Stable Model Reference Adaptive Control Research Articles

Simple adaptive control – a stable direct model reference adaptive control methodology – brief survey

SUMMARYIn spite of successful proofs of stability and even successful demonstrations of performance, the eventual use of model reference adaptive control methodologies in practical real‐world systems has met a rather strong resistance from practitioners and has remained very limited. Apparently, the practitioners have a hard time understanding the conditions that can guarantee stable operations of adaptive control systems under realistic operational environments. Besides, it is difficult to measure the robustness of adaptive control system stability and allow it to be compared with the common and widely used measure of phase margin and gain margin that is utilized by present, mainly LTI, controllers. Furthermore, recent counterexamples seem to show that adaptive systems may diverge even when all required conditions are fulfilled. This paper attempts to revisit the fundamental qualities of the common direct model reference adaptive control methodology based on gradient and to show that some of its basic drawbacks have been addressed and eliminated within the so‐called simple adaptive control methodology. The sufficient conditions that guarantee stability are clearly stated and lead to similarly clear proofs of stability. The main claim of the paper is that if sufficient information exists for a robust classical design control, same information can be used to implement robust simple adaptive controllers. As many real world applications show, the added value of using add‐on adaptive control techniques, the use is pushing the desired performance beyond any previous limits. The paper also shows that the previous counterexamples to model reference adaptive control become just simple, successful, and stable applications of simple adaptive control. Copyright © 2013 John Wiley & Sons, Ltd.

PdE-based model reference adaptive control of uncertain heterogeneous multiagent networks

Robust globally stable model reference adaptive control (MRAC) laws recently derived for systems described by parabolic and hyperbolic partial differential equations (PDEs) with spatially-varying coefficients under distributed sensing and actuation are extended to heterogeneous multiagent networks characterized by parameter uncertainty. The extension is carried out using partial difference equations (PdEs) on graphs that preserve parabolic- and hyperbolic-like cumulative network behavior. Unlike in the PDE case, only boundary input is specified for the reference model. The algorithms proposed directly incorporate this boundary reference input into the reference PdE to generate the distributed admissible reference evolution profile followed by the agents. The agent evolution thus depends only on the interaction with the adjacent agents, making the system fully decentralized. Numerical examples are presented as well, including the case of the switched topology associated with a sudden loss of an agent. The resulting PdE MRAC laws inherit the robust linear structure of their PDE counterparts.

Constant specific growth rate in fed-batch cultivation of Bordetella pertussis using adaptive control

Monitoring and control of production processes for biopharmaceuticals have become standard requirements to support consistency and quality. In this paper, a constant specific growth rate in fed-batch cultivation of Bordetella pertussis is achieved by a newly designed specific growth rate controller. The performance of standard control methods is limited because of the time-varying characteristics due to the exponentially increasing biomass and volume. To cope with the changing dynamics, a stable model reference adaptive controller is designed which adapts the controller settings as volume and biomass increase. An important asset of the design is that dissolved oxygen is the only required online measurement. An original design without considering the dissolved oxygen dynamics resulted experimentally in oscillatory behaviour. Hence, in contrast to common believes, it is essential to include dissolved oxygen dynamics. The robustness of this novel design was tested in simulation. The validity of the design was confirmed by laboratory experiments for small-scale production of B. pertussis. The controller was able to regulate the specific growth rate at the desired set point, even during a long fed-batch cultivation time with exponentially increasing demands for substrates and oxygen.

Is normalization necessary for stable model reference adaptive control?

One of the longest standing open questions in adaptive control concerns the correctness of the stability claim of the un-normalized model reference scheme proposed by Monopoli in 1974. Although provably correct solutions to the problem now abound, in particular, it is well known that adding a normalization to Monopoli's original scheme ensures global convergence, it is interesting to know whether this technique-driven modification is really necessary or only required to complete the stability proof in the absence of more elaborate arguments. In this note, we construct a counterexample that provides a definite-unfortunately, negative-answer to the claim. Instrumental for the establishment of this result is a technical lemma that shows that, under some conditions on the regressor that may appear in Monopoli's scheme, the parameter error freezes as the adaptation gain goes to infinity. On the lighter side, we also prove that Monopoli's scheme is semiglobally stable, underscoring the relevance of this important contribution.

