The steady-state invertibility and feedforward control of linear time-invariant systems

The notion of steady-state invertibility of a system is introduced, which is concerned about the problem of being able to find an input so that the output of a stable system is asymptotically equal to a specified output of a certain class of functions. Necessary and sufficient conditions are found for a linear time-invariant system to be steady-state invertible. Application of these results is then made to find necessary and sufficient conditions for a feedforward controller to exist for a general linear time-invariant system, so that asymptotic tracking, in the presence of a general class of measurable disturbances occurs. Explicit feedforward controllers which will accomplish this are obtained. Properties of the steady-state invertibility condition are then obtained; in particular, it is shown that a system is "almost always" steady-state invertible if the number of plant inputs is not less than the number of outputs; if the number of plant inputs is less than the number of outputs, then a system is "almost never" steady-state invertible. It is then shown that a system which is minimum phase and which has at least the same number of inputs as outputs is always steady-state invertible.     

Casual systems and feedforward loops

It is shown that a causal linear system with an equal number of inputs and outputs can be made to be simultaneously invertible and output reproducible by employing a feedforward loop. This feed-forward loop does not affect the stability, of the system one way or the other.     

The robust control of a servomechanism problem for linear time-invariant multivariable systems

Necessary and sufficient conditions are found for there to exist a robust controller for a linear, time-invariant, multivariable system (plant) so that asymptotic tracking/regulation occurs independent of input disturbances and arbitrary perturbations in the plant parameters of the system. In this problem, the class of plant parameter perturbations allowed is quite large; in particular, any perturbations in the plant data are allowed as long as the resultant closed-loop system remains stable. A complete characterization of all such robust controllers is made. It is shown that any robust controller must consist of two devices 1) a servocompensator and 2) a stabilizing compensator. The servocompensator is a feedback compensator with error input consisting of a number of unstable subsystems (equal to the number of outputs to be regulated) with identical dynamics which depend on the disturbances and reference inputs to the system. The sorvocompensator is a compensator in its own right, quite distinct from an observer and corresponds to a generalization of the integral controller of classical control theory. The sole purpose of the stabilizing compensator is to stabilize the resultant system obtained by applying the servocompensator to the plant. It is shown that there exists a robust controller for "almost all" systems provided that the number of independent plant inputs is not less than the number of independent plant outputs to be regulated, and that the outputs to be regulated are contained in the measurable outputs of the system; if either of these two conditions is not satisfied, there exists no robust controller for the system.     

Properties and calculation of transmission zeros of linear multivariable systems

A new definition of transmission zeros for a linear, multivariable, time-invariant system is made which is shown to be equivalent to previous definitions. Based on this new definition of transmission zeros, new properties of transmission zeros of a system are then obtained; in particular, it is shown that a system with an unequal number of inputs and outputs almost always has no transmission zeros and that a system with an equal number of inputs and outputs almost always has either n−1 or n transmission zeros, where n is the order of the system; transmission zeros of cascade systems are then studied, and it is shown how the transmission zeros of a system relate to the poles of a closed loop system subject to high gain output feedback. An application of transmission zeros to the servomechanism problem is also included. A fast, efficient, numerically stable algorithm is then obtained which enables the transmission zeros of high order multivariable systems to be readily obtained. Some numerical examples for a 9th order system are given to illustrate the algorithm.     

Unified adaptive control of non-minimum-phase systems Part 1. Weighted control effort and pole placement for stochastic systems

Global convergence of a class of unified adaptive control algorithms for discrete time non-minimum-phase stochastic linear systems is established. It is shown that the algorithms ensure that the system inputs and outputs are sample mean bounded, the minimum variance cost function with weighted control effort achieves its minimum possible values, the weighted factors adjusted on-line eventually converge to expected values, and the closed-loop poles arise at prespecified locations.     

Linear estimator stabilization of nonlinear sampled-data systems

It has been shown that the state of a linear system can be constructed from observations of its inputs and outputs. The observer which performs the construction is itself a linear system with time constants which may be chosen by the designer. This linear asymptotic estimator has been used previously to stabilize certain types of contimuous nonlinear systems. In this paper, a linear sampled-data estimator is developed. This estimator is used first to stabilize a linear sampled-data system and then it is used to stabilize a class of nonlinear sampled-data systems by the choice of the estimator's time constants and the feedback gain. Typical example applications are analyzed to illustrate the theoretical investigations.     

