prof. RNDr. Sergej Čelikovský, CSc.

In this paper, a continuous-time consensus algorithm with guaranteed finite-time convergence is proposed. Using homogeneity theory, finite-time consensus is proved for fixed topologies. The proposed algorithm is computationally simpler than other reported finite-time consensus algorithms, which is an important feature in scenarios of energy efficient nodes with limited computing resources such as sensor networks. Additionally, the proposed approach is compared on simulations with existing consensus algorithms, namely, the standard asymptotic consensus algorithm and the finite-time and fixed-time convergent algorithms, showing, in cycle graph topology, better robustness features on the convergence with respect to the network growth with less control effort. Indeed, the convergence time of other previously proposed consensus algorithms grows faster as the network grows than the one herein proposed whereas the control effort of the proposed algorithm is lower.

This paper focuses on the development of an optimization algorithm for car motion predictive control that addresses both hybrid car dynamics and hybrid minimization criterion. Instead of solving computationally demanding nonlinear mixed-integer programming task or approximating the hybrid dynamics/criterion, the Hamiltonian-switching Hybrid Nonlinear Predictive Control algorithm developed in this paper incorporates the information about hybridity directly into the optimization routine. To decrease the time-complexity, several adaptive prediction horizon approaches are proposed and for some of them, it is shown that they preserve maneuverabilityrelated properties of the car. All developed alternatives are verified on an example of a motion control of a racing car and compared with the approximation-based nonlinear predictive control and a commercial product. Moreover, sensitivity analysis examining robustness of the algorithm is included as well.

Switched Affine Systems (SAS'. s) is a class of Hybrid Systems composed of a collection of Affine Systems (AS'. s) and a switching signal that determines, at each time instant, the evolving affine subsystem. This paper is concerned with the observability and observer design for single-input single-output (SISO) SAS'. s under unknown perturbation, for the case that no information about the switching signal is available. It is firstly demonstrated that in the presence of disturbances every pair of AS'. s is always indistinguishable from the continuous output, meaning that it is not possible to infer the evolving AS by using only the information provided by the output of the SAS. Nevertheless, by taking advantage of the knowledge on the disturbance bound, new distinguishability conditions are derived, making possible to distinguish the evolving AS. By using these new distinguishability conditions, an observer scheme for SISO SAS'. s, subject to unknown switching signal and unknown perturbations, is presented. Such an observer scheme determines in finite-time the evolving AS. Furthermore, it estimates both the state of the system and the disturbance. Finally, the proposed observer scheme is effectively applied for a non-autonomous chaotic modulation application, which is an attractive method for spread-spectrum secure communication in which the message is fed as a disturbance to a chaotic SAS and the output is then transmitted through an open channel to a receiver, which is an observer algorithm that recovers the message (the disturbance) from the output signal.

This paper presents a computationally tractable algorithm focusing on overall optimization of a production process. The proposed algorithm embraces both the input profile and the state initial conditions optimization and consists of three stages: (i) optimization of the input profile with constant initial conditions, (ii) reduction of the input profile complexity and (iii) joint optimization of the input profile parameters and state initial conditions. The newly proposed algorithm is compared with several alternatives on a series of numerical experiments representing penicillin cultivation process. As a part of the evaluation, a broader range of optimization periods is considered and not only the criterion but also the complexity of the provided input profiles is inspected. The obtained encouraging results show the superiority of the newly proposed solution and demonstrate the usefulness of the joint-optimization algorithm.

The paper describes a simple time-optimal control strategy for a class of second-order bilinear systems with nonnegative inputs. The structure of the model is motivated by the task of noncontact manipulation of an object in a planar force field generated by a single source; such setup constitutes a basic building block for a planar manipulation by an array of force field sources. The nonnegative-in-control property means that an object (particle) placed freely in the field can only feel an attractive force towards the source. In this paper we further restrict the control inputs to a binary signal---the field can be switched on and off. The control objective is to bring the object to the origin (where the source of the force field is located) as fast as possible. The optimal switching strategy is proposed using geometric arguments and verified using numerical simulations and experiments with a laboratory platform for noncontact magnetic manipulation.

In this paper, maximization of efficiency of the penicillin production is addressed. To overcome the issues caused by the terminal time being an optimizable parameter, two adaptations of the Hamiltonian-based gradient method are proposed. The first algorithm incorporates the terminal time directly into the set of optimized variables while the second one makes use of a time scaling with optimizable time-dependent time scale being a virtual input. Both algorithms are verified on a set of numerical experiments and the obtained results show significant improvement of the penicillin production efficiency gained by their use, which suggests their exploitation within the industrial nonlinear predictive controllers for bioprocess optimization.

