Non-minimum Phase Zeros Research Articles

An adaptive cascade compensator for a wide class of stable minimum or nonminimum phase plants through time moment estimation

Model Reference Adaptive Control (MRAC) of a wide repertoire of stable Linear Time Invariant (LTI) systems is addressed here. Even an upper bound on the order of the finite-dimensional system is unavailable. Further, the unknown plant is permitted to have both minimum phase and nonminimum phase zeros. Model following with reference to a completely specified reference model excited by a class of piecewise continuous bounded signals is the goal. The problem is approached by taking recourse to the time moments representation of an LTI system. The treatment here is confined to Single-Input Single-Output (SISO) systems. The adaptive controller is built upon an on-line scheme for time moment estimation of a system given no more than its input and output. As a first step, a cascade compensator is devised. The primary contribution lies in developing a unified framework to eventually address with more finesse the problem of adaptive control of a large family of plants allowed to be minimum or nonminimum phase. Thus, the scheme presented in this paper is confined to lay the basis for more refined compensators--cascade, feedback and both--initially for SISO systems and progressively for Multi-Input Multi-Output (MIMO) systems. Simulations are presented.

On the synthesis of controllers for continuous time LTI systems that achieve a non-negative impulse response

The problem of controlling the oscillatory response of a system has many practical applications. In this paper, I consider the problem of controlling the sign of the closed loop impulse response for a delay-free LTI system. If the impulse response of a stable system does not change sign, then its step response will neither undershoot nor overshoot. In this paper, I will show that such a synthesis is possible iff the open loop system does not have real, non-minimum phase zeros. In addition, I provide a stable compensator that achieves a stable, non-negative impulse response, if there exists one.

Cheap control tracking performance for non‐right‐invertible systems

AbstractThere exists a well‐known fundamental limitation upon the achievable setpoint tracking performance of a non‐right‐invertible plant. This limitation manifests itself, for example, in the cost associated with the cheap control tracking problem. In this paper, we provide a new interpretation of this limitation. We show that the cheap control cost may be decomposed into the sum of two terms. The first of these depends upon certain non‐minimum phase zeroes that include the non‐minimum phase plant zeroes. The second term depends upon the extent to which the plant direction varies with frequency. To state these results, we first develop a co‐ordinate transformation that may be used to define the notion of frequency dependent plant direction and to display the relevant non‐minimum phase zeroes. We also show that the cheap control cost is connected to an integral relation that constrains the performance of any stable closed‐loop system (not necessarily under cheap control) for which the plant has a single control input and two performance outputs. Copyright © 2002 John Wiley & Sons, Ltd.

Optimal tracking performance for SIMO systems

This note studies the optimal tracking problem for linear time-invariant single-input-multiple-output systems responding to a step reference signal. An integral square error criterion is used as the measure for tracking performance. Explicit expressions are developed for the optimal tracking error. These results characterize how nonsquare plants may pose additional difficulties for tracking, other than those resulted from nonminimum phase zeros and unstable poles.

Non-interacting control design for multivariable industrial processes

Abstract In this paper, a systematic method is proposed for the design of general multivariable controller for complex processes to achieve the goal of fast loop responses with acceptable overshoots and minimum loop interaction while maintaining low complexity of the feedback controller. The design of general transfer function type controller is based on the fundamental relations under decoupling of a multivariable process, and the characterization of the unavoidable time delays and non-minimum phase zeros that are inherent in the decoupled loops. The objective loop transfer functions are then suitably specified to achieve fast loop response taking into account the performance limitation imposed by those non-minimum phase zeros and time delays. The ideal controller is then obtained which is in general a complicated irrational transfer matrix, for which model reduction with recursive least squares is applied in the frequency domain to obtain a much simpler transfer matrix with its elements in the form of rational transfer function plus delay. Simulations show that very satisfactory control performance is achieved.

Effects of control structure on performance for an automotive powertrain with a continuously variable transmission

The wheel speed control problem of an automotive powertrain equipped with a conventional spark-ignition engine directly connected to a continuously variable transmission (CVT) and an electronic throttle is considered. We revisit the work by Guzzella and Schmid (1995) and show that the control structure that dedicates the throttle actuator to maintaining engine operation at the maximum fuel efficient operating points results in a single-input-two-output system that presents a fundamental limitation in the achievable wheel speed response. The limitation arises from the nonminimum phase (NMP) zero in the transfer function from the CVT ratio rate to the vehicle wheel speed. We relax the requirements on the fuel efficient operation and employ the electronic throttle as a second actuator for the wheel speed regulation problem. The resulting two-input-two-output control structure is then analyzed to determine how to mitigate the limitations associated with the NMP zero. Simulations show that the multivariable strategy improves the system performance because it produces minimum phase behavior without large transient deviations from the optimal fuel economy operation.

