Interaction bounds in multivariable control systems

Time-domain limitations due to right half-plane zeros and poles in linear multivariable control systems are studied. Lower bounds on the interaction are derived. They show not only how the location of zeros and poles are critical in multivariable systems, but also how the zero and pole directions influence the performance. The results are illustrated on the quadruple-tank process, which is a new multivariable laboratory process.     

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     

纯量反馈系统稳定零极近似相消的积分约束

导出了单输入单输出反馈控制系统误差响应基于名义系统稳定的零、极点近似相消的时间域积分约束，此积分约束是任何反馈控制系统均应满足的．这一约束给出了单输入单输出反馈系统固有折中的新的观点．名义系统稳定的零、极点近似相消的存在导致反馈控制系统的调节时间延长或者误差响应的无穷范数变大．因此，在反馈控制系统设计中，尽量避免补偿器的零、极点与名义系统的极、零点近似相消(即使这些零、极点是稳定的)．     

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     

Optimal tracking and two-channel disturbance rejection under control energy constraint

In this paper, the optimal tracking problem under control energy constraint is studied. The disturbance in the upstream channel and down channel is adequately considered. Some new results are derived and it is shown that the performance limitation is tightly dependent on the non-minimum phase zeros and the unstable poles. In addition, the performance limitation, which is greater than that of the disturbance only in the down channel, depends on other new non-minimum phase zeros. Finally, the example explains and validates the conclusion.     

Dynamic behaviour of integrated plants

The effect of material recycle and heat integration on the dynamics and control of chemical processing plants is considered. In analogy to linear control theory, one may consider how plant interconnections affect the fundamental properties of the dynamics, such as the poles and zeros. This implies that recycle of mass and energy, which are feedback mechanisms, affects the poles and thus possibly the plant stability, whereas parallel interconnections in a plant affect the zeros and thus the achievable performance of the plant under feedback control.     

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.     

Ripple-free deadbeat control of SISO discrete systems

The ripple-free deadbeat control problem for arbitrary (not necessarily stable) SISO linear discrete plants and reference signals is treated. It is established that a causal, stabilizing ripple-free deadbeat controller exists if and only if the zeros of the plant and the poles of the reference signal are disjoint, and a complete characterization of all such controllers is obtained. Solutions to two illustrative problems are presented.     

On the stability of solutions to minimal and nonminimal design problems

A partial resolution to the question of stability of solutions to the minimal design problem is given in terms of transfer matrix factorizations employing the new notions of common system poles and common systems zeros as well as the fixed poles of all solutions and the fixed poles of minimal solutions. The results are employed to more directly and easily resolve questions involving the attainment of stable solutions to the model matching problem and stable minimal-order state observers.     

Order estimation for linear time-invariant systems using frequencydomain identification methods

A two-step coarse-fine order estimation technique is proposed to determine the order of the numerator and the denominator polynomials of rational transfer function models for single-input/single-output (SISO) linear time-invariant systems. The coarse order estimation is based on rank detection by verification of the stochastic significance of the singular values of a linearized problem. The fine order estimation is based on a statistical analysis of the maximum likelihood cost function. The method is tested on measurements of low-(4 zeros, 6 poles) and high- (58 poles, 58 zeros) order systems     

Accurate Computation of Vocal Tract Filter Parameters Using a Hybrid Genetic Algorithm

This paper reports an original technique for accurately estimating the parameters of human vocal tract filters for vowels in English for speech processing applications such as voice recognition. In this paper, the vocal tract filter design problem is reformulated as a general nonlinear optimization problem and solved using a hybrid Genetic Algorithm (GA). The hybrid GA computes a rough estimate of the global minimum using GA and refines using computationally cheap local search. Issues that are of concern in digital filtering such as achieving stability and overcoming finite precision effects are addressed. The objective function for optimization used in this paper is formulated in terms of poles and zeros of the filters to avoid ill-conditioning and to take advantage of symmetries in the location of poles and zeros. Simulation results indicate that the approach presented in this paper provides a close fit in terms of mean square error between the experimental and designed filters.     

