Stabilising PID tuning for a class of fourth-order integrating nonminimum-phase systems

Fed-batch fermentation processes are commonly used in bioprocessing industry. A fed-batch fermentation process often exhibits integrating/unstable type of dynamics with multiple right-half plane zeros. A class of fourth-order integrating model can be used to adequately represent such a complex dynamics of the fed-batch fermentation process. In this paper, rigorous stability analysis of proportional-integral-derivative (PID) controller based on the Routh-Hurwitz criteria for the fourth-order integrating system is presented. A set of all stabilising PID controller parameter regions is established. Based on these stabilising regions, a general PID controller tuning procedure is proposed for the fourth-order integrating system with two right-half plane zeros. Numerical study shows that based on the proposed tuning procedure, a low-order PID controller can outperform a fifth-order optimal LQG controller in terms of servo and regulatory controls.     

Approximate decoupling and asymptotic tracking for MIMO systems

This paper presents an algorithm for approximate input-output decoupling of nonlinear MIMO systems that are either numerically ill-posed or exhibit nearly singular behavior in the application of decoupling algorithms. Although the systems considered are regular, so that the exact decoupling algorithms are applicable in this case, they require inversion of an ill-conditioned matrix, and yield high gain feedback solutions that may result in actuator saturation and cancellation of high frequency zeros. The approximate algorithms of this paper are numerically robust, and provide solutions that do not cancel far off right-half plane zeros. This latter characteristic is especially valuable when some of the far off right-half plane zeros are unstable. The algorithms are inspired by and are generalizations of some examples in the flight control literature     

Properties of optimal weighted sensitivity designs

Weighted sensitivity designs for a class of single-input/single-output, linear time-invariant systems are shown to have the property that the optimal controller has fewer right-half plane poles than the plant has right-half plane zeros. While weighted complementary sensitivity designs are shown to have the property that the optimal controller has fewer right-half plane zeros than the plant has right-half plane poles. Effects of various choices of weighting functions on the optimal solutions are described     

Low order integral-action controller synthesis

A simple controller synthesis method is developed for certain classes of linear, time-invariant, multi-input multi-output plants. The number of poles in each entry of these controllers depends on the number of right-half plane plant zeros, and is independent of the number of poles of the plant to be stabilized. Furthermore, these controllers have integral-action so that they achieve asymptotic tracking of step input references with zero steady-state error. The designed controller’s poles and zeros are all in the stable region with the exception of one pole at the origin for the integral-action design requirement. The freedom available in the design parameters may be used for additional performance objectives, although the only goal here is stabilization and tracking of constant references.     

A parameter adaptive control structure for linear multivariable systems

This paper presents an adaptive control structure which can be used to assign all poles and zeros of a continuous-time linear multivariable system represented by an (m times m) strictly proper transfer matrixT(s), providedT(s)has no right-half plane zeros. The controller parameters can be directly estimated from input-output data. The paper also serves to point out the type of a priori information necessary for multivariable adaptive controller design. This information is a natural extension of that required in the scalar case.     

On the order of stable compensators for a class of time-delay system

The stabilization using a stable compensator does not introduce additional unstable zeros into the closed-loop transfer function beyond those of the original plant, so it is a desirable compensator, the price is that the compensator‘s order will go up. This note considered the order of stable compensators for a class of time-delay systems. First, it is shown that for single-loop plants with at most one real fight-half plane zero, a special upper bound for the minimal order of a strongly stabilizing compensator can be obtained in terms of the plant order; Second, it is shown that approximate unstable pole-zero cancellation does not occur,and the distances between distinct unstable zeroes are bounded below by a positive constant, then it is possible to find an upper bound for the minimal order of a strongly stabilizing compensator.     

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.     

Optimal feedforward compensators for systems with right-half plane zeros

Feedforward from measurable disturbances is a powerful complement to feedback control to improve disturbance rejection capability. Recent works have remarked the necessity of a design strategy for those cases where the ideal feedforward controller is not realizable. In this paper, a simple shaping design procedure is presented together with straightforward rules to obtain optimal feedforward controllers for the case when the ideal compensator is not realizable due to right-half plane zeros in the process dynamics. Finally, some simulations and a robustness analysis demonstrate the benefits of the proposed tuning rules.     

The synthesis of linear multivariable systems by state-variable feedback

Design procedures are presented for the compensation of linear multivariable-feedback systems. The design objectives which are noninteraction and low-order compensators are obtained by state-variable feedback. Aqth order plant withminputs andmoutputs whereq>mis treated. All the state variables are assumed to be available and measurable. It is shown that it is often possible by state-variable feedback to obtain noninteraction without an increase in system order. When the determinant of the plant transfer-function matrix has right half plane zeros the state-variable feedback method cannot be used to obtain noninteraction since the resulting compensated system would be unstable. It is shown that it is not possible to change these right-half plane zeros by either the well-known transfer-function design methods or by the state-variable feedback method. A combination of both transfer-function and state-variable techniques is discussed and shown to lower the order of the compensators required for the compensation of certain systems.     

The maximally achievable accuracy of linear optimal regulators and linear optimal filters

A linear system with a quadratic cost function, which is a weighted sum of the integral square regulation error and the integral square input, is considered. What happens to the integral square regulation error as the relative weight of the integral square input reduces to zero is investigated. In other words, what is the maximum accuracy one can achieve when there are no limitations on the input? It turns out that the necessary and sufficient condition for reducing the regulation error to zero is that 1) the number of inputs be at least as large as the number of controlled variables, and 2) the system possess no right-half plane zeros. These results are also "dualized" to the optimal filtering problem.     

