Digital iterative learning control of linear multivariable plants

It is shown that digital iterative learning controllers can be designed for linear multivariable plants using only the step-response matrices of such plants. This demonstration is effected by proving a fundamental theorem which establishes precise sufficient conditions under which iterative learning control is achieved by such digital controllers. These general results are illustrated by the presentation of numerical results for the digital iterative learning control of a third-order linear multivariable plant with two inputs and two outputs.     

Learning rates in the digital iterative learning control of linear multivariable plants

The learning rates achievable in the digital iterative learning control of linear multi-variable plants are investigated. It is shown that the irregularity and stability characteristics of the plants under control impose severe constraints on the achievable learning rates. These results are not only significant in their own right but also strongly motivate the introduction of compensators to increase the learning rates achievable in irregular plants. These general results are illustrated by the presentation of numerical results for the iterative learning digital control of an uncompensated fourth-order completely irregular plant with two inputs and two outputs     

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.     

Internal-model-based design of repetitive and iterative learning controllers for linear multivariable systems

Repetitive and iterative learning control are two modern control strategies for tracking systems in which the signals are periodic in nature. This paper discusses repetitive and iterative learning control from an internal model principle point of view. This allows the formulation of existence conditions for multivariable implementations of repetitive and learning control. It is shown that repetitive control can be realized by an implementation of a robust servomechanism controller that uses the appropriate internal model for periodic distrubances. The design of such controllers is discussed. Next it is shown that iterative learning control can be implemented in the format of a disturbance observer/compensator. It is shown that the resulting control structure is dual to the repetitive controller, and that both constitute an implementation of the internal model principle. Consequently, the analysis and design of repetitive and iterative learning control can be generalized to the powerful analysis and design procedure of the internal model framework, allowing to trade-off the convergence speed for periodic-disturbance cancellation versus other control objectives, such as stochastic disturbance suppression.     

Adaptive and non-adaptive 'pole-placement' control of multivariable linear time-varying plants

In this paper, the adaptive and non-adaptive 'pole-placement' control problems are addressed for a class of indexinvariant multivariable linear time-varying plants. In the case where the plant parameters are completely known, it is shown that a 'pole-placement'-like control algorithm can be designed by solving a time-varying Diophantine equation. Furthermore, the tracking performance of such a controller can be improved by incorporating the internal model principle into the design. In the case where the plant parameters are only partially known, a gradient-based adaptive law with projection and normalization is employed to estimate the plant parameters on-line. An adaptive controller is then designed, based on these parameter estimates, and the stability properties of the adaptive closed-loop plant are established. The design and realization of both the adaptive and non-adaptive control laws is illustrated by means of a simple example.     

Monotonically convergent iterative learning control for linear discrete-time systems

In iterative learning control schemes for linear discrete time systems, conditions to guarantee the monotonic convergence of the tracking error norms are derived. By using the Markov parameters, it is shown in the time-domain that there exists a non-increasing function such that when the properly chosen constant learning gain is multiplied by this function, the convergence of the tracking error norms is monotonic, without resort to high-gain feedback.     

An inverse-model approach to multivariable norm optimal iterative learning control with auxiliary optimisation

Motivated by the commonly encountered problem in which tracking is only required at selected intermediate points within the time interval, a general optimisation-based iterative learning control (ILC) algorithm is derived that ensures convergence of tracking errors to zero whilst simultaneously minimising a specified quadratic objective function of the input signals and chosen auxiliary (state) variables. In practice, the proposed solutions enable a repeated tracking task to be accurately completed whilst simultaneously reducing undesirable effects such as payload spillage, vibration tendencies and actuator wear. The theory is developed using the well-known norm optimal ILC (NOILC) framework, using general linear, functional operators between real Hilbert spaces. Solutions are derived using feedforward action, convergence is proved and robustness bounds are presented using both norm bounds and positivity conditions. Algorithms are specified for both continuous and discrete-time state-space representations, with the latter including application to multi-rate sampled systems. Experimental results using a robotic manipulator confirm the practical utility of the algorithms and the closeness with which observed results match theoretical predictions.     

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.     

Genetic algorithms in norm-optimal linear and non-linear iterative learning control

In this paper it is analysed whether or not it is possible to apply the norm-optimal iterative learning control algorithm to non-linear plant models. As a new theoretical result it is shown that if the non-linear plant meets a certain technical invertibility condition, the sequence of tracking errors generated by the norm-optimal algorithm will converge geometrically to zero. However, due to the non-linear nature of the plant, it is typically impossible to calculate analytically the sequence of input functions produced by the norm-optimal algorithm. Therefore it is proposed that genetic algorithms can be used as a computational tool to calculate the sequence of norm-optimal inputs. The proposed approach benefits from the design of a low-pass FIR filter. This filter successfully removes unwanted high frequency components of the input signal, which are generated by the genetic algorithm method due to the random nature of the genetic algorithm search. Simulations are used to illustrate the performance of this new approach, and they demonstrate good results in terms of convergence speed and tracking of the reference signal regardless of the nature of the plant.     

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     

Experimental evaluation of iterative learning control algorithms for non-minimum phase plants

The purpose of this paper is two-fold, firstly it describes the development and modelling of an experimental test facility as a platform on which to assess the performance of Iterative Learning Control (ILC) schemes. This facility includes a non-minimum phase component. Secondly, P-Type, D-Type and phase-lead types of the algorithm have been implemented on the test-bed, results are presented for each method and their performance is compared. Although all the ILC strategies tested experience eventual divergence when applied to a non-minimum phase system, it is found that there is an optimum phase-lead ILC design that maximizes convergence and minimizes error. A general method of arriving at this phase-lead from knowledge of the plant model is described. A variety of filters have been applied and assessed in order to improve the overall performance of the algorithm.     

