Unstable Zeros Research Articles

Modification of Infinite and Unstable Invariant Zeros in Linear Systems Using Stability-Preserving State Feedback

Modification of Infinite and Unstable Invariant Zeros in Linear Systems Using Stability-Preserving State Feedback

Effects of Two-Channel Noise and Packet Loss on Performance of Information Time Delay Systems.

The performance limitations of multiple-input multiple-output (MIMO) information time delay system (ITDS) with packet loss, codec and white Gaussian noise (WGN) are investigated in this article. By using the spectrum decomposition technique, inner-outer factorization, and partial factorization techniques, the expression of performance limitations is obtained under the two-degree-of-freedom (2DOF) compensator. The theoretical analysis results demonstrate that the system performance is related to the time delay, non-minimum phase (NMP) zeros, unstable zeros and their directions in a given device. In addition, WGN, packet loss and codec also impact the performance. Finally, the theoretical results are verified by simulation examples. Simulation results show that packet loss rate and encoding and decoding have a greater impact on system performance.

Asymptotic Output Tracking Control of a Class of Linear Systems by Finite-and-Quantized Output Feedback

This paper presents a novel finite-and-quantized output feedback asymptotic tracking control method for a general class of continuous-time linear time-invariant systems. First, we construct a finite quantizer with time-varying thresholds and design a pole placement control law that exclusively utilizes the finite-and-quantized output signal and an external reference signal. Then, we establish the boundedness of all closed-loop signals and prove the asymptotic convergence of the output tracking error to zero. The proposed method combines the advantages of classical pole placement control technique and finite quantization feedback technique. It not only reduces the requirement for feedback information compared with existing tracking control methods but also effectively handles unstable poles and zeros in controlled systems, thereby achieving asymptotic output tracking. Finally, we provide a representative example to validate the effectiveness and new features of our proposed method.

Cascade tri-neuron hopfield neural network: Dynamical analysis and analog circuit implementation

This paper studies a cascade tri-neuron Hopfield neural network (CTN-HNN) with no connection between the first neuron and the third neuron. Such incompletely connected neuronal structure may be part of complex neural networks with negligible inputs, and may be present in abnormal neural networks with some neurological diseases. It was proved that the stable point, limit cycle, single-scroll chaotic attractor and double-scroll chaotic attractor were observed in the network by adjusting the synaptic weight from the second neuron to itself. To further dynamic analysis, we first demonstrate that this three-dimensional (3D) autonomous nonlinear dynamical system is symmetric about the origin and bounded ultimately, and has one unstable saddle-focus zero point and two symmetric nonzero points with different stability types. Afterwards, taking two synaptic weights as adjustable parameters, the numerical analysis reveals that the CTN-HNN exhibits coexisting chaotic and periodic attractors, coexisting bursting firings and long-term transient chaos by employing multiple numerical methods. Finally, an analog printed circuit is built using one main circuit module and three activation function modules, thus effectively demonstrating the complex dynamics of the network.

Implementation Methods of Controllers Designed by $H_\infty$ Loop Shaping

When the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> loop-shaping controller is implemented as a unity feedback system, its step responses are sometimes unsuitable for tracking control, typically, displaying large overshoots. In this paper, the implementation method is studied for tracking control with small overshoots. Three implementation methods that are described by a unity feedback system, a previous simple two-degrees-of-freedom(2DoF) system, and a new simple 2DoF system proposed in this paper are examined. For the closed-loop transfer function of each system, the unstable zeros are analyzed and an upper bound of the peak gains is presented. These results show that the two simple 2DoF systems have adequate properties for tracking control with small overshoots. The new 2DoF system has good features that the system is minimum phase if the plant is minimum phase, and that the resonance peak gain is one. Their tracking performances are compared in several numerical examples.

The limiting zero dynamics of discrete-time system based on forward triangle sample-and-hold.

