This paper proposes a smooth control Lyapunov function with maximal controlled domain of attraction for the state-feedback control of constrained uncertain linear systems describing the dynamics of multivariable processes. Constructive algorithms are shown to build such a control Lyapunov function within a merging procedure due to the so-called R-functions. Robustness with respect to polytopic model uncertainties and disturbance rejection are also discussed. The controlled dynamics of two chemical reactions are simulated to show the benefits of the proposed strategy.

This paper introduces a simple modification to classic PI controllers to achieve the ideal behaviour of the regulator where the proportional component is used to get close to the reference signal, and the integral component to achieve zero steady-state error. As a consequence, the control effort is reduced and overshoots/wind-up phenomena are mitigated, while performance and simplicity of the PI controller are preserved. The proposed nonlinear PI is extensively compared with similar controllers from the literature to validate the theoretical expectations.

The constrained stabilization of linear uncertain systems is investigated via the set-theoretic framework of control Lyapunov R-functions. A novel composition rule allows the design of a composite control Lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones, generalizing both polyhedral and truncated ellipsoidal control Lyapunov functions. The feasibility test of the proposed smooth control Lyapunov functions can be cast into matrix inequalities conditions. The constrained linear quadratic control is addressed as an application.

A novel statistical indicator is introduced to evaluate the nonlinear behaviour of dynamical systems, as a support to other classic methods (e.g., Lyapunov exponents, entropy of curve methods). According to the theoretical properties of the proposed method, it is possible to infer the actual dimension of the phase space where the system dynamics evolve; it is possible to differentiate linear from nonlinear systems also in the presence of measurement noise; different dynamical systems that appear similar according to other indicators are further correctly distinguished. Theoretical expectations are confirmed by several examples of classic benchmark dynamical systems.

This article introduces the use of R-functions to compose single Lyapunov functions (LFs) via classic Boolean operators, with the aim to obtain a rich family of non-conventional, generally non-convex functions. The main benefit of the proposed composition is the nice geometric interpretation, since it corresponds to intersection and union operations in the phase space region. The composition of LFs is parameterised through a variable γ and classic compositions of LFs through min and max operations are recovered as a special case for a particular value of γ. The proposed logical composition is applied to region of asymptotic stability (RAS) estimation problems, where the union of several LFs corresponds to the union of the RAS estimates obtained from the separate use of each LF. Likewise, the intersection of several LFs defined on independent subsets of the state space variables provides a single LF for the overall dynamical system. Sufficient conditions for the composition function to be an LF are provided and results are described through several examples of classic nonlinear dynamical systems.

A nonlinear proximal point algorithm is proposed in the setting of adaptive control. Based on the Bregman generalized distance induced by a convex function, forward non-orthogonal projections provide a more general approach to classic optimization algorithms. In this paper convergence theorems are provided for the application of the Bregman algorithm to the adaptive control of both linear systems and nonlinear control-affine systems. It is shown that the proposed algorithm is particularly suited for adaptive control applications because it outperforms the gradient algorithm for on-line parameter estimation. Simulations on an autonomous underwater vehicle and a robotic manipulator show the benefits of the proposed adaptive control scheme when compared to the classic ones.

The stabilization problem of constrained uncertain linear systems is addressed via the class of control Lyapunov R-functions that are obtained reformulating the classic geometric intersection operator in terms of R-functions. The feasibility test of the proposed smooth control Lyapunov functions can be casted into (bi)linear matrix inequalities conditions. Like polyhedral Lyapunov functions, the maximal estimate of the controlled invariant state space set is achieved. The advantage of the proposed approach is that the inner sublevel sets are smooth and can be made everywhere differentiable. This smoothing technique is very general and it can be used to smooth both polyhedral and truncated ellipsoidal control Lyapunov functions to improve the control performances, as shown in some benchmark examples.

Abstract Polyhedral Lyapunov functions are convenient to solve the constrained stabilization problem of linear systems as non-conservative estimates of the domain of attraction can be obtained. Alternatively, truncated ellipsoids can be used to find an under-estimate of the feasible region, with a considerably reduced number of parameters. This paper reformulates classic geometric intersection operators in terms of R-functions, leading to a new family of smooth Lyapunov functions. This approach can be used to smooth both polyhedral and truncated ellipsoids Lyapunov functions improving control performances, as shown in several benchmark examples.

The constrained stabilization of the multivariable nonlinear model of a continuous stirred tank reactor is addressed. In this control setting, the novel class of control Lyapunov R-functions is proposed to design a smooth control Lyapunov function having external level set in accordance to the state constraints and non-homothetical inner level sets arbitrarily close to the desired (optimal) ones. The flexibility in shaping the composite control Lyapunov function is obtained by introducing a novel composition rule. The simulation of an exothermic chemical process shows the benefits of using a control Lyapunov R-function together with a gradient-based nonlinear controller, in fact a much larger controlled invariant state space set is obtained, moreover better state convergence and smoother control inputs are recovered.

