time-varying systems with a quadratic cost functional when not all the states of the plant are accessible for measurement. The procedure involves choosing a constant set of eigenvalues and ensures closed-loop stability, even when the plant is originally unstable. The performance of the resulting feedback system can be made arbitrarily close to the optimum which results when all states of the plant can be measured.