Multivariable Adaptive Control Using High-Frequency Gain Matrix Factorization

In this note, we extend the application of a less restrictive condition about the high-frequency gain matrix to design stable direct model reference adaptive control for a class of multivariable plants with relative degree greater than one. The new approach is based on a control parametrization derived from a factorization of the high-frequency gain matrix K/sub p/ in the form of a product of three matrices, one of them being diagonal. Three possible factorizations are presented. Only the signs of the diagonal factor or, equivalently, the signs of the leading principal minors of K/sub p/, are assumed known.

Design of a stable adaptive controller for driving aerobic fermentation processes near maximum oxygen transfer capacity

In many industrial fermentation processes oxygen availability is the main limiting factor for product production. Typically the dissolved oxygen (DO) concentration decreases continuously at the beginning of the batch until it reaches a critical level where the oxygen transfer rate is very close to the vessel's maximum transfer capacity. The process may be further driven close to this sensitive operating point with a controller that manipulates the carbon source feed rate. This operating strategy is linked with important productivity issues and is still frequently realised in open-loop at production scale. The main purpose of the present study is to derive an effective closed-loop control solution and to demonstrate its economical advantage in relation to the open-loop form of operation. A stable model reference adaptive controller (MRAC) was designed based on a phenomenological model of the process. The implementation requires two on-line measurements: the DO tension and oxygen transfer rate (OTR) between gas–liquid phases, which are nowadays standard and easily available in production facilities. The controller performance is accessed with a simulation case study. The main results show that the adaptive controller is precise, stable and robust to disturbances and to inaccuracies like variability in raw materials typical in fermentations run in complex media. The controller is simple, easy to implement, and could possibly improve productivity in processes for which oxygen transfer capacity is limiting growth and product production.

Globally stable direct fuzzy model reference adaptive control

In the paper a fuzzy adaptive control algorithm is presented. It belongs to the class of direct model reference adaptive techniques based on a fuzzy (Takagi–Sugeno) model of the plant. The plant to be controlled is assumed to be nonlinear and predominantly of the first order. Consequently, the resulting adaptive and control laws are very simple and thus interesting for use in practical applications. The system remains stable in the presence of unmodelled dynamics (disturbances, parasitic high-order dynamics and reconstruction errors are treated explicitly). The global stability of the overall system is proven in the paper, i.e. it is shown that all signals remain bounded while the tracking error and estimated parameters converge to some residual set that depends on the size of disturbance and high-order parasitic dynamics. The proposed algorithm is tested on a simulated three-tank system. Its performance is compared to the performance of a classical MRAC.

Globally stable model reference adaptive control based on fuzzy description of the plant

A novel fuzzy adaptive control algorithm is presented that belongs to direct model reference adaptive techniques based on a fuzzy (Takagi-Sugeno) model of the plant. The global stability of the overall system is proven, namely all the signals in the system remain bounded while the tracking error and estimated parameters converge to some residual set that depends on the size of disturbance and high-order parasitic dynamics. The hallmarks of the approach are its simplicity and transparency. The proposed algorithm is a straightforward extension of classical model reference adaptive control (MRAC) with a robust adaptive law to nonlinear systems described by fuzzy models. The performance of the approach was tested on a simulated plant and compared with the performance of a PI controller and a classical MRAC.

Stable model reference adaptive fuzzy control of a class of nonlinear systems

In this paper, we propose a new adaptive fuzzy control scheme called model reference adaptive fuzzy control (MRAFC). The MRAFC scheme employs a reference model to provide closed-loop performance feedback for generating or modifying a fuzzy controller's knowledge base. The MRAFC scheme grew from ideas in conventional model reference adaptive control (MRAC). The MRAFC scheme is developed to perform adaptive feedback linearization to a class of nonlinear systems. A class of fuzzy controllers, which can be expressed in an explicit form, is used as the primary controller. Based on Lyapunov's second method, we have developed MRAFC schemes and derived fuzzy rule adaptive laws. Hence, not only the stability of the system can be assured but also the performance, such as the issues of robustness and parameter convergence, of the MRAFC system can be analyzed explicitly. We showed that in the case of no modeling error, the state error converges to zero asymptotically. In the case that persistent excitation is satisfied, we showed that the MRAFC system is asymptotically stable. By considering the periodic signal as reference input signal, we showed that the square wave can make the MRAFC system be persistently excited. The feasibility of applying these techniques has been demonstrated by considering the control of an inverted pendulum in following a reference model response.

Asymptotically stable multiple-input multiple-output direct model reference adaptive controller for processes not necessarily satisfying a positive real constraint

This paper presents an asymptotic output tracking stability proof for a new modification to a direct model reference adaptive control procedure. This modification, which alleviates a very restrictive positive real constraint, greatly expands the class of processes that can now be controlled with zero output error. This paper presents illustrative examples demonstrating the utility of the algorithm.