Zero dynamics have crucial effect on system analysis and controller design. In the control analysis process, system performance is influenced by the unstable zero dynamics, greatly. This study concerns with the properties of limiting zero dynamics when the signal of controlled continuous-time systems was reconstructed by forward triangle sample-and-hold (FTSH). FTSH is a newly sample and hold method in the signal reconstruction area. However, more theoretical details about the limiting zero dynamics of the resulting discrete-time systems need to be revealed. Firstly, the framework for the limiting zero dynamics in the situation of sufficiently small or large sample period is introduced. Furthermore, this study provides the stable conditions of limiting zeros in the two different sampling situations. The results indicate that one can select a suitable variable parameter value of FTSH to replace the sampling zeros of discrete-time system locate inside the stable region. This paper also reveals the truth that FTSH has the outstanding advantage compared with the BTSH using theoretical analysis method. Finally, example simulations verify the effectiveness of the results in this study.

Performance Analysis of MIMO Information Time-Delay System Under Bandwidth, Cyber-Attack, and Gaussian White Noise

The performance limitations of multi-input–multi-output (MIMO) information time-delay system (ITDS) due to bandwidth, additive white Gaussian noise (AWGN), and cyber-attacks are investigated in this work. The game theory methods are used to analyze the impact of intrusion detection systems on systems performance in order to deal with network attacks under incomplete information. The expression of performance limitations is obtained by using the spectrum decomposition technique, inner–outer factorization, and partial factorization techniques under the two-degree-of-freedom (2DOF) compensator. The results demonstrate that the performance limitations of the system are not only influenced by the inherent properties, including time delay, nonminimum phase (NMP) zeros, and unstable zeros and their directions of the given object but are also influenced by the communication parameters, such as AWGN and bandwidth. In addition, the results show that under the cyber-attacks with incomplete information, the performance is only influenced by the payoff matrix of the defense, and not by the payoff matrix of the offense. Finally, the theoretical results are verified by performing simulations.

Non-Minimum Phase Behavior Analysis of the Weak Grid-Tied VSC's Q-Axis Current Control Dynamics

In this letter, the phenomenon is found that there are non-minimum phase and “undershoot” behavior in grid-tied VSC, even if the outer loop is not applied. Based on the small-signal model, the mechanism that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -axis-related non-minimum phase behavior reduces VSC's stability and the “undershoot” behavior are analyzed in the letter. It is found that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -axis current control dynamics of weak grid-tied VSC introduce an open-loop unstable zero when the reactive power is supplied from the AC system, which is the key factor for the non-minimum phase behavior.

Modeling Method for Sampling Zeros of Sampled-data System with Time Delay in Generalized Sample Hold

Unstable zeros are widely known to dramatically curb the performance of controllers and the direct applications of several control algorithms. Time delay is a crucial research topic due to the prevalence of communication and computational delays in the control process. In particular, time delay exacerbates the possibility of continuous-time systems with stable zeros transforming into discrete-time systems with unstable zeros. For continuous-time systems with a relative degree greater than or equal to two, at least one zero of the corresponding discrete-time system converges to either marginally stable or unstable locations as the sampling time approaches zero. This study investigates the sampling zeros of discrete-time systems in the case of a generalized sample hold function (GSHF) with time delay. Moreover, it presents a novel modeling method and stability conditions of the limiting zeros for the discrete-time system. The results demonstrate that the stable properties of zeros for the time delay systems can be preserved in the discretization process when GSHF is applied for signal reconstruction when zero-order hold fails to do so. Finally, numerical examples are provided to illustrate the effectiveness of the proposed method in this study.

Closed Loop Identification and Control for Input Time Delay Systems

In this research, the identification results for open-loop time delay systems are extended to closed-loop systems. It is shown that a subspace identification method used in open-loop systems can be applied to closed-loop systems. When the subspace identification method was used directly, problems such as increasing poles and unstable zeros caused by numerical errors were encountered; however, these problems can be prevented by a shift operation. Finally, the simulated identification results for the time delay system and the improved control performance using the estimated time delay are shown.

Some issues about FL and LAB servo control

In this paper, the servo control design methodologies based on Feedback Linearization (FL) and Linear Algebra Based (LAB) are applied to nonlinear systems with and without zero dynamics, analysing the case of unstable zero dynamics. Their advantages and drawbacks are discussed. Some examples are developed to illustrate the results.

