Non-minimum Phase Zeros Research Articles

Exact and Causal Inversion of Nonminimum-Phase Systems: A Squaring-Down Approach for Overactuated Systems

Nonminimum-phase zeros pose challenges for controller design, i.e., in inversion-based control approaches where inverting these zeros may result in “unstable” poles. The aim of this paper is to exploit the additional freedom in overactuated systems to eliminate these zeros, facilitating subsequent inversion. In particular, an approach for causal and exact inversion of systems with nonminimum-phase behavior is presented. In an either static or dynamic squaring-down step prior to inversion, the approach exploits the fact that nonsquare systems typically have no invariant zeros. The proposed approach is successfully demonstrated in experiments on an overactuated motion system. The method enables exact inversion for nonsquare systems without requiring preview or preactuation.

Local Loop Shaping for Rejecting Band-Limited Disturbances in Nonminimum-Phase Systems With Application to Laser Beam Steering for Additive Manufacturing

Closed-loop disturbance rejection without sacrificing overall system performance is a fundamental issue in a wide range of applications from precision motion control, active noise cancellation, to advanced manufacturing. The core of rejecting band-limited disturbances is the shaping of feedback loops to actively and flexibly respond to different disturbance spectra. However, such strong and flexible local loop shaping (LLS) has remained underdeveloped for systems with nonminimum-phase zeros due to challenges to invert the system dynamics. This article proposes an LLS with prescribed performance requirements in systems with nonminimum-phase zeros. Pioneering an integration of the interpolation theory with a model-based parameterization of the closed loop, the proposed solution provides a filter design to match the inverse plant dynamics locally and, as a result, creates a highly effective framework for controlling both narrowband and wideband vibrations. From there, we discuss methods to control the fundamental waterbed limitation, verify the algorithm on a laser beam steering platform in selective laser sintering additive manufacturing, and compare the benefits and tradeoffs over the conventional direct inverse-based loop-shaping method. The results are supported by both simulation and experimentation.

Experimental validation of complex non-minimum phase zeros in a flexure mechanism

In this paper, we present the design of an experimental setup that validates the existence of complex non-minimum phase (CNMP) zeros in a flexure mechanism under specific conditions. Using this setup, the dynamics of a lightly damped, nominally symmetric XY flexure mechanism comprising two double parallelogram flexure modules (DPFM) is experimentally studied while varying the operating point and parametric asymmetry. While CNMP zeros have been reported in analytical studies, a direct experimental validation of model-based prediction poses several challenges and is rare in the literature. The challenges addressed in the presented experimental setup design include isolating the specific dynamics of interest while minimizing extraneous dynamics, avoiding friction and backlash in the system to offer clean and repeatable measurements, and employing simple and practical sensing and actuation methods. This experimental setup is used to obtain X direction frequency response measurements and thereby record the presence or absence of CNMP zeros, while varying the Y direction operating point and the mass asymmetry between two DPFMs. This experimental evidence of the existence of CNMP zeros under specific conditions agrees well with the previous model-based predictions.

Reset control approximates complex order transfer functions

A controller with the frequency response of a complex order derivative may have a gain that decreases with frequency, while the phase increases. This behaviour may be desirable to ensure simultaneous rejection of high-frequency noise and robustness to variations of the open-loop gain. Implementations of such complex order controllers found in the literature are unsatisfactory for several reasons: the desired behaviour of the gain may be difficult or impossible to obtain, or non-minimum phase zeros may appear, or even unstable open-loop poles. We propose an alternative nonlinear approximation, combining a CRONE approximation of a fractional derivative with reset control, which does not suffer from these problems. An experimental proof of concept confirms the good results of this approximation and shows that nonlinear effects do not preclude the desired performance.

New results on the trade-off performance of LTI system with colored noise and encoder-decoder strategy

This paper studies the trade-off performance between tracking performance and control input energy of the multi-input multi-output(MIMO), linear and time-invariant(LTI) system over an additive coloured Gaussian noise(ACGN) channel and the encoder-decoder strategies. The restriction that filter in the encoder-decoder strategy must be diagonal matrix is not necessary. And some new results are derived according to the inner-outer factorization. The results show that the trade-off performance is correlated to the unstable pole, non-minimum phase zero of the system. Also new poles and zeros generated by the non-diagonal encoder-decoder strategies may affect the trade-off performance. At last, two examples with different filters and different encoder-decoder strategies are discussed to validate the conclusions. The various encoder-decoder strategies revealed by the simulations may enhance or deteriorate the trade-off performance proposed in this paper.