Tracking performance limitations in LTI multivariable discrete-time systems

In this paper, we investigate tracking properties of linear shift-invariant feedback control systems. We consider the standard unity feedback configuration, and use the energy of an error signal as a measure of tracking ability. Our main goal is to understand the fundamental limitation on tracking performance, which can arise due to the nonminimum phase zeros, unstable poles, and time delays in the plant, and which varies with input reference signals. We consider step, ramp, and sinusoidal signals, and for each type of the signals we derive a closed form expression for the minimum tracking error attainable by any stabilizing controller. Our results display an explicit dependence of the tracking error on nonminimum phase zeros, unstable poles, and in particular the coupling between the directions of the poles and zeros, and those of the input reference signal, upon which a number of useful conclusions can be drawn. One interesting outcome is that not only zero and pole locations affect tracking performance, but their directional properties also play an important role. The paper provides a nontrivial extension of the previously available results to discrete-time systems, with a consideration on broader classes of reference inputs.

IDENTIFICATION OF PERFORMANCE LIMITATIONS IN CONTROL USING ARX-MODELS

Non-minimum phase zeros and poles of a process put upper and lower constraints on the bandwidth of a closed loop system. It is thus of great interest to be able to identify these quantities. In this contribution it is shown that non-minimum phase zeros and unstable poles can be identified using high order ARX-models without the standard o(n) (n is the model order) variance penalty for over modeling. An asymptotic, in the model order and the number of data, expression for the variance of non-minimum phase zeros is derived. This result shows that the problem of determining the performance limits of a system from experimental data is considerably easier than identifying the complete system.

BEST TRACKING AND REGULATION PERFORMANCE UNDER CONTROL EFFORT CONSTRAINT: TWO-PARAMETER CONTROLLER CASE

This paper studies optimal tracking and regulation control problems by two-parameter controllers, in which objective functions of tracking error and regulated response, defined by integral square measures, are to be minimized jointly with the control effort, where the latter is measured by the plant input energy. The approach here is similar to our recent study for one-parameter controller case, and the problems are solved explicitly by deriving analytical expressions for the best achievable performance. Besides the plant non-minimum phase zeros, time delays, and unstable poles, the results reveal and quantify how the gain characteristics of the plant may all affect the performance. These effects are nonexistent when the control effort is not taken into account.

Iterative learning control for linear discrete time nonminimum phase systems

This paper investigates iterative learning control for linear discrete time nonminimum phase systems. First, iterative learning control with advanced output data is considered for maximum phase systems. Next, the results are extended to nonminimum phase systems. The stability of the inverse mapping from the desired output to the input is proven based on the results for maximum phase systems. The input should be updated with the output which is more advanced than the input by the sum of the relative degree of the system and the number of nonminimum phase zeros. An example is given to indicate the importance of proper advances of output in the input update law.

Decoupling internal model control for multivariable systems with multiple time delays

In this paper, the decoupling internal model control (IMC) with stability is investigated for multivariable stable processes with multiple time delays. All the stabilizing IMC controllers which solve this decoupling problem and the resulting closed-loop systems are characterized in terms of the open-loop system's time delays and non-minimum phase zeros. It shows that the inclusion of some time delays and non-minimum phase zeros might be necessary to make a decoupling solution realizable and stabilizing. Based on this characterization, a control design method for best achievable performance is presented. However, owing to the high complexity of the theoretical controller, a practical controller design procedure is developed with the help of the proposed model reduction algorithm. Examples are given to illustrate our analysis and design. Significant performance improvement over the existing multivariable Smith predictor control has been achieved with the proposed approach.

Optimal tracking performance: preview control and exponential signals

We study tracking performance limitation problems. Two issues are addressed concerning how earlier results developed elsewhere may be extended to more general classes of reference signals, and how tracking performance may be further improved beyond that offered by feedback control. Toward these issues we consider exponentially increasing reference inputs and examine the use of preview control for tracking. Using an optimal interpolation approach, we develop analytical expressions and a conceptual insight which will aid in the understanding of these issues. To this effect, we derive explicit expressions for the optimal tracking error, either as exact solutions or bounds. It is found that for the exponential signals the earlier results can be directly extended, and similar conclusive statements can be drawn. It is also shown that, in general, preview can be used to advantage for improving tracking performance, especially in dealing with the effect resulted from plant nonminimum phase zeros.

Precise Speed Control of a Travelling Wave Ultrasonic Motor for Prosthetic Hands

A repetitive speed controller with zero phase error tracking compensation (ZPETC) is proposed to reduce the intrinsic speed ripples of travelling wave ultrasonic motors (TWUMs). The proposed control scheme is also intended to compensate the speed variations due to load and temperature changes. This controller is expected to improve the performance of the impedance control system of a prosthetic hand, which is partially powered by a TWUM. Repetitive controllers are capable of improving the response of dynamical systems under periodic disturbances by learning from the stored error values of preceding periods. Since it is desirable for the stabilizing compensator of repetitive controllers to be zero phase and a perfect error tracker, the ZPET controller is a natural choice when the servo’s closed-loop transfer function has non-minimum phase zeros. The performance of the proposed controller was compared against that of a Pi-repetitive controller and its effectiveness experimentally demonstrated.

