Two-delay robust digital control and its applications-avoiding theproblem on unstable limiting zeros

In the design of linear control systems, the existence of unstable zeros makes it difficult to construct many control systems. When the usual digital control scheme is used, unstable zeros appear in the discrete-time model due to the existence of limiting zeros even though the continuous-time plant is minimal phase. To avoid this unstable zero problem, two new digital control schemes, called two-delay output control and two-delay input control, are proposed. These control systems are proved to have no finite zeros. Asymptotic inverse systems, pole assignable unknown input observers, and output feedback controllers having the LTR (loop transfer recovery) property are developed using the two-delay output control. Zero assignable control systems and model matching control systems are developed using the two-delay input control     

Minimal inversion,command matching and disturbance decoupling in multivariable systems

This paper addresses the two basic and related problems of minimal inversion and perfect output control in linear multivariable systems. A simple analytical expression is obtained for the inverse of the transfer-function matrix of a square multivariable system. This expression is then used to construct a minimal-order inverse of the system. It is shown that the poles of the minimal-order inverse are the transmission zeros of the system. As a result, necessary and sufficient conditions for existence and stability of the inverse system are stated simply in terms of the zero polynomial of the original system. Furthermore, the minimal-order inverse is shown to be proper provided the original system has a full rank feedthrough matrix. The related problem of perfect output control, namely command matching and disturbance decoupling, in linear multivariable systems by means of feedforward controllers is also formulated and solved in a transfer-function setting. It is shown that a necessary and sufficient condition for existence of the required controllers is that the plant zero polynomial is not identical to zero or unstable. The order of the required controllers is equal to the number of plant transmission zeros. The control scheme proposed in this paper is composed of a feedback controller to enhance system stability and robustness, a feedforward controller to ensure command matching, and another feedforward controller to achieve disturbance decoupling. The three controllers have no effect on each other and can therefore be designed independently. A number of numerical examples are discussed for illustration.     

Compensation of uncertain continuous systems by using the internal model control principle

In this paper the design of compensators for uncertain continuous plants is investigated. The standard derived compensators are based on the application of the internal model control (IMC) method. The required a priori knowledge on the plant is rather weak, namely, an upper bound of the plant relative order, the numbers of the strictly unstable and critically unstable plant poles being integrators and upper and lower bounds of the amplitude-versus-frequency plot over the low frequency band in the case of minimum-phase open-loop systems. If the open-loop system has unstable zeros and/or poles then the above bounds are required to be known for a modified magnitude plot which substitutes the unstable zeros (poles) by stable poles (zeros) which are their complex-conjugate reflections on the left-hand plane. An absolute upper bound of the open-loop phase plot is obtained on a finite frequency interval which allows the closed-loop system to guarantee a prescribed relative stability in many practical situations. The method is dependent on the alternative design of phase lead/lag classical compensators and to indirect adaptive control situations where the adaptive identifier is used for the parametrization of the adaptive controller.     

Feedforward digital tracking controllers for motion control applications

This paper reviews digital tracking control algorithms for motion control applications. In tracking control, the control objective is to steer the control object along the time-varying desired output. Two design approaches are presented for the case where the desired signal is known in advance, i.e. previewable. One approach is based on the mathematical inverse of a closed-loop system consisting of a controlled plant and a feedback controller. If the mathematical inverse is asymptotically stable, i.e. the closed-loop system does not possess zeros outside the unit circle (unstable zeros), it is an ideal feedforward controller for achieving perfect tracking under the preview assumption. For closed-loop systems with unstable zeros, a cancellation technique for the phase shift induced by unstable zeros is introduced. Another approach is based on linear quadratic optimal control and is known as finite optimal preview control. In this approach, the feedback controller and feedforward controller are determined simultaneously by minimizing a quadratic performance index which involves a tracking error term and a term related to the control effort. Applications of these tracking control algorithms to mechanical systems control are described.     

Adaptive actuator failure compensation control for MIMO systems*

Two adaptive failure compensation control schemes based on MRAC are developed for a class of MIMO LTI systems with unknown actuator failures. An effective controller structure is proposed to achieve the desired plant-model output matching when implemented with matching parameters. Design conditions are specified for such nominal plant-model output matching. Two adaptive versions of the nominal controller are proposed and stable adaptive laws are derived for updating the controller parameters when plant parameters and failure parameters are unknown. All closed-loop signals are bounded and the plant outputs track the given reference outputs asymptotically, despite the uncertainties in actuator failures and plant parameters. Simulation results for an aircraft lateral dynamic model verify the desired adaptive control system performance in the presence of unknown rudder and aileron failures.     