A Novel Feedback Linearisation Control of Flyback Converter

Abstract This paper presents a novel approach for feedback linearisation in a continuous conduction mode (CCM) of the flyback converter. Due to the unstable zero dynamics, a flyback converter has highly non-linear behaviour. Flyback converters mostly use the indirect (current) control mechanism. In contrast, this paper shows a direct control of the output voltage of a flyback converter with feedback linearisation (a non-linear control method). In the designed controller, an error integrator is applied to improve the dynamic and steady-state behaviour of the controller. To design the feedback linearisation method, the state-space averaged model is determined. The converter and the proposed control are tested in a MatLab/Simulink environment, and the results are compared with other optimal controller methods. The results provide feedback about the efficiency and practical implementation of the proposed method.

Output feedback sliding-mode control based on dynamic-gain observer for non-minimum phase systems

This paper considers the output feedback sliding-mode control for an uncertain linear system with unstable zeros. Based on a frequency shaping design, a dynamic-gain observer is used for state estimation of an uncertain system. This paper confirms that (1) state estimation is globally stable in a practical sense, (2) the resultant error can be arbitrarily small with respect to the system uncertainties, and (3) the proposed sliding-mode control can drive the uncertain system state into an arbitrarily small residual set around the origin, such that the size of residual set is controlled by the filter design. Moreover, the proposed control design is inherently robust to measurement noise; the effect of measurement noise can effectively be attenuated without any additional work.

Leader-Following Almost Output Consensus for Linear Heterogeneous Multiagent Systems With Disturbance-Affected Unstable Zero Dynamics by Output Feedback

A novel design of observer-based output feedback consensus protocols in a low-and-high gain feedback framework is proposed for a linear heterogeneous multiagent system, where the unstable zero dynamics of the follower agents is affected by disturbances. Conditions on the graph that represents the connectivity among the agents and on how the disturbances enter the zero dynamics of the follower agents are identified, under which the leader-following almost output consensus can be attained with the proposed protocols. More specifically, the proposed protocols achieve leader-following output consensus to an arbitrarily high level of accuracy, and the effect of the disturbances on the consensus errors is attenuated to an arbitrarily low level.

Output tracking control of non‐minimum phase systems using on‐line output redefinition approach based on barrier function method

This paper considers the output tracking problem of non-minimum phase non-linear systems using the inverse dynamics control which is impossible in conventional methods due to unstable zero dynamics. The proposed method is based on the output redefinition approach defined as a parametric function. The tracking error is decomposed into two terms. The first one named redefined output tracking error is the difference between the redefined output and the reference signal, and the second one is the distance between the redefined output and the system output. For the first term, due to minimum phase property of the system with the redefined output, an inverse dynamics control is used to guarantee the convergence of the redefined output tracking error to zero. To minimize the second term, the parameters of the redefined output are set by solving an optimization problem whose constraints include minimum phase conditions. To solve the optimization problem which satisfies all constraints, the barrier function method as an interior point method is adopted. The optimization parameters are set using steepest descent algorithm. Due to on-line updating the redefined output parameters, the related zero dynamics is time varying. Therefore, sufficient conditions for exponential stability of the resulting time-varying zero dynamics are obtained.

Parallel compensator for zero dynamics stabilization in fluidized bed spray granulation

This contribution is concerned with the stabilization of continuous fluidized bed spray granulation processes with external sieve mill cycle. To keep the overall mass of particles inside the granulation chamber in well-defined bounds these processes are in general operated with a mass controller. This configuration is well-known to become unstable for certain ranges of the milling diameter. Here, the instability results in a nonlinear limit cycle of the particle size distribution. As has been shown in a previous contribution the stability problem is not inherent to the process, which is open-loop stable over a wide range of operating conditions. In fact, the root cause for the instability is the mass controller, which interacts with the unstable zero dynamics of the granulation process. To overcome this problem a parallel compensator will be designed, which allows to stabilize the process zero dynamics and thus solves the problem of control induced instabilities.