On the properties of reduced order Kalman filters

Several known results are unified by considering properties of reduced-order Kalman filters. For the case in which the number of noise sources equals the number of observations, it is shown that the reduced-order Kalman filter achieves zero steady-state variance of the estimation error if and only if the plant has no transmission zeros in the right-half plane, since these would be among the poles of the Kalman filter. The reduced-order Kalman filter cannot achieve zero variance of the estimation error if the number of independent noise sources exceeds the number of observations. It is also shown that the reduced-order Kalman filter achieves the generalized Doyle-Stein condition for robustness when the noise sources are colocated with the control inputs. When there are more observations than noise sources, additional noise sources can be postulated to improve the observer frequency response without diminishing robustness     

非最小相位系统控制器的优化设计

抑制非最小相位系统的右半平面零点所造成的负调至今仍是个开放问题.以同时抑制 非最小相位系统的超调、负调和调整时间为目的,合理融合遗传算法的并行搜索结构和模拟退 火的可控性概率突跳,提出了设计一类PID控制器的混合优化策略.基于典型系统的数值仿真 验证了混合策略的有效性和对系统的适应性,以及所设计控制器的优越性.     

一种用于非最小相位系统的模糊控制器设计

针对非最小相位系统具有右半平面零点,使系统阶跃响应存在负调,以及常规方法控制不理想等问题,在分析常规模糊控制器的基础上,提出了一种新型的模糊积分控制器的设计方法,改善了常规模糊控制器的动态和稳态性能。对输入变量设计了一种简单可调整宽度的三角形隶属度函数;并针对非最小相位系统,提出了一般情况下控制规则基的设计原则。计算机仿真结果证实了这种设计方法的有效性。     

Design of stable controllers attaining low H/sup infinity / weighted sensitivity

Strong stabilization and sensitivity reduction for a linear time-invariant single-input single-output system are considered. Stable controllers are practically desirable in the presence of possible actuator or sensor failures. It is verified precisely that the optimal controller of H/sup infinity / weighted sensitivity minimization is almost always unstable when the plant has more than two zeros in the right-half plane. An algorithm for attaining the low sensitivity property for stable controllers is presented. It is based on Nevanlinna-Pick interpolation.<>     

D-stability of polynomial matrices

Necessary and sufficient conditions are formulated for the zeros of an arbitrary polynomial matrix to belong to a given region D of the complex plane. The conditions stem from a general optimization methodology mixing quadratic and semidefinite programming, LFRs and rank-one LMIs. They are expressed as an LMI feasibility problem that can be tackled with widespread powerful interior-point methods. Most importantly, the D-stability conditions can be combined with other LMI conditions arising in robust stability analysis.     

Characterization and compensation of discrete-time non-minimum-phase systems for precision tracking control

The objective of this study is to characterize the unstable zeros of discrete-time non-minimum-phase systems and to compensate the gain and phase errors induced by unstable zeros for tracking control. These two objectives are attempted via examining the frequency response of unstable zeros. The gain error and phase shift induced by unstable zeros are first presented. It is well known that they are undesirable and cannot be cancelled directly via the method of pole-zero cancellation. Through investigating the characteristics of unstable zeros, a systematic approach is then given to compensate the gain and phase errors induced by unstable zeros. Using this approach, both a simple tracking control method and a precision tracking control method are proposed. The design procedure is presented and the design formulae are given. Simulation results are presented to illustrate the effectiveness of the proposed methods.     

Application of facial reduction to H ∞ state feedback control problem

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H _∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H _∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.     

Bounded tracking for non-minimum phase nonlinear systems with fast zero dynamics

In this paper, we derive tracking control laws for non-minimum phase nonlinear systems with both fast and slow, possibly unstable, zero dynamics. The fast zero dynamics arise from a perturbation of a nominal system. These fast zeros can be problematic in that they may be in the right half plane and may cause large magnitude tracking control inputs. In this paper, we combine the ideas from some recent work of Hunt, Meyer and Su with that of Devasia, Chen and Paden on an asymptotic tracking procedure for non-minimum phase nonlinear systems. We give (somewhat subtle) conditions under which the tracking control input is bounded as the magnitude of the perturbation of the nominal system becomes zero. Explicit bounds on the control inputs are calculated for both SISO and MIMO systems using some interesting non-standard singular perturbation techniques. The method is applied to a suite of examples, including the simplified planar dynamics of VTOL and CTOL aircraft.     

Quantitative design method for MIMO uncertain plants to achieve prescribed diagonal dominant closed-loop minimum-phase tolerances

A quantitative design method for multi-input multi-output linear time-invariant feedback systems for plants with large uncertainty has been presented by Horowitz ( 1982), and by Yaniv and Horowitz ( 1986). This design method is developed here to guarantee minimum-phase closed-loop diagonal elements for systems with basically non-interacting (Horowitz and Loecher 1981) off-diagonal closed-loop tolerances. The advantage of this design is that with minimum-phase transfer functions, a very important class of time-domain specifications can be translated to the frequency domain, as shown by Krishman and Cruickshanks ( 1977) and by Horowitz ( 1976). The attractive properties of this design method are: (a) the problem is reduced to a successive single-loop design with no interaction between the loops, and no iterations are necessary; (b) the technique can be applied to all n × n plants P with P^?1 having no poles in the right-half plane, and satisfying some conditions described in § 5; (c) the procedure is interactive with n steps for an n × n MIMO plant, and in each step, one of the elements of the diagonal feedback compensation and one row of the prefilter matrix are designed.     

Periodic Compensation of Continuous-Time Plants

This note presents a periodic compensator which achieves robust stability for single-input-single-output (SISO), linear time invariant (LTI) plants having both right-half plane (RHP) poles and zeros, a job LTI controllers fail to do. In addition, for strictly proper plants this controller achieves model matching ensuring at the same time that the periodic oscillations present in the plant output are insignificant in magnitude. The design steps are straightforward and linear algebraic in nature