Scaling and redundancy for ill-conditioned two input, two output plants

It is well known in the multivariable control literature that ill-conditioned plants may pose significant difficulties in control design. In this paper, we explore two causes of ill-conditioning. The first arises when one input has much smaller gain than does the other. The second occurs when the inputs are almost redundant, in the sense that they affect the plant outputs in approximately the same manner. To quantify these two sources of ill-conditioning, we define input and output redundancy angles, and relate these angles to the size of the plant inverse and condition number. We also investigate the dependence of the redundancy angles upon scaling, and show that certain plants have nearly redundant inputs regardless of the choice of units.     

Iterative methods for ill-conditioned Toeplitz matrices

In this paper we study the use of the Sine Transform for preconditioning linear Toeplitz systems. We consider Toeplitz matrices with a real generating function that is nonnegative with only a small number of zeros. Then we can define a preconditioner of the formS _n ΛS _n whereS _n is the matrix describing the discrete Sine transform and Λ is a diagonal matrix. If we have full knowledge aboutf then we can show that the preconditioned system is of bounded condition number independly ofn. We can obtain the same result for the case that we know only the position and order of the zeros off. If we only know the matrix and its coefficientst _j , we present Sine transform preconditioners that show in many examples the same numerical behaviour.     

A Riccati equation for block-diagonalization of ill-conditioned systems

A simple transformation, originally introduced for singularly perturbed systems, is now applicable to a larger class of time-invariant systems.     

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     

Multimodel identification for control of an ill-conditioned distillation column

Identification for control of an ill-conditioned system requires special techniques. The directionality of such a system should be taken into account in the design of identification experiments. In distillation, information about the directionality properties can be obtained from certain flow gains, which are easy to determine in practice. Based on such information, the high- and low-gain directions of the plant can be explicitly excited. In this paper, a pilot-scale distillation column is identified by this approach at two different operating points. At each operating point, a nominal second-order plus time-delay model with logarithmic outputs is determined. This model structure makes it possible to capture the dynamic directionality of the plant. In addition, models describing variations and uncertainties in the high- and low-gain directions are determined by a special technique. The models obtained are superior to models determined via traditional step tests. The former satisfy integral controllability requirements, while the latter tend to violate them.     

An improved Lanczos algorithm for solving ill-conditioned linear equations

In the existing Lanczos algorithms for solving systems of linear equations,the estimate for the residual is effective for well-conditioned systems. However, in actual implementation on a computer we find that the estimate is no longer reliable for ill-conditioned cases. We first analyze in theory this observation, then develop an improved Lanczos algorithm. Numerical examples are also given to demonstrate the effectiveness of the present algorithm.     

The design of servomechanism controllers for uncertain plants

The design of multivariable servomechanism controllers is considered here for plants whose parameters are uncertain or perturbed within a cone-bounding criterion. The solution comes from a controller design problem as an LQ optimal control problem in which the gain matrices result in a quadratic performance upper bound (similar to that described by Gilman and Rhodes (1976() which depends on the uncertainty. Under conditions on the nominal plant and on the uncertainty, a resulting linear time-invariant controller is guaranteed to achieve the objectives of robust tracking and regulation with closed-loop stability.     

Adaptive backstepping control design for linear multivariable plants

A nonlinear backstepping scheme is developed for adaptive control of linear plants with multiple inputs and multiple outputs. Solutions to plant parametrization, state observer, and adaptive control law for the multivariable backstepping design are proposed. The developed adaptive controller has the desired properties for ensuring closed-loop signal boundedness and asymptotical tracking.     

QFT design for uncertain non-minimum phase and unstable plants revisited

Design method for uncertain non-minimum phase and unstable plants in the quantitative feedback theory (QFT) developed by Horowitz and Sidi is revisited in this paper. It is illustrated that the existing method may not work since some design rules have not been clearly specified by several examples including non-minimum phase plants and unstable plants. Then stability of a new nominal plant is carefully examined and analysed, and an improved design method is presented. The result in this paper provides mathematical justification of the QFT design procedure for nonminimum phase and unstable plants in Horowitz and Sidi (1978) and Horowitz (1992).     

Improved solutions for ill-conditioned problems involved in set-membership estimation for fault detection and isolation

Set-membership (SM) estimation implies that the computed solution sets are guaranteed to contain all the feasible estimates consistent with the bounds specified in the model. Two issues often involved in the solution of SM estimation problems and their application to engineering case studies are considered in this paper. The first one is the estimation of derivatives from noisy signals, which in a bounded uncertainty framework means obtaining an enclosure by lower and upper bounds. In this paper, we improve existing methods for enclosing derivatives using Higher-Order Sliding Modes (HOSM) differentiators combining filtering. Our approach turns the use of high order derivatives more efficiently especially when the signal to differentiate has slow dynamics. The second issue of interest is solving linear interval equation systems, which is often an ill-conditioned problem. This problem is reformulated as a Constraint Satisfaction Problem and solved by the combination of the constraint propagation Forward Backward algorithm and the SIVIA algorithm. The two proposed methods are tested on illustrative examples. The two methods are then used in a fault detection and isolation algorithm based on SM parameter estimation that is applied to detect abnormal parameter values in a biological case study.     