A Differential-Flatness-Based Approach for Autonomous Helicopter Shipboard Landing

This paper presents a differential flatness-based approach to trajectory planning and tracking for helicopters and demonstrates this strategy for time-optimal autonomous helicopter shipboard landing. For planning and outer-loop tracking control, helicopter dynamics are captured by a simplified nonlinear model that is shown to be differentially flat, which enables the transformation of the original nonlinear system dynamics into an equivalent linear form with endogenous nonlinear constraints. For time-optimal trajectory planning, this equivalent model is used to construct a computationally-efficient optimization problem for free end-time end-state problems, where the decision-space is that of the trajectories of the flat outputs and their derivatives. While the attitude dynamics are stabilized by an inner loop controller, for tracking the planned trajectory a flatness-based outer loop control law is designed to stabilize the unstable zero dynamics. Further, analytical guarantees that relate inner loop tracking error to outer loop robust stability and tracking performance are derived. Simulations are performed on a high-fidelity numerical UH-60 helicopter model under ground effect and ship airwake disturbances, together with ship motion profiles collected from a naval environment to validate the proposed approach for realistic landing scenarios.

Optimization of Observer Feedback Gains for Stable Sensorless IM Drives at Very Low Frequencies: A Comparative Study between GA and PSO

Instability of an adaptive flux observer (AFO) in the regenerating mode at low frequencies is a great challenge of sensorless induction motor (SIM) drives. Zero observer feedback gains (OFGs) in the regenerating mode at low frequencies are the main reasons for moving the dominant zero of the speed estimators to the unstable region. OFGs should be appropriately selected to transfer the unstable dominant zero to the stable region. In this paper, genetic algorithm (GA) and particle swarm optimization (PSO) techniques were used to design the OFGs for a stable observer. A fair comparison of the dominant zero location between the two approaches using the optimized OFGs is presented under parameter deviation. Analytical results and the design procedure of the OFGs using the two approaches are presented under deviations of stator resistance and mutual inductance to guarantee a stable dominant zero in the regenerating mode of IM. The dominant zeros obtained by PSO had a superior location to that obtained by GA for both stator resistance and mutual inductance deviations. It was observed that one of the gains had an almost constant value over a wide range of parameter deviations. However, the value of the other gain was dependent on the deviation of machine parameters. The advantage of using PSO over GA is that the relation between the gain and parameter deviation can be represented by a deterministic and mostly linear relationship. Simulation and experimental work of the SIM drive are presented and evaluated under the optimized OFGs.

Discrete sliding mode control with optimal generalized sliding function for non-minimum phase system

Discrete sliding mode control with classical sliding function for the input–output representation is not applicable in non-minimum phase systems, due to the presence of unstable zero. In fact, the presence of the unstable zero risks deteriorating or saturating the control law. In order to surmount this problem, a new discrete generalized sliding mode controller for non-minimum phase systems is proposed with a new sliding function called discrete generalized sliding function. Here, the determination of optimal coefficients of the discrete generalized sliding function is proposed in terms of linear matrix inequalities. Using the proposed method, a discrete generalized sliding mode control for non-minimum phase systems with input–output model is presented and compared with the discrete generalized sliding mode controller using the pole placement method. An illustrative example verifies the effectiveness of the proposed technique.

Suboptimal output consensus of a group of discrete-time heterogeneous linear non-minimum phase systems

This paper formulates the suboptimal output consensus problem for a discrete-time heterogeneous linear multi-agent system with unstable zero dynamics, where each agent possesses its own objective function and the sum of all these private objective functions, called the overall objective function, is to be minimized. Mild assumptions on the communication topology and the agent dynamics are made under which a parameterized distributed consensus protocol based on low gain feedback is proposed for each agent. The multi-agent system is shown to achieve suboptimal output consensus under the proposed protocols in the sense that the states of all agents remain bounded while their outputs converge to a pre-specified arbitrarily small neighborhood of the optimal point as long as the design parameter is chosen small enough. Simulation results are presented that demonstrate the proposed design.