The history of PIDs dates back to the beginning of the twentieth century when preliminary works of [Sperry (1922)] and [Minorski (1922)] provided mathematical results for the control of the ship motion and of automatically steered bodies in general. In particular, Minorski was the first to introduce three-term controllers with Proportional-Integral-Derivative (PID) actions. The success of PID was fast and solid, as nowadays they still represent the most popular choice in most industrial process control applications. The main reasons for their success are:

This paper provides a new approach for the analysis and eventually the classification of dynamical systems. The objective is obtained by extending the theory of the entropy of plane curves to R-n space. Properties of a dynamical system are inferred by investigating how a curve connecting a set of initial conditions in the phase space evolves with time, according to its generalised entropy. In particular all linear dynamical systems are characterised by a constant zero entropy, while higher asymptotic values indicate nonlinear behaviours. An algorithmic procedure to evaluate the entropy at each time step is outlined and it proves to be very efficient to describe chaotic systems as well. In this case the generalised entropy is proved to be linked to other conventional indicators known from literature. The entropy based approach is extensively tested for the analysis of several benchmark dynamical systems.

A novel state feedback control technique to stabilise linear differential inclusions through composite quadratic Lyapunov functions is presented. By using a gradient-based control technique, the minimum effort control is composed through intersection and union operations, derived from the theory of R-functions. While conventional min and max compositions are recovered as a special case, it is shown that smoother sublevel sets and everywhere differentiability are obtained tuning the composition parameter. Examples of both intersection and union compositions are provided to show that intermediate control performances in terms of convergence time are obtained, while improved performances in the control signal can be achieved.

This paper revisits the problem of decentralised control of Multiple Input Multiple Output (MIMO) systems. In industrial process control this problem is typically solved through the use of Proportional Integral and Derivative (PID) regulators that are used to close single loops, according to the implicit assumption that interactions between input and output variables can be neglected. However special care and conservatism are required for the choice of the regulator gains so that stability can be preserved also in the presence of the interactions. The decentralised MIMO control problem is solved here by extending the Variable Structure - Proportional Controller (VS-PI) proposed in to the MIMO case, and considering a priority-based approach for the sequential control of the single loops. In this paper the general idea is presented and preliminary simulation results including a robot control using brushless DC motor drives are included.

In this paper a novel solution is proposed for the stabilisation of Linear Differential Inclusions (LDI) with a continuous state feedback minimum effort control that is designed according to a smoothed polyhedral Lyapunov function. Algorithms from the literature are available to build polyhedral Lyapunov functions, which are convenient as they constitute a universal class for the solution of the stabilizability problem. It is however desirable to smooth the piecewise linear function, so that gradient based controllers, like the minimum effort one, provide continuous control signals. Polyhedral functions are here reinterpreted as intersections of linear functions through the use of R-functions, and the corresponding level curves can be easily smoothed progressively towards the origin using a free parameter. Results are compared with another similar method on some benchmark control problems from literature.

In the last few years the number of telelaboratories all over the world has significantly increased as they have proved to represent a valid support to conventional teaching tools. Telelaboratories allow a large number of people to access a limited number of resources, for instance students can learn the direct kinematics of a serial manipulator by moving a robotic arm that can be physically placed on the other side of the world. This paper proposes a Distributed Control Lab (DCL) which extends the conventional idea of a telelaboratory, as it can be used as a portable telelaboratory, completely contained inside a Live Linux-RTAI DVD, which can be connected to a large number of real plant to perform a process supervision task. The control panel consists of a user-friendly interface where a wide variety of graphical effects (including 3D models) can be associated to plant variables, so that it is very easy to interact with the underlying process. The software architecture of the DCL, the didactic benefits of the DCL and innovative uses of the proposed telelaboratory are described in detail. Interesting future ideas along this line of research are also outlined.

This paper describes the current evolution of the telelaboratory facilities at the University of Pisa. In particular, starting from a standard environment providing remote access to a set of experiments, the telelaboratory is now organized as a collection of learning objects, i.e., modular didactic units designed following specific learning objectives within control systems and robotic fields. The telelaboratory has a remote Web-based access which can be used both as a simulation environment and as a remote way of performing real physical experiments. The developed telelaboratory is based on free open-source software such as Scilab/Scicos, Comedi, and real-time application interface patch for Linux kernels. In-house software tools, such as a Java Web hyper modular interface system, graphic environment tools, and a virtual laboratory interface based on Java applets, have been developed as a support for the learning process.

This paper provides a new method to estimate the Region of Asymptotic Stability (RAS) via computing the union of several regions, each one of them is known to be a subset of the RAS. The union is computed analytically by using R-functions, which represent a natural extension of Boolean functions to real-valued functions. The benefit of the proposed approach is that usually the best Lyapunov function to estimate the RAS is not known, but each trial of a different candidate Lyapunov function provides a subset of the RAS. Therefore R-functions can be used to compute the union of all the estimations, taking into account all the performed trials. Sufficient conditions for the union R-function to be a new composite Lyapunov function are provided, and performances of the proposed approach are tested over classic benchmark problems.

With the advance of digital control hardware the simple but effective proportional-integral-derivative (PID) control technology is moving towards a higher level of performance and robustness. A more general class of Variable Structure (VS) PID has derived from original PIDs to improve their performances and capabilities. This paper revises the main properties of VS regulators and proposes a novel VS PI controller that combines the advantages of popular PIs with the more flexibility of VS controllers. The proposed regulator is compared with the classic PI over several examples taken from the literature, including first, second and fourth order dynamical systems. An experimental set-up is implemented on purpose using AVR® 32 Microcontrollers in a hardware in the loop approach to validate the simulation results in a more realistic environment.