Quantitative non-linear compensation design for saturating unstable uncertain plants

Consider an unstable uncertain plant controlled by an element Np, which saturates when its input |x| ≥ M. The system can be stabilized by means of feedback, which, however, is absent during Np saturation. If the saturation interval is long enough, it is impossible to recover system stability via Np. This paper presents a synthesis technique for ensuring that Np does not saturate despite very large command inputs. The basic idea is to prevent |x|>M, via an added saturating element N with saturation level m, which in turn is determined by |x|. A systematic, quantitative design technique is presented for unstable plants with large uncertainty, to achieve (a) desired performance tolerances over the linear range (small command inputs) and (b)acceptable but unavoidably slower response for large command inputs. Both (a) and (b) are achieved over the specified extent of plant uncertainty. The design technique makes use of several previously developed quantitative synthesis theories for minimum- and non-minimum-phase uncertain plants in linear operation, and for uncertain minimum-phase stable plants subject to saturation.     

Constructive nonlinear anti-windup design for exponentially unstable linear plants

In this paper we give a constructive method for anti-windup design for general linear saturated plants with exponentially unstable modes. The constructive solution is independent of the controller dynamics so that the size of the (necessarily bounded) operating region in the exponentially unstable directions of the plant state space is large. Desirable properties of the closed-loop are formally proved and shown to induce a very desirable behavior on a MIMO example with two exponentially unstable modes.     

Decomposed control design for non-stationary plants subject to disturbances

This paper combines the block (Luk'yanov and Utkin 1982, 1998), sliding mode and high gain control techniques (Utkin 1992) to form a new decomposed control law suitable for general multivariable non-stationary plants. Different versions of the control law are developed with various dependencies on plant parameters and state variables. The new control law is demonstrated by simulation of its application to an electrical drive employing a dc motor with controlled flux.     

The design of servomechanism controllers for uncertain plants and input signals

This paper summarizes a methodology for the design of servomechanism controllers for multivariate plants and reference/disturbance signals which may be uncertain or weakly non-linear. This framework represents a departure from the idealized tracking/disturbance rejection problem in the direction of realistic design considerations. The objective becomes ‘good’, though not exact, error regulation for applied signals which may be unknown or random in nature.     

Flight controller design for uncertain and nonlinear plants with pilot compensation

Nonlinear quantitative feedback theory (QFT) is used to design a flight control system for the nonlinear model of the YF-16 aircraft (A/C) with C* as the controlled output. The resulting closed loop stability augmentation system (SAS), P_e(S), becomes part of the outer loop containing the pilot. The Neal-Smith pilot model for a compensatory tracking task is used to develop a technique which allows the designer to synthesize compensation in the outer loop, which includes a free compensator F_p(S). The latter is chosen to minimize pilot workload, increase system bandwidth, and improve handling qualities ratings as per the Neal–Smith criteria, for the tracking task. The available pilot compensation abilities are then available for further increasing of system bandwidth to improve overall capabilities. This approach can be used at the early stages of flight control design, thus saving time and money over the current practice. Simulations in the time and frequency domains demonstrate that the desired performance is attained.     

Simple control-relevant identification test methods for a class of ill-conditioned processes

The identification of a class of ill-conditioned processes is considered. The goal of identification is model predictive control (MPC). For this class of processes, it is essential to have good estimation of the very small difference between certain transfer functions, or, low gain direction. Two simple and practical test methods will be proposed that can enhance the model quality of low gain direction. The main idea is to use the high amplitude and high correlation test signals that excite the low gain direction more and that will not disturb process operation. The test signals can be used in both open-loop and closed-loop tests. A high purity distillation column model is used to show the effectiveness of the methods.     

混合矩阵为病态情况下的盲信号提取快速算法

在盲信号提取技术中,当混合矩阵是病态情况（混合矩阵奇异或欠定）时,根据混合矩阵秩的性质,提出了在该情况下的盲信号可提取性判据,并给出了一种可提取信号的快速提取算法。该算法中只使用了待提取信号与其他观测量之间的四阶累积量,而不需要计算各个观测量之间的四阶累积量,因此大大减少了算法的复杂度。先提取信号的误差没有对后提取的信号产生累积,因此提高了提取信号的精确度。该算法简单。仿真结果表明该算法有效,并且有很好的性能。