Robust controllability and observability degrees of polynomially uncertain systems

This paper deals with the class of polynomially uncertain continuous-time linear time-invariant (LTI) systems whose uncertainties belong to a semi-algebraic set. The objective is to determine the minimum of the smallest singular value of the controllability or observability Gramian over the uncertainty region. This provides a quantitative measure for the robust controllability or observability degree of the system. To this end, it is shown that the problem can be recast as a sum-of-squares (SOS) problem. In the special case when the uncertainty region is polytopic, the corresponding SOS formulation can be simplified significantly. One can apply the proposed method to any large-scale interconnected system in order to identify those inputs and outputs that are more effective in controlling the system, in a robust manner. This enables the designer to simplify the control structure by ignoring those inputs and outputs whose contribution to the overall control operation is relatively weak. A numerical example is presented to demonstrate the efficacy of the results.     

A constrained minimum variance input-output estimator for linear dynamic systems

A stationary estimator is presented that provides estimates for both the outputs and the inputs of linear time-invariant systems. The estimates satisfy the input-output equations and are optimal in a weighted minimum variance sense. It is shown that the true optimum problem is too complicated to allow a general closed-form solution and a sub-optimal approach is suggested that leads to a simple computational algorithm.     

Closed-loop stable model predictive control of integrating systems with dead time

Model predictive control (MPC) applications in the process industry usually deal with process systems that show time delays (dead times) between the system inputs and outputs. Also, in many industrial applications of MPC, integrating outputs resulting from liquid level control or recycle streams need to be considered as controlled outputs. Conventional MPC packages can be applied to time-delay systems but stability of the closed loop system will depend on the tuning parameters of the controller and cannot be guaranteed even in the nominal case. In this work, a state space model based on the analytical step response model is extended to the case of integrating time systems with time delays. This model is applied to the development of two versions of a nominally stable MPC, which is designed to the practical scenario in which one has targets for some of the inputs and/or outputs that may be unreachable and zone control (or interval tracking) for the remaining outputs. The controller is tested through simulation of a multivariable industrial reactor system.     

Dynamic decoupling for right-invertible nonlinear systems

It is shown that every system with more inputs than outputs that is right invertible, in the sense of [5], can be decoupled via a dynamic state feedback that can be computed using a given algorithm.This result generalizes the one of [2] and completes the results of [3].     

Periodic tracking adaptive control for multivariable systems having more outputs than inputs

This note presents an adaptive control algorithm for multivariable systems in which the number of outputs is greater than the number of inputs. The algorithm can force the outputs to track arbitrary given reference signals periodically. This is the best tracking performance for systems lacking output function controllability. It has been shown that the tracking period is the upper bound on the controllability index of the controlled system. The proposed algorithm is applicable to multivariable systems with arbitrary interactor matrix but no knowledge of the interactor matrix is required.     

Observer-based feedback stabilization of linear systems with event-triggered sampling and dynamic quantization

We consider the problem of output feedback stabilization in linear systems when the measured outputs and control inputs are subject to event-triggered sampling and dynamic quantization. A new sampling algorithm is proposed for outputs which does not lead to accumulation of sampling times and results in asymptotic stabilization of the system. The approach for output sampling is based on defining an event function that compares the difference between the current output and the most recently transmitted output sample not only with the current value of the output, but also takes into account a certain number of previously transmitted output samples. This allows us to reconstruct the state using an observer with sample-and-hold measurements. The estimated states are used to generate a control input, which is subjected to a different event-triggered sampling routine; hence the sampling times of inputs and outputs are asynchronous. Using Lyapunov-based approach, we prove the asymptotic stabilization of the closed-loop system and show that there exists a minimum inter-sampling time for control inputs and for outputs. To show that these sampling routines are robust with respect to transmission errors, only the quantized (in space) values of outputs and inputs are transmitted to the controller and the plant, respectively. A dynamic quantizer is adopted for this purpose, and an algorithm is proposed to update the range and the centre of the quantizer that results in an asymptotically stable closed-loop system.     

Finite-dimensional adaptive control for infinite-dimensional systems

This paper deals with the design of a finite-dimensional model reference adaptive controller for a distributed parameter system. The space of inputs and the space of outputs are both one dimensional. First an input—output representation is introduced, using filtered values generated from the inputs and outputs. Next it is shown that we can design a finite-dimensional adaptive control system by a new adaptive law. The new adaptive law guarantees the existence of a large region of attraction from which all signals are bounded and the tracking error converges to a small residual set.     