A multivariable MRAC scheme with application to a nonlinear aircraft model

This paper revisits the multivariable model reference adaptive control (MRAC) problem, by studying adaptive state feedback control for output tracking of multi-input multi-output (MIMO) systems. With such a control scheme, the plant-model matching conditions are much less restrictive than those for state tracking, while the controller has a simpler structure than that of an output feedback design. Such a control scheme is useful when the plant-model matching conditions for state tracking cannot be satisfied. A stable adaptive control scheme is developed based on LDS decomposition of the high-frequency gain matrix, which ensures closed-loop stability and asymptotic output tracking. A simulation study of a linearized lateral-directional dynamics model of a realistic nonlinear aircraft system model is conducted to demonstrate the scheme. This linear design based MRAC scheme is subsequently applied to a nonlinear aircraft system, and the results indicate that this linearization-based adaptive scheme can provide acceptable system performance for the nonlinear systems in a neighborhood of an operating point.     

An adaptive disturbance rejection control scheme for multivariable nonlinear systems

An adaptive disturbance rejection control scheme is developed for uncertain multi-input multi-output nonlinear systems in the presence of unmatched input disturbances. The nominal output rejection scheme is first developed, for which the relative degree characterisation of the control and disturbance system models from multivariable nonlinear systems is specified as a key design condition for this disturbance output rejection design. The adaptive disturbance rejection control design is then completed by deriving an error model in terms of parameter errors and tracking error, and constructing adaptive parameter-updated laws and adaptive parameter projection algorithms. All closed-loop signals are guaranteed to be bounded and the plant output tracks a given reference output asymptotically despite the uncertainties of system and disturbance parameters. The developed adaptive disturbance rejection scheme is applied to turbulence compensation for aircraft fight control. Simulation results from a benchmark aircraft model verify the desired system performance.     

Globally stable adaptive system design for minimum phase SISO plants with input saturation

Model reference adaptive control problem for single-input single-output time-invariant continuous-time plants with input saturation is considered with main attention focused on global properties. A sufficient condition is presented and a new design method of adaptive control systems is proposed. If a priori information about the plant is available to choose the reference model and the reference input so that the sufficient condition holds, the closed-loop adaptive control system designed by the proposed method can have global stability and globally output tracking property. It is shown that the sufficient condition becomes necessary in some cases.     

Global stability and performance of a simplified adaptive algorithm†

Most self-tuning and adaptive control algorithms usually use reference models, controllers, or identifiers of about the same order as the controlled plant. Since the dimension of the plants in the real world may be very large or unknown, implementation of adaptive control procedures may be difficult, or sometimes impossible. In this paper we prove global stability for a simple adaptive algorithm that can use low-order model reference and controllers, since no observers or identifiers are used in the adaptation process. The algorithm is basically fitted for systems that are denominated as ‘almost positive real’. It is shown that, at the price of bounded rather than vanishing output tracking errors, the simple algorithm can be applied in systems that can be stabilized via constant output feedback. These procedures are believed to reduce considerably the effort required for implementation of adaptive control in practical applications, especially in multivariable large-scale systems.     

Decentralized blocking zeros and the decentralized strongstabilization problem

This paper is concerned with a new system theoretic concept, decentralized blocking zeros, and its applications in the design of decentralized controllers for linear time-invariant finite-dimensional systems. The concept of decentralized blocking zeros is a generalization of its centralized counterpart to multichannel systems under decentralized control. Decentralized blocking zeros are defined as the common blocking zeros of the main diagonal transfer matrices and various complementary transfer matrices of a given plant. As an application of this concept, we consider the decentralized strong stabilization problem (DSSP) where the objective is to stabilize a plant using a stable decentralized controller. It is shown that a parity interlacing property should be satisfied among the real unstable poles and real unstable decentralized blocking zeros of the plant for the DSSP to be solvable. That parity interlacing property is also sufficient for the solution of the DSSP for a large class of plants satisfying a certain connectivity condition. The DSSP is exploited in the solution of a special decentralized simultaneous stabilization problem, called the decentralized concurrent stabilization problem (DCSP). Various applications of the DCSP in the design of controllers for large-scale systems are also discussed     

The internal model principle of control theory

The classical regulator problem is posed in the context of linear, time-invariant, finite-dimensional systems with deterministic disturbance and reference signals. Control action is generated by a compensator which is required to provide closed loop stability and output regulation in the face of small variations in certain system parameters. It is shown, using the geometric approach, that such a structurally stable synthesis must utilize feedback of the regulated variable, and incorporate in the feedback path a suitably reduplicated model of the dynamic structure of the disturbance and reference signals. The necessity of this control structure constitutes the Internal Model Principle. It is shown that, in the frequency domain, the purpose of the internal model is to supply closed loop transmission zeros which cancel the unstable poles of the disturbance and reference signals. Finally, the Internal Model Principle is extended to weakly nonlinear systems subjected to step disturbances and reference signals.     

Virtual Reference Feedback Tuning for non-minimum phase plants

Model reference control design methods fail when the plant has one or more non-minimum phase zeros that are not included in the reference model, leading possibly to an unstable closed loop. This is a very serious problem for data-based control design methods, where the plant is typically unknown. In this paper, we extend the Virtual Reference Feedback Tuning method to non-minimum phase plants. This extension is based on the idea proposed in Lecchini and Gevers (2002) for Iterative Feedback Tuning. We present a simple two-step procedure that can cope with the situation where the unknown plant may or may not have non-minimum phase zeros.     