The linear model predictive control which is frequently used for building climate control benefits from the fact that the resulting optimization task is convex (thus easily and quickly solvable). On the other hand, the nonlinear model predictive control enables the use of a more detailed nonlinear model and it takes advantage of the fact that it addresses the optimization task more directly, however, it requires a more computationally complex algorithm for solving the non-convex optimization problem. In this paper, the gap between the linear and the nonlinear one is bridged by introducing a predictive controller with linear time-dependent model. Making use of linear time-dependent model of the building, the newly proposed controller obtains predictions which are closer to reality than those of linear time invariant model, however, the computational complexity is still kept low since the optimization task remains convex. The concept of linear time-dependent predictive controller is verified on a set of numerical experiments performed using a high fidelity model created in a building simulation environment and compared to the previously mentioned alternatives. Furthermore, the model for the nonlinear variant is identified using an adaptation of the existing model predictive control relevant identification method and the optimization algorithm for the nonlinear predictive controller is adapted such that it can handle also restrictions on discrete-valued nature of the manipulated variables. The presented comparisons show that the current adaptations lead to more efficient building climate control.

This paper brings structured Lyapunov functions guaranteeing cooperative state synchronization of identical agents. Versatile synchronizing region methods for identical linear systems motivate the structure of proposed Lyapunov functions. The obtained structured functions are applied to cooperative synchronization problems for affine-in-control nonlinear agents. For irreducible graphs a virtual leader is used to analyze synchronization. For reducible graphs a combination of cooperative tracking and irreducible graph cooperative synchronization is used to address cooperative dynamics by Lyapunov methods. This provides a connection between the synchronizing region analysis, incremental stability and Lyapunov cooperative stability conditions. A class of affine-in-control systems is singled out based on their contraction properties that allow for cooperative stability via the presented Lyapunov designs.

This paper deals with identification of a building model based on real-life data and subsequent temperature controller design. For the identification, advanced identification technique - namely MPC Relevant Identification method - is used. This approach has the capability of providing models with better prediction performance compared to the commonly used methods. Regarding the controller part, several alternatives are proposed. First, both linear and nonlinear MPC controlling the zone temperature are designed. Although highly attractive due to promising energetic savings and thermal comfort satisfaction, MPCs demand high computational power. To overcome this issue and preserve the attractive properties of the MPC, two MPC-learned feedback controllers are proposed, one learned from LMPC and the other learned from NMPC. While remaining computationally low-cost, they improve the performance of the classical controllers towards the high-performance MPC standards. The results exploiting data from real operation of an office equipped with air handling unit situated in Lakeshore building, Michigan Tech, are presented and discussed.

This paper addresses the problem of model predictive control for a class of nonlinear systems which satisfies per- sistent excitation condition. The conditions under which a non- linear system description can be handled are specified and two algorithms (one optimizing the first input sample and the other considering optimization of an M -sample subsequence of the input profile) solving the persistent excitation condition within a predictive controller for nonlinear systems are developed, both maximizing the smallest eigenvalue of the information matrix increase. The numerical experiments performed on a test-bed system demonstrate that the algorithms are able to successfully improve identifiability of a nonlinear system description while keeping the original controller performance degradation lower than arbitrarily chosen level.

In this paper, the task of quantized nonlinear predictive control is addressed. In such case, values of some inputs can be from a continuous interval while for the others, it is required that the optimized values belong to a countable set of discrete values. Instead of very straightforward a posteriori quantization, an alternative algorithm is developed incorpo- rating the quantization aspects directly into the optimization routine. The newly proposed quaNPC algorithm is tested on an example of building temperature control. The results for a broad range of number of quantization steps show that (unlike the naive a posteriori quantization) the quaNPC is able to maintain the control performance close to the performance of the original continuous-valued nonlinear predictive controller and at the same time it significantly decreases the undesirable oscillations of the discrete-valued input.

The mechanical chain consisting of 4 links with 3 actuators placed between them is considered in this paper. It will be further referred to as the so-called 4-link. The 4-link can be used to study underactuated walking-like movement resembling walking of a pair of legs with knees and hips without a torso. Underactuated mechanical systems have less actuators than degrees of freedom. When a point of contact with surface is fixed, the above 4-link clearly has 4 degrees of freedom while 3 actuators only. The control of the 4-link model in a way resembling a human walk will be designed here based on the embedding of a simpler two degrees of freedom model with one actuator into the model of 4-link. This simpler model is referred to as the so-called generalized Acrobot. The embedding is performed via imposing two virtual constraints fixing dependence between knees angles and hip angle via suitable selection of a pair of constraining functions.

