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     

具有随机延时的网络控制系统的均方可镇定性分析

针对网络系统的可镇定性问题,研究整数步随机延时离散时间线性系统的均方可镇定性.利用Youla参数化与内外分解方法,结合均方小增益定理得到系统输出反馈均方可镇定的充分必要条件.该条件明确给出系统可镇定性与被控对象特性(不稳定极点、非最小相位零点、相对阶)和信道特性(频域信噪比函数)的关系,其中频域信噪比函数在被控对象不稳定极点的取值对可镇定影响甚大.利用仿真算例量化被控对象的非最小相位零点及相对阶对可镇定性的影响,验证可镇定性条件的正确性.     

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.     

Robust stabilization of systems with multiple real poleuncertainties

The robust stabilization of systems with several uncertain real poles is discussed. Necessary and sufficient conditions of existence of a controller in terms is obtained of an interpolation problem are given. The interpolating function is obtained directly for a particular case. Two examples are shown for the synthesis of a controller. An explicit solution of the controller for plants without zeros in the right hand plane (RHP) and without known poles in the RHP is given. This shows the practical utility of the method to the design of robust stabilizers     

Controller synthesis for sign-invariant impulse response

In this paper, we consider the problem of designing controllers for discrete-time linear time-invariant (LTI) plants that render the closed-loop impulse response nonnegative. Such systems have a non-undershooting and non-overshooting step response. We first show that the impulse response of any discrete-time LTI system changes sign at least "r" times if it has "r" real, positive zeros outside a circular disk centered at the origin and containing all its poles. We then show that a necessary and sufficient condition on the plant for the existence of a compensator that makes the closed loop impulse response sign invariant is that there be no real, positive, nonminimum phase plant zeros. Finally, we show, by construction, how such a compensator may be synthesized when the plant does satisfy the existence condition.     

Decoupling controller design for linear multivariable plants

Decoupling controller design for linear time-invariant square multiple-input multiple-output (MIMO) plants under the unity-feedback configuration is discussed. For plants with no coincidences of unstable poles and zeros, a simplified necessary and sufficient condition for closed-loop stability is given. The simplified condition leads to a simple parameterization of the set of all achievable decoupled input/output (I/O) maps and an algorithm which allows the design of decoupling controllers to achieve preassigned closed-loop poles     

The solution of a transcendental problem and its applications insimultaneous stabilization problems

A complete solution for a special transcendental problem in system theory is given. One of the interpretations of this transcendental problem is the following simultaneous stabilization problem: under what conditions can two linear plants be simultaneously stabilized by a single compensator having no real poles in the closed right half of the complex plane? These conditions are completely general and therefore encompass unstable non-minimum-phase plants as well. It is shown that the simultaneous stabilizability of two given plants via such a restricted compensator depends solely on the locations of the real zeros of the two plants and their difference plant in the closed right half of the complex plane. For two plants satisfying the stabilizability conditions, a computational design procedure for constructing a desired compensator is provided and illustrated. A computationally checkable necessary condition for simultaneous stabilization of three plants is also given     

Conditions for stable zeros of sampled systems

A criterion for there being no unstable discrete zeros is derived in terms of coefficients of a continuous-time transfer function and of a sampling period when a continuous-time, stable, and strictly proper plant is sampled. It is also shown that the shortest sampling period ensuring minimum-phase behavior can be calculated from a simple equation     

Single-loop feedback-stabilization of linear multivariable dynamical plants

This paper derives the necessary and sufficient conditions for a multivariable plant P(s) with asymptotically stable hidden modes to be stabilizable by means of singleloop feedback employing an asymptotically stable controller and feedback sensor. These conditions are completely general and therefore encompass unstable, nonminimum-phase plants as well. For single-input-output plants with zero gain at infinite frequency the conditions reduce to the sole requirement that no plant zeros on the nonnegative real axis of the complex s-plane lie to the left of an odd number of real plant poles, multiple poles counted according to their multiplicities.It has also been possible to derive simple necessary and sufficient conditions for a closed-loop transfer function T(s) to be realizable by asymptotically stable compensation around a prescribed single-input-output plant P(s). It is expected that this latter result can be suitably generalized to the multivariable case.In a real sense, this paper constitutes a continuation of some earlier unpublished work [8] by the first author.     

SIMO和MISO系统绝对稳定性问题的一个猜想和几个推论

首先分析了具有多个非线性特性的 SIMO 和 MISO 系统的绝对稳定性问题, 指出应用已知的频域判据来解决上述问题很难奏效. 然后, 基于所有孤立部分传递函数都正实的充分必要条件给出了上述系统为稳定的一个猜想, 当传递函数的零极点都位于虚轴上时, 由这一猜想得到了一个已知的结论; 当传递函数的零极点都位于实轴上时, 由这一猜想得到了一个新的结论, 本文证明该结论是正确的; 最后, 根据这一猜想, 给出了传递函数极点位于复平面的一个例子, 它涉及到一类系数矩阵为时变正定矩阵的振动方程的稳定性问题, 值得去深入研究.     

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.     

稳定广义预测控制与性能分析

基于已有的一种稳定广义预测控制(SGPC)结构,以参考信号为优化变量对目标函数进行寻优,求出了控制器的滚动最优控制律,该控制律可以无静差地跟踪常数设定值.文中给出并证明了SGPC闭环控制系统的稳定性充分条件.数值仿真表明,该SGPC控制器可以保证闭环系统无稳态误差,并且适用于具有近似相消零极点的被控对象.     

A note on the Youla-Bongiorno-Lu condition

Youla, Bongiorno and Lu have presented a condition for the stabilizability using stable controllers of an unstable plant. We show this condition may be reformulated in terms of Cauchy indices, allowing its checking in a finite number of rational operations, should the plant not be given in factored form, i.e. should its poles and zeros be unknown. The results applies to discrete and continuous-time, and scalar and multivariable plants.     

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     

On undershoot in SISO systems

We show that the step response of an asymptotically stable strictly proper nonminimum phase continuous time system with m₁ real open right half plane zeros, no conjugate complex zeros and poles, and with real open left half plane zeros and poles satisfying a given “bracketing” condition, exhibits undershoot exactly int m₁+½ times. Both qualitative and quantitative arguments are also introduced to highlight the main issues under discussion. Simple examples are given to illustrate and complement the main result of this paper     

Robust stabilization of a SISO system with uncertainty in timedelay

Sufficient conditions are developed for the robust stabilization of an unstable strictly proper linear time-invariant (LTI) and single-input-single-output (SISO) plant with uncertainty in the time delay and with arbitrary poles and zeros in the open right half of the complex plane. The technique developed shows that a tradeoff exists between the open-loop gain of the control system and the robust stability with the time delay uncertainty     

On undershoot and nonminimum phase zeros

In this note we derive a simple necessary and sufficient condition for a stable system to exhibit an undershooting step response. Specifically, we show that undershoot occurs if and only if the plant has an odd number of real right-half plane zeros.     

Limiting zero distribution of sampled systems

In this paper, we consider the sampling of a continuous time system with possibly a non-rational transfer function. Limiting zero distribution of the sampled system represented by an FIR model is derived. It is shown that the zeros of the FIR sampled system converge evenly to the unit circle or to the unit disk plus unstable roots of the reciprocal polynomials depending on assumptions.     

Simple criterion for testing non-minimum-phase systems

Simple sufficient conditions for non-minimum-phase systems are derived using the fundamental theorem of algebra. It is shown that the unstable poles or zeros of a control system can be tested directly from the coefficients of the characteristic polynomials without any calculation. This criterion is useful for the computer-aided design of control systems.