Minimization of the Bode sensitivity integral

In this paper we present two new results on Bode-type integrals. Firstly, we obtain, for a given scalar or multivariable continuous-time plant, the infimum of the Bode sensitivity integral which can be obtained with any stabilizing controller. The result involves the unstable plant poles and, perhaps surprisingly, a subset of the plant nonminimum phase zeros. Secondly, we obtain an apparently new expression for the Bode integral for the complementary sensitivity for a stable discrete-time scalar system.

Logarithmic integrals, interpolation bounds, and performance limitations in MIMO feedback systems

We study performance limitation issues found in linear multivariable feedback systems. Our main contributions include Bode and Poisson type integral inequalities and performance limits for the sensitivity and complementary sensitivity functions. These results characterize and quantify explicitly how open-loop unstable poles and nonminimum phase zeros may impose inherent limitations on feedback design and fundamental limits on the best achievable performance. The role of time delay is also studied in this context. Most notably, we show that the performance and design limitations in multivariable systems intrinsically depend on the locations as well as directions of unstable poles and nonminimum phase zeros, and in particular, on how pole and zero directions are aligned. The latter is characterized by angles measuring the mutual orientation between zero and pole directions, and it is shown to play a crucial role in multivariable system design.

Integral constraints and performance limits on complementary sensitivity: discrete-time systems

We study performance limitation issues for multivariable discrete-time feedback systems. The complementary sensitivity function is employed as a performance measure, and Bode and Poisson-type integral inequalities and H ∞ -type performance limits are derived. The results exhibit frequency-dependent constraints as well as best achievable limits on the complementary sensitivity function, which are shown to be determined by nonminimum phase zeros, unstable poles, and time delays. In particular, the directions of such zeros and poles are seen to play a central role to this effect.

Limitations on maximal tracking accuracy

This paper studies optimal tracking performance issues pertaining to finite-dimensional, linear, time-invariant feedback control systems. The problem under consideration amounts to determining the minimal tracking error between the output and reference signals of a feedback system, attainable by all possible stabilizing compensators. An integral square error criterion is used as a measure for the tracking error, and explicit expressions are derived for this minimal tracking error with respect to step reference signals. It is shown that plant nonminimum phase zeros have a negative effect on a feedback system's ability to reduce the tracking error, and that in a multivariable system this effect results in a way depending on not only the zero locations, but also the zero directions. It is also shown that if unity feedback structure is used for tracking purposes, plant nonminimum phase zeros and unstable poles can together play a particularly detrimental role in the achievable tracking performance, especially when the zeros and poles are nearby and their directions are closely aligned. On the other hand, if a two parameter controller structure is used, the achievable tracking performance depends only on the plant nonminimum phase zeros.

Feedback limitations of linear sampled-data periodic digital control

We present a set of feedback limitations for linear time-invariant systems controlled by periodic digital controllers based upon an analysis of the inter-sample response of the closed-loop system to sinusoidal inputs. Fundamental sensitivity and complementary sensitivity functions govern the fundamental and harmonic components of the continuous closed-loop response. The continuous and discrete response of the system is sensitive to variations in the analog plant at frequencies integer multiples of ωs/N away from the excitation frequency, where ωs is the sampling frequency and N is the period of the controller. These functions satisfy interpolation and integral constraints due to open-loop non-minimum phase zeros and unstable poles. In addition, the use of periodic digital control may result in a reduction in closed-loop bandwidth. Copyright © 2000 John Wiley & Sons, Ltd.

Integral constraints and performance limits on complementary sensitivity: Discrete-time systems

We study performance limitation issues for multivariable discrete-time feedback systems. The complementary sensitivity function is employed as a performance measure, and Bode and Poisson type integral inequalities and H ∞ type performance limits are derived. The results exhibit frequency-dependent constraints as well as best achievable limits on the complementary sensitivity function, which are shown to be constrained by nonminimum phase zeros and unstable poles. In particular, the directions of such zeros and poles are seen to play a central role to this effect.

Robust controller design for a skid to turn missile

The design and analysis of Robust Autopilot for skid-to-turn missile is presented. Two of the popular Robust Controller design approaches, The Loop Shaping Design Procedure (LSDP) and The Robust Eigenstructure Assignment are considered. The missile model considered in the present work is having lightly damped modes and non-minimum phase zeros, with stringent performance requirements. Numerical results are presented to evaluate the Robustness of stability and performance of the controller. Merits and demerits of the above said methodologies are brought out clearly as applied to this specific plant.