Modified tracking performance limitation of networked time-delay systems with two-channel constraints

This paper investigates the modified tracking performance limitation of the networked time-delay systems with two-channel constraints. We consider both the white Gaussian noise and packet dropout constraints in the communication channels. In the plant, the non-minimum phase, unstable poles and time-delay are considered. The modified tracking performance limitation expressions will be achieved using the co-prime factorization and the spectral decomposition technique, and the two-parameter controller is adopted. The results show that the modified tracking performance limitation is related to the intrinsic properties of the given plant, including the non-minimum phase zeroes, the unstable poles and the time-delay. Furthermore, the network communication parameters, e.g. the white Gaussian noise, the packet-dropouts probability and the modified factor affect the modified tracking performance limitation of the networked time-delay systems. Finally, some particular examples are provided to illustrate the efficiency of the proposed method.

Optimal performance of LTI systems over power constrained erasure channels

In this paper, the optimal performance of multiple-input multi-output (MIMO) linear time-invariant (LTI) plant is investigated. The communication channel is modeled as a power constrained channel with packet dropouts. The covariance of error signal between reference input and system’s output is chosen as the performance index. Based on the frequency domain analysis method, the exact expressions of the tracking performance limitation are derived. The results reveal that the best performance of NCSs not only has strong connection with both the nonminimum phase zeros and unstable poles of the plant, but also has close relation with the essential feature of reference input signal and communication parameters. Finally, a simulation example is discussed to validate the conclusions.

Tracking Performance Limitations of MIMO Networked Control Systems With Multiple Communication Constraints.

In this paper, the tracking performance limitation of networked control systems (NCSs) is studied. The NCSs are considered as continuous-time linear multi-input multioutput (MIMO) systems with random reference noises. The controlled plants include unstable poles and nonminimum phase (NMP) zeros. The output feedback path is affected by multiple communication constraints. We focus on some basic communication constraints, including additive white noise (AWN), quantization noise, bandwidth, as well as encoder-decoder. The system performance is evaluated with the tracking error energy, and used a two-degree-of-freedom (2DOF) controller. The explicit representation of the tracking performance is given in this paper. The results indicate the tracking performance limitations rely to internal characteristics of the plant (unstable poles and NMP zeros), reference noises [the reference noise power distribution (RNPD) and its directions], and the characteristics of communication constraints. The characteristics of communication constraints include communication noise power distribution (CNPD); quantization noise power distribution (QNPD), and their distribution directions; transform bandwidth allocation (TBA); transform encoder-decoder allocation (TEA), and their allocation directions; and NMP zeros and MP part of bandwidth. Moreover, the tracking performance limitations are also affected by the angles between the each transform NMP zero direction and RNPD direction, and these angles between each transform unstable poles direction and the direction of communication constraint distribution/allocation. In addition, for MIMO NCSs, bandwidth (there are not identical two channels) can always affect the direction of unstable poles, and the channel allocation of bandwidth and encode-decode may be used for a feasible method for the performance allocation of each channel. Finally, an instance is given for verifying the effectiveness of the theoretical outcomes.

Delay Margin of Low-Order Systems Achievable by PID Controllers

This paper concerns the delay margin achievable using proportional-integral-derivative (PID) controllers for linear time-invariant (LTI) systems subject to variable, unknown time delays. The basic issue under investigation addresses the question: What is the largest range of time delay so that there exists a single PID controller to stabilize the delay plants within the entire range? Delay margin is a fundamental measure of robust stabilization against uncertain time delays and poses a fundamental, longstanding problem that remains open except in simple, isolated cases. In this paper, we develop explicit expressions of the exact delay margin and its upper bounds achievable by a PID controller for low-order delay systems, notably the first- and second-order unstable systems with unknown constant and possibly time-varying delays. The effect of nonminimum phase zeros is also examined. PID controllers have been extensively used to control and regulate industrial processes that are typically modeled by first- and second-order dynamics. While furnishing the fundamental limits of delay within which a PID controller may robustly stabilize a delay process, our results should also provide useful guidelines in tuning PID parameters and in the analytical design of PID controllers.

Mean-square stabilizability via output feedback for a non-minimum phase networked feedback system

This work studies mean-square stabilizability via output feedback for a networked linear time invariant (LTI) feedback system with a non-minimum phase plant. In the feedback system, the control signals are transmitted to the plant over a set of parallel communication channels with possible packet dropout. Our goal is to analytically describe intrinsic constraints among channel packet dropout probabilities and the plant’s characteristics in the mean-square stabilizability of the system. It turns out that this is a very hard problem. Here, we focus on the case in which the plant has relative degree one and each non-minimum zero of the plant is only associated with one of control input channels. Then, the admissible region of packet dropout probabilities in the mean-square stabilizability of the system is obtained. Moreover, a set of hyper-rectangles in this region is presented in terms of the plant’s non-minimum phase zeros, unstable poles and Wonham decomposition forms which is related to the structure of controllable subspace of the plant. A numerical example is presented to illustrate the fundamental constraints in the mean-square stabilizability of the networked system.