Discrete reachability of hybrid systems

This paper concerns hybrid systems subject to discrete-event supervisory control. It investigates the discrete reachability, where the hybrid system should be moved from a discrete initial state z init into a given discrete goal state z goal by a sequence of discrete inputs. The reachability analysis is carried out in three steps. First, a discrete-event model of the hybrid system is set up. Stochastic automata are used to describe all state sequences which may be generated by the hybrid system. Such a model is called complete. Second, methods for the reachability analysis of stochastic automata are elaborated which concern a weak and a strong condition for reachability. Third, these methods are applied to the hybrid system. It is shown that the reachability of the automaton implies the discrete reachability of the hybrid system, because the model is complete. Therefore, the weak and the strong condition for reachability of the stochastic automaton yield a necessary and a sufficient condition for discrete reachability of the hybrid system.     

On the optimality of two‐stage Kalman filtering for systems with unknown inputs

This paper is concerned with the optimal solution of two‐stage Kalman filtering for linear discrete‐time stochastic time‐varying systems with unknown inputs affecting both the system state and the outputs. By means of a newly‐presented modified unbiased minimum‐variance filter (MUMVF), which appears to be the optimal solution to the addressed problem, the optimality of two‐stage Kalman filtering for systems with unknown inputs is defined and explored. Two extended versions of the previously proposed robust two‐stage Kalman filter (RTSKF), augmented‐unknown‐input RTSKF (ARTSKF) and decoupled‐unknown‐input RTSKF (DRTSKF), are presented to solve the general unknown input filtering problem. It is shown that under less restricted conditions, the proposed ARTSKF and DRTSKF are equivalent to the corresponding MUMVFs. An example is given to illustrate the usefulness of the proposed results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Deadbeat output regulators for decouplable recursive non-linear systems

The deadbeat output regulator problem is formulated and solved in a straight-forward way for a class of decouplable multivarialbe recursive non-linear systems. It is shown that the decoupling control sequences drive each regulated output to zero and remain at zero thereafter in a minimum number of control iterations; this number is equal to the relative degree of the corresponding output. In addition, we show that the internal stability of a deadbeat closed-loop system is guaranteed only if the system is minimum phase.     

Global convergence for adaptive one-step-ahead optimal controllers based on input matching

This paper establishes global convergence for adaptive one-step-ahead optimal controllers applied to a class of linear discrete time single-input single-output systems. The class of systems includes all stable systems whether they are minimum phase or not, all minimum phase systems whether they are stable or not, and some unstable nonminimum phase systems. The key substantive assumption is that the one-step-ahead optimal controller designed using the true system parameters leads to a stable closed-loop system. Subject to this natural restriction, it is shown that a simple adaptive control algorithm based on input matching is globally convergent in the sense that the system inputs and outputs remain bounded for all time and the input converges to the one-step-ahead optimal input. Both deterministic and stochastic cases are treated.     

Output feedback adaptive stabilization and command following for minimum phase dynamical systems with unmatched uncertainties and disturbances

In this article, we develop an output feedback adaptive control framework for continuous-time minimum phase multivariable dynamical systems for output stabilisation and command following. The approach is based on a nonminimal state-space realisation that generates an expanded set of states using the filtered inputs and filtered outputs and their derivatives of the original system. Specifically, a direct adaptive controller for the nonminimal state-space model is constructed using the expanded states of the nonminimal realisation and is shown to be effective for multi-input, multi-output linear dynamical systems with unmatched disturbances, unmatched uncertainties and unstable dynamics. The proposed adaptive control architecture requires only knowledge of the open-loop system's relative degree as well as a bound on the system's order. Several illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach.     

Identification of linear, multivariable systems operating under linear feedback control

The possibility of estimating process parameters using input-output data collected when the system operates in closed loop is discussed in this paper. Concepts that are useful for a systematic treatment of the problem are introduced. The results refer to the case where the regulator is a linear feedback law or alternates between several such laws. It is shown that a straightforwardly applied identification scheme has the same identifiability properties as the more complex method in which the parameters of the closed-loop system are estimated first. It is also shown that it is always possible to achieve the same identifiability properties as for open-loop systems by shifting between different linear regulators. The required number of regulators depends only on the number of inputs and outputs. The results obtained are illustrated by a numerical example.     

一类非线性MIMO系统的直接自适应模糊鲁棒控制

针对一类未知的非线性MIMO系统, 本文提出了一种直接自适应模糊鲁棒控制设计方法. 理论分析和仿真实验都已证明, 该方法确保闭环系统全局稳定, 获得H_∞跟踪性能指标, 外部干扰、模糊逻辑逼近误差和输入对输出的交叉耦合可衰减到给定的水平, 系统鲁棒性好.