Iterative learning of model reference adaptive controller for uncertain nonlinear systems with only output measurement

In this paper, a model reference adaptive control strategy is used to design an iterative learning controller for a class of repeatable nonlinear systems with uncertain parameters, high relative degree, initial output resetting error, input disturbance and output noise. The class of nonlinear systems should satisfy some differential geometric conditions such that the plant can be transformed via a state transformation into an output feedback canonical form. A suitable error model is derived based on signals filtered from plant input and output. The learning controller compensates for the unknown parameters, uncertainties and nonlinearity via projection type adaptation laws which update control parameters along the iteration domain. It is shown that the internal signals remain bounded for all iterations. The output tracking error will converge to a profile which can be tuned by design parameters and the learning speed is improved if the learning gain is large.     

H∞-Based Selective Inversion of Nonminimum-phase Systems for Feedback Controls

Stably inverting a dynamic system model is fundamental to subsequent servo designs. Current inversion techniques have provided effective model matching for feedforward controls.However, when the inverse models are to be implemented in feedback systems, additional considerations are demanded for assuring causality, robustness, and stability under closed-loop constraints. To bridge the gap between accurate model approximations and robust feedback performances, this paper provides a new treatment of unstable zeros in inverse design. We provide first an intuitive pole-zero-map-based inverse tuning to verify the basic principle of the unstable-zero treatment. From there, for general nonminimum-phase and unstable systems, we propose an optimal inversion algorithm that can attain model accuracy at the frequency regions of interest while constraining noise amplification elsewhere to guarantee system robustness. Along the way, we also provide a modern review of model inversion techniques. The proposed algorithm is validated on motion control systems and complex high-order systems.     

IMPLICIT TRIANGULAR OBSERVER FORM DEDICATED TO A SLIDING MODE OBSERVER FOR SYSTEMS WITH UNKNOWN INPUTS

It has been presented in previous works that every uniformly observable single‐output system can be put on a triangular observation form. For this structure a special kind of sliding mode observer has been designed by authors, which ensures a finite‐time state reconstruction using a step by step observation algorithm. In this paper, we show that the multi‐output case is more delicate to study especially when the system has some unknown inputs. Thus, in order to generalizes the triangular observer form, from single to multi‐output case, we define an Implicit Triangular Observer (ITO) form. For such a form, two results are given. Firstly, we design a finite time converging observer for all values of the unknown inputs. Secondly, we give the necessary and sufficient condition, including a matching condition, for the existence of a coordinate change to put the system into this form. It is also shown that this class of systems is a subset of the uniform observable class of systems.     

Solvability conditions and solutions to perfect regulation problem under measurement output feedback

The problem of perfect regulation is to design a family of control laws for a given plant such that the resulting overall closed-loop system is internally stable and its controlled output can be reduced to zero arbitrarily fast from any initial condition. Such a problem was heavily studied by many researchers in the 1970s and early 1980s. However, to the best of our knowledge, all of the earlier results deal only with the problem under full state feedback. In this paper, we solve the long-standing problem of perfect regulation via measurement output feedback for general linear time-invariant multivariable systems. In particular, we derive necessary and sufficient conditions under which the problem of perfect regulation via measurement output feedback is solvable for general systems, and, under these conditions, construct two families of feedback laws, one of full order and the other reduced order, that solve the problem.     

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     

A stability theorem for identification and adaptive control

Establishes a globally convergent algorithm for discrete time model reference adaptive systems, an extension of the classical decreasing gain algorithm. Using this approach, the process, in adaptive control, may contain zeros outside the unit circle if the external reference signal is a solution of a linear homogeneous difference equation     

基于线性网络的一类带扰动线性对象的逆控制研究

将线性网络应用于一类带扰动的线性对象,提出了一种基于该线性网络的自适应逆控制方案,该方案由辨识器、控制器和扰动消除器三部分构成,合理选择三个线性网络的输入,通过辨识器的在线学习,同时更新控制器和扰动消除器的权值,文章研究了该方案的收敛性和方案的跟踪性。根据可变步长权值收敛条件,设计了输入解相关变步长LMS算法调整辨识器权值方法。通过仿真研究了逆控制方法的有效性。     

Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics

This paper examines the robustness properties of existing adaptive control algorithms to unmodeled plant high-frequency dynamics and unmeasurable output disturbances. It is demonstrated thai there exist two infinite-gain operators in the nonlinear dynamic system which determines the time-evolution of output and parameter errors. The pragmatic implication of the existence of such infinite-gain operators is that 1) sinusoidal reference inputs at specific frequencies and/or 2) sinusoidal output disturbances at any frequency (including dc), can cause the loop gain to increase without bound, thereby exciting the unmodeled high-frequency dynamics, and yielding an unstable control system. Hence, it is concluded that existing adaptive control algorithms as they are presented in the literature referenced in this paper, cannot be used with confidence in practical designs where the plant contains unmodeled dynamics because instability is likely to result. Further understanding is required to ascertain how the currently implemented adaptive systems differ from the theoretical systems studied here and how further theoretical development can improve the robustness of adaptive controllers.