The optimal linear-quadratic time-invariant regulator with cheap control

The infinite-time linear-quadratic regulator is considered as the weighting on the control energy tends to zero (cheap control). First, a study is made of the qualitative behavior of the limiting optimal state and control trajectories. In particular, the orders of initial singularity are found and related to the excess of poles over zeros in the plant. Secondly, it is found for which initial conditions the limiting minimum cost is zero (perfect regulation). This generalizes an earlier result of Kwakernaak and Sivan. Finally, a simple extension is made to the steady-state LQG problem with cheap control and accurate observations.     

The role of transmission zeros in linear multivariable regulators

The problem is considered of designing compensators to regulate linear multivariable systems. It is shown that the essential mechanism employed by such a compensator is the multivariable analogue of pole-zero cancellation: the compensator supplies transmission zeros to cancel the unstable poles of the disturbance and reference signals. If the compensator is additionally required to function in the presence of small variations in system parameters then it must supply transmission zeros in greater multiplicity. Finally it is shown that the transmission zeros are generated by an ‘ internal model ’, incorporated in the compensator, of the dynamics of the disturbance and reference signals.     

用时域法分析增加零极点对控制系统性能的影响

主要用时域法分析增加零极点对系统性能的影响。给出原系统的阶跃响应曲线和主要性能指标,然后在原系统闭环传递函数的基础上增加一个零点,或者增加一个极点,并不断改变零极点的位置来分析,从而得出增加零极点后对系统的性能有哪些影响。     

Inversion of sampled-data system approximates the continuous-time counterpart in a noncausal framework

Although correspondence between the poles of a continuous-time and sampled-data system with a piecewise constant input is simple and desirable from the stability viewpoint, the relationship between zeros is intricate. Inversion of a sampled-data system is mostly unstable irrespective of the stability of the continuous-time counterpart. This makes it difficult to apply inversion-based control techniques such as perfect tracking, transient response shaping or iterative learning control to sampled-data systems. Although recently developed noncausal inversion techniques help us to circumvent unboundedness of the inversion caused by unstable zeros, whether the inversion of sampled-data systems approximates the continuous-time counterpart or not as the sample period is shortened is still to be determined. This article gives a positive conclusion to this problem.     

Variance-error quantification for identified poles and zeros

This paper deals with quantification of noise induced errors in identified discrete-time models of causal linear time-invariant systems, where the model error is described by the asymptotic (in data length) variance of the estimated poles and zeros. The main conclusion is that there is a fundamental difference in the accuracy of the estimates depending on whether the zeros and poles lie inside or outside the unit circle. As the model order goes to infinity, the asymptotic variance approaches a finite limit for estimates of zeros and poles having magnitude larger than one, but for zeros and poles strictly inside the unit circle the asymptotic variance grows exponentially with the model order. We analyze how the variance of poles and zeros is affected by model order, model structure and input excitation. We treat general black-box model structures including ARMAX and Box–Jenkins models.     

Analysis of parametric sensitivity and structural optimization of modal control systems with state controllers

The problem of synthesis of parametrically robust modal control systems with inertialess and dynamic state controllers is considered. The dependence of the domain of parametric robustness of the system with the state controller on the mutual position of zeros and poles and the structure of the control plant is studied. It is demonstrated that the analysis of norms of coordinate transformation matrices of a plant and the construction of zones of parametric robustness make it possible to choose the most efficient structure of the modal control system.     

Pole and zero assignment by state vector feedback

A procedure is described which enables specified poles and zeros of a scalar transfer function of a controllable, observable, linear system to be obtained by using state vector feedback to two inputs. The number of zeros is equal to the number of zeros in the transfer function, before feedback is applied, from one input or the other, whichever is the greater. Those zeros which can be changed, and those which cannot, are identified. The former can be made equal to, or arbitrarily close to, any assigned values, and the poles can be assigned arbitrarily.     

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