Decentralized control of linear multivariable systems

This paper studies the effect of decentralized feedback on the closed-loop properties of jointly controllable, jointly observablek-channel linear systems. Channel interactions within such systems are described by means of suitably defined directed graphs. The concept of a complete system is introduced. Complete systems prove to be precisely those systems which can be made both controllable and observable through a single channel by applying nondynamic decentralized feedback to all channels. Explicit conditions are derived for determining when the closed-loop spectrum of ak-channel linear system can be freely assigned or stabilized with decentralized control.     

Synthesis of limit-tracking systems incorporating linear multivariable plants

The steady states of tracking systems incorporating self-selecting controllers and linear multivariable plants exhibit limit tracking when none of the plant outputs exceeds its corresponding set point (i.e. its limit value) and the number of pairs of equal plant outputs and set points is not less than the rank of the plant steady-state transfer-function matrix. The condition for the feasibility of limit tracking is given. An approach to the synthesis of limit-tracking systems is proposed that utilizes an order-reduction technique to reduce the number of separate set-point tracking controllers from _pC_m to p ? m + 1 in the case of m-input/p-output plants while guaranteeing the existence of the steady states of such tracking systems.     

Robust iterative learning control for linear systems with multiple time-invariant parametric uncertainties

This article presents a novel robust iterative learning control algorithm (ILC) for linear systems in the presence of multiple time-invariant parametric uncertainties.The robust design problem is formulated as a min–max problem with a quadratic performance criterion subject to constraints of the iterative control input update. Then, we propose a new methodology to find a sub-optimal solution of the min–max problem. By finding an upper bound of the worst-case performance, the min–max problem is relaxed to be a minimisation problem. Applying Lagrangian duality to this minimisation problem leads to a dual problem which can be reformulated as a convex optimisation problem over linear matrix inequalities (LMIs). An LMI-based ILC algorithm is given afterward and the convergence of the control input as well as the system error are proved. Finally, we apply the proposed ILC to a generic example and a distillation column. The numerical results reveal the effectiveness of the LMI-based algorithm.     

Deadbeat control of linear multivariable generalized state-spacesystems

The classic idea of deadbeat control is extended to linear multivariable discrete-time generalized state-space systems using algebraic methods. The asymptotic properties of the linear quadratic regulator theory are used to obtain the classes of deadbeat controllers using stabilizing full semistate feedback. The solution is constructed from a `cheap control' problem. Both semistate and output deadbeat control laws are considered. The main design criteria are to drive the semistate and/or outputs of the system to zero in minimum time and that the closed-loop system be internally stable. Unique properties of these types of control laws are discussed. For semistate deadbeat control, all the (dynamic) poles including the ones at infinity are moved to the origin, whereas for output deadbeat, some of the finite transmission zeros are canceled. Numerically reliable algorithms are developed to solve both problems     

Generalized characterization of tracking systems and linear multivariable plants

In order to design tracking systems incorporating linear multivariable plants with more controlled outputs than manipulated inputs, it is shown that a more'general tracking concept than set-point tracking is necessary. The inclusion of inequalities in tracking conditions facilitates the characterization of tracking systems and linear multivariable plants. It is shown that the possibility of undertracking (i.e. tracking with non-negative errors) is characterized by the separation theorem of convex analysis, that linear multivariable plants can be classified into Class I and Class II plants based upon their steady-state transfer-function matrices, and that, in the case of Class I plants, undertracking is possible for any set-point commands. Furthermore, the necessary and/or sufficient conditions for Class I plants are given. Finally, it is shown that, in the case of Class 1 plants, undertracking is also possible for any set-point commands and any constant disturbances.     

A pseudoinverse-based iterative learning control

Learning control is a very effective approach for tracking control in processes occurring repetitively over a fixed interval of time. In this paper, an iterative learning control (ILC) algorithm is proposed to accommodate a general class of nonlinear, nonminimum-phase plants with disturbances and initialization errors. The algorithm requires the computation of an approximate inverse of the linearized plant rather than the exact inverse. An advantage of this approach is that the output of the plant need not be differentiated. A bound on the asymptotic trajectory error is exhibited via a concise proof and is shown to grow continuously with a bound on the disturbances. The structure of the controller is such that the low frequency components of the trajectory converge faster than the high frequency components     

Time-domain identification of transmission zero locations of linear multivariable plants

In the case of asymptotically stable multivariable plants, the location in the complex plane of the transmission zeros is identified on the basis of the initial and ultimate time-domain behavior of the step-response matrix.     

Decentralized iterative learning control for large-scale interconnected linear systems with fixed initial shifts

This paper deals with the problem of iterative learning control for large-scale interconnected linear systems in the presence of fixed initial shifts. According to the characteristics of the systems, iterative learning control laws are proposed for such large-scale interconnected linear systems based on the PD-type learning schemes. The proposed controller of each subsystem only relies on local output variables without any information exchanges with other subsystems. Using the contraction mapping method, we show that the schemes can guarantee the output of the system converges uniformly to the corresponding output limiting trajectory over the whole time interval along the iteration axis. Simulation examples illustrate the effectiveness of the proposed method.