Non-Linear Dynamic Inversion Control Design for Rotorcraft

Flight control design for rotorcraft is challenging due to high-order dynamics, cross-coupling effects, and inherent instability of the flight dynamics. Dynamic inversion design offers a desirable solution to rotorcraft flight control as it effectively decouples the plant model and effectively handles non-linearity. However, the method has limitations for rotorcraft due to the requirement for full-state feedback and issues with non-minimum phase zeros. A control design study is performed using dynamic inversion with reduced order models of the rotorcraft dynamics, which alleviates the full-state feedback requirement. The design is analyzed using full order linear analysis and non-linear simulations of a utility helicopter. Simulation results show desired command tracking when the controller is applied to the full-order system. Classical stability margin analysis is used to achieve desired tradeoffs in robust stability and disturbance rejection. Results indicate the feasibility of applying dynamic inversion to rotorcraft control design, as long as full order linear analysis is applied to ensure stability and adequate modelling of low-frequency dynamics.

Formula omitted] adaptive sampled-data control for uncertain multi-input multi-output systems

This paper develops a sampled-data (SD) controller for uncertain multi-input multi-output (MIMO) systems, possibly with non-minimum phase zeros, using the L1 adaptive control architecture. The proposed controller compensates for disturbances and uncertainties within the bandwidth of a low-pass filter. Sufficient conditions for robust stability are obtained for the closed-loop system with SD controller, where the input/output signals are held constant over a sampling period. It is shown that the hybrid closed-loop system can recover the performance of a continuous-time reference system as the sampling time of the SD controller tends to zero. Simulation examples are provided to validate the theoretical findings.

On the achievable tracking performance of NCSs over SNR limited channels

In this paper, the achievable tracking performance limitations of discrete-time, multi-input multi-output (MIMO) networked control systems (NCSs) are studied. The channel is modeled as an additive white Gaussian noise and signal-to-noise ratio (SNR) limited channel with feedback. Under this framework, the closed relationships among stabilization, tracking performance, and SNR limited are quantitatively revealed. Some new results a.erived according to the allpass factorization and Youla parameterization of two degrees of freedom controller. The results show that the best tracking performance is in connection with the unstable poles, non-minimum phase zeros of the system. It is also demonstrated that the tracking performance will be badly degraded by feedback channel noise and due to the SNR limited. Finally, a simulation example is presented to validate the conclusions.

Simultaneous stabilization of discrete-time delay systems and bounds on delay margin

This paper studies the delay margin problem of linear time-invariant (LTI) systems. For discrete-time systems, this problem can also be posed as one of simultaneous stabilization of multiple unstable systems with different lengths of delay. Our contribution is threefold. First, for general LTI plants with a number of distinct unstable poles and nonminimum phase zeros, we employ analytic function interpolation and rational approximation techniques to derive bounds on the delay margin. We show that readily computable and explicit lower bounds can be found by computing the real eigenvalues of a constant matrix, and LTI controllers, potentially of a low order, can be synthesized to achieve the bounds based on the H∞ control theory. Second, we show that these results can be coherently extended to systems with time-varying delays, also resulting in bounds on the delay range and delay variation that ensure simultaneous stabilization. Finally, we investigate the delay margin problem in the context of PID control. For first-order unstable plants, we obtain bounds achievable by PID controllers. It is worth noting that unlike its continuous-time counterpart, the discrete-time delay margin problem is fundamentally more difficult, due to the intrinsic difficulty in simultaneously stabilizing several systems. Nevertheless, while previous works on the discrete-time delay margin led to largely negative results, the bounds developed in this paper provide instead conditions that guarantee the simultaneous stabilization of multiple delay plants, or ranges within which the delay plants can be robustly stabilized.

Mean Square Stabilization Over SNR-Constrained Channels With Colored and Spatially Correlated Additive Noises

This note addresses the problem of stabilizing a multi-input multi-output (MIMO) discrete-time linear time invariant (LTI) system over a MIMO additive noise channel. To treat this problem, we assume that the communication link between the plant output and the controller input consists of multiple additive colored mutually correlated noise channels subject to independent signal-to-noise ratio (SNR) constraints. We derive analytical conditions for which mean square stabilization (MSS) can be achieved under such constraints. We also formulate numerical methods to test these conditions when the noise is white and correlated. Moreover, for simpler plant models, a characterization of the set of power constraints compatible with MSS is obtained. Our results show that the frequency response of the spectral factor related to the channel noise affects the minimum SNR for stability depending mostly on the unstable poles and their directions. This is aggravated by the existence of nonminimum phase zeros and higher relative degree of the plant. On the other hand, we detect that systems under highly correlated noise show lower SNR requirements for stability compared to ones with independent noise channels. Numerical simulations are provided to illustrate the theoretical results.