Stable model reference adaptive control for periodically LTV systems based on the maximal error evaluation

Abstract This paper presents a model reference adaptive control system (MRACS) which does not depend on the degree of the plant when the plant is the continuous-time periodically linear time varying (LTV) system. A priori information is only the relative degree and the time-varying period. The controller has adjustable scalar functions. The optimal functions of these are estimated directly, synchronizing with the time-varying period of the plant by adaptive laws. The main contribution of this paper is that the computational burden of the adaptive law for the direct function estimation is reduced. It is accomplished with the maximum normalized error evaluation typed adaptive law. The design method and the global stability analysis of the proposed system are shown and a simple numerical simulation result is indicated to verify the viability of the proposed system.

Model Reference Adaptive Control for Non-Minimum Phase System by 2-Delay Feedback

In most of the studies of model reference adaptive control (MRAC), the controlled systems are confined to minimum phase systems, since the MRAC techniques utilize the control laws involving cancellations of zeros of the systems. It makes the scope of application of MRAC too restrictive, for nonminimum phase discrete-time systems can often appear. In this paper, we propose a design method of model reference adaptive control systems for non-minimum phase systems by utilizing 2-delay input control. In our method, the poles and zeros of the controlled systems are relocated by the 2-delay feedback control laws with 2-delay sampling. Since no cancellation of zeros occurs, the stable model reference adaptive control systems can be constructed even if unstable zeros exist. Some simulation results for non-minimum phase systems also shown the effectiveness of the proposed method.

Discrete-time model reference adaptive control for plants having deterministic disturbances

Based on the second method of Lyapunov, a globally stable discrete model reference adaptive controller design method with unknown deterministic disturbances is derived. The basic idea is to parameterize the disturbances by a ramp function using two parameters. It includes the SISO and MIMO cases. A state variable filter (SVF) is added to ensure that the proposed method is applicable to a plant with moderate measurement noise. The performance of these designs is illustrated by simulation.

A stable MRAC design for discrete plants with unmodelled dynamics

This paper presents a model reference adaptive control (MRAC) design which maintains stability around the equilibrium of the tracking error in the presence of unmodelled dynamics. Only a slight a priori knowledge in terms of lower and upper bounds for the absolute values of the parameters and orders of the unmodelled dynamics is required. The key to achieving signal boundedness is to use an additional parameter in the adaptive controller which involves the use of the a priori knowledge about the unmodelled dynamics.

Discrete Stable Model Reference Adaptive Control in the Presence of Unmodeled Dynamics

Abstract A standard adaptive control algorithm is modified using a time-dependent deat-beat parameter-adaptive structure. Under certain weakness assumptions about the influence of the unmodelled plant parameters on the response, stability of the overall scheme is guaranteed.

A discrete stable robust MRAC design with an extra adaptation parameter

This paper presents a stable model reference adaptive control (MRAC) scheme with robust properties against the presence of unmodeled dynamics. The key of the design is the use of an extra adaptation parameter which corresponds to an extra component in the regression vector, which is built using an absolute bound of the contribution to the output of the unmodeled dynamics.

Discrete model-reference adaptive control using only input and output measurements

A globally stable discrete model-reference adaptive controller design using only input and output measurements is presented. The rejection of both measurable and unmeasurable load disturbances is included in the adaptive system. The use of state-variable filters ensures that the basic scheme is applicable to a process with moderate measurement noise. With large measurement noise, an extended scheme using the technique of instrumental variable is introduced. The performance of these designs is substantiated by simulation.

Stable model reference adaptive control in the presence of bounded disturbances

The adaptive control of a linear time-invariant plant in the presence of bounded disturbances is considered. In addition to the usual assumptions made regarding the plant transfer function, it is also assumed that the high-frequency gain k <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> of the plant and an upper bound on the magnitude of the controller parameters are known. Under these conditions the adaptive controller suggested assures the boundedness of all signals in the overall system.

Computational aspects of discrete time adaptive control

Stable model reference adaptive control schemes for discrete time systems can be designed which take into account computational time and limited range of computing machines. An algorithm is developed using Lyapunov second method and properties of strictly positive real functions.

Discrete model reference adaptive control systems

Abstract A synthesis technique is presented for stable discrete model reference adaptive control systems. In the adaptive system the process parameters are adjusted indirectly according to a discrete gradient law. The stability of the adaptive scheme is demonstrated by Liapunov's second method.