Performance limitation of networked control systems with networked delay and two-channel noises constraints

ABSTRACTThe performance limitation of networked control systems (NCSs) with networked delay and two-channel noises constraints are studied in this paper. The networked parameters mainly consider the networked delay in the forward channel and white Gaussian noise constraint existing in forward channel and feedback channel. The tracking performance limitation expression is achieved by using the spectral decomposition technique and selecting the optimal one-parameter structure. The obtained result shows that the tracking performance limitation of NCSs is related to the position of the non-minimum phase zeroes, the position of the unstable poles, networked delay, two-channel noises and input signal. We can also know that the networked delay and white Gaussian noise have influence on the tracking performance limitation of NCSs. Finally, a simulation example is given to demonstrate effectiveness of the theoretical results.

Optimal modified performance of MIMO networked control systems with multi-parameter constraints

The optimal modified performance of the multi-input multi-output (MIMO) networked control systems (NCSs) with encoding–decoding, channel noise in the forward channel and packet dropouts, quantization in the feedback channel is investigated in this paper. A new and efficient tracking performance index for the NCSs is presented which prevents variations in the tracking error where there is no integrator in the plant. The optimal modified performance is obtained by the method of coprime factorization and partial fraction. The results demonstrate that the optimal modified performance is related to the locations of the non-minimum phase (NMP) zeros, unstable poles of the given plant as well as their directions. In addition, the modified factor, packet dropouts probability, channel noise and encoding–decoding are also closely related to optimal modified performance of the NCSs. Finally, we present some particular examples to illustrate the theoretical results.

On Avoiding the Effect of Non-Minimum Phase Zeros Over the Signal-to-Noise Ratio Limitation

Fundamental limitations in networked control systems have been researched for the last years with many stability results, from channel transmission minimal rate based on information theory, to minimal signal-to-noise (SNR). For the latter, it has been confirmed early on that the usual culprits increase the minimal SNR for stability, such as unstable plant poles, non-minimum phase (NMP) zeros, time delay, together with new ones such as communication channel bandwidth and colored transmission noise. In this letter we specifically propose two approaches to completely avoid the effect of NMP zeros on the minimal SNR for stability in discrete-time. The first approach stems from the observation that the controller that achieves the minimal SNR for a minimum phase plant in discretetime, has all its zeros located outside the unit circle (NMP zeros). Such observation allow then to characterize this set of NMP zeros as candidate NMP zeros of alternative plant models with the same unstable poles, which in turn will not increase the minimal SNR for stability above the expression imposed only by the plant unstable poles. Our second approach extends the set of possible NMP zeros that do not increase the SNR limitation to any location, by proposing the synthesis of a two-degree of freedom linear time invariant controller solution that achieves the minimal SNR for stability imposed only by the plant unstable poles.

Boosting Speed and Accuracy in Precision Motion Control

Feedforward (FF) control uses a priori knowledge about a given system and its disturbances to influence the system’s behavior in a pre-defined way. However, unlike feedback (FB) control, it does not adjust the control signal (or manipulated variable) in response to how the system actually reacts. In other words, FF control is proactive while FB control is reactive. Model-inversion-based FF (MIBFF) control is a specific type of FF control where the control variable is determined by inverting a model of the system to be controlled. It is very popular for its use in output tracking control problems, where the goal is to force the system’s output to follow a desired trajectory. Output tracking is critical in several different applications, e.g., in manufacturing, robotics, automotive, aerospace, and is typically achieved using a combination of FF and FB control. As motivating examples, consider MIBFF in ultra-precise wafer scanners used in photolithography (see Sidebar S1) and in desktop 3D printing (see Sidebar S2). As these motivating examples illustrate, non-minimum phase (NMP) zeros often arise in practical engineering applications due to non-collocated sensors and actuators, fast sampling, etc. MIBFF for systems with NMP zeros is arguably the most important (and certainly the most researched) issue related to MIBFF.

A numerical investigation of direct and indirect closed-loop architectures for estimating nonminimum-phase zeros

ABSTRACTThis paper presents a numerical investigation of three direct architectures and three indirect architectures for identifying a plant operating in closed loop. Motivated by adaptive control of systems with nonminimum-phase (NMP) zeros and taking advantage of the fact that zeros are not moved by feedback, the performance metric is the accuracy of the estimates of the NMP zeros of the plant. Assuming known plant order, single-input, single-output, infinite-impulse-response models are constructed in the presence of process and sensor noise. Least squares provides the baseline estimation technique, and prediction error methods are used to account for correlation between the model input and noise. The goal is to compare the accuracy of the NMP-zero estimates obtained from each method and for each architecture.