In this paper, two alternatives approaches to model predictive control (MPC) are compared and contrasted for the role of zone temperature controller - the commonly used linear formulation (LMPC) and rather unconventional nonlinear formulation (NMPC). The economical focus is reflected by the performance criterion being a combination of the comfort requirements and the monetary cost penalties (price of the consumed hot water and the electricity needed to deliver the water to the building) of the controlled inputs. With this formulation, the optimal controller drives toward minimization of the real price rather than minimization of abstract quantities. It turns out that the NMPC is able to attack the cost minimization directly while retaining a compact optimization formulation as opposed to the suboptimal linear alternative. A considerable part of the superiority of the NMPC can be owed also to the use of a nonlinear model that captures the nonlinear building dynamics much more accurately than the linear models. Thanks to an enhanced search step choice introduced in this paper, the NMPC outperforms both the LMPC and the conventional controller significantly even under severe computational restrictions which demonstrates its strong practical applicability.

In the building climate control area, the linear model predictive control (LMPC)| nowadays considered a mature technique - benefits from the fact that the resulting optimization task is convex (thus easily and quickly solvable). On the other hand, while nonlinear model predictive control (NMPC) using a more detailed nonlinear model of a building takes advantage of its more accurate predictions and the fact that it attacks the optimization task more directly, it requires more involved ways of solving the non-convex optimization problem. In this paper, the gap between LMPC and NMPC is bridged by introducing several variants of linear time- varying model predictive controller (LTVMPC). Making use of linear time-varying model of the controlled building, LTVMPC obtains predictions which are closer to reality than those of linear time invariant model while still keeping the optimization task convex and less computationally demanding than in the case of NMPC. The concept of LTVMPC is verified on a set of numerical experiments performed using a high fidelity model created in a building simulation environment and compared to the previously mentioned alternatives (LMPC and NMPC) looking at both the control performance and the computational requirements.

Underactuated mechanical systems are systems with less actuators than degrees of freedom. Therefore, it is complicated to measure all states, i.e. angular positions or angular velocities, of the mechanical system. Alternative solution consists in an observer design such that unmeasurable states are estimated. For this purpose, a high gain observer for an Acrobot was introduced. By virtue of an Acrobot embedding into a 4-link model, the high gain observer was simply extended and applied to the 4-link model. The main aim of this paper consists in a coupling of a method of the Acrobot embedding into the 4-link model and the high gain observer design for the Acrobot. The coupling results in an observer for the 4-link model with the same structure as the already developed high gain observer for the Acrobot model.

In this article we present a novel approach to identification of a class of mechanical systems that can be written in a form of Euler-Lagrage equations. This identification procedure is based on a polynomial regression that is employed for data smoothing and estimation. Procedure requires only the measurements of torques or forces that are used to control the system and measurements of angles or other co-ordinates that are used to specify the location of the system. This identification algorithm is demonstrated on the problem of identification of three link bipedal robot and the results indicate that the procedure is effective and over-performs classical approaches used to solve the problem.

In this paper, the task of finding an algorithm providing sufficiently excited data within the MPC framework is tackled. Such algorithm is expected to take action only when the re-identification is needed and it shall be used as the “least costly” closed loop identification experiment for MPC. The already existing approach based on maximization of the smallest eigenvalue of the information matrix increase is revised and an adaptation by introducing a semi-receding horizon principle is performed. Further, the optimization algorithm used for the maximization of the provided information is adapted such that the constraints on the maximal allowed control performance deterioration are handled more carefully and are incorporated directly into the process instead of using them just as a termination condition. The effect of the performed adaptations is inspected using a numerical example. The example shows that the employment of the semi-receding horizon brings major improvement of the identification properties of the obtained data and the proposed adaptive-search step algorithm used for the “informativeness” optimization brings further significant increase of the contained information while the aggravation of the economical and tracking aspects of the control are kept at aceptable level.

This article presents a procedure that can be used to identify parameters of robotic manipulators that can be described by Euler-Lagrange equations of motion. This approach requires only measurements of angular position and torques acting in joints of the robot. To obtain smooth data for identification, angular measurements are smoothed by cubic splines. This allows analytical calculations of corresponding angular velocities and accelerations. Estimation of robot parameters can be then posed as an overdetermined linear problem. This approach is demonstrated on a simulation of a two-degree of freedom robotic manipulator. Results show that the procedure is much more robust compared to other classically used techniques while preserving high accuracy.

This paper makes a step towards practical applicability of the optimal control for industrial penicillin production. Using the nonlinear gradient method as the key optimization tool, two ways of measurement feedback incorporation into the optimization procedure are proposed. Firstly, the receding horizon approach (whose linear variant is widely spreading in the field of operation of various industrial processes) is investigated considering different lengths of optimization horizon. Secondly, the shrinking horizon approach inspired by the character of the solved task with terminal criterion is examined. In order to make the latter comparable to the receding horizon approach, various sampling periods of the input signal are considered. Utilization of the nonlinear continuous time model of the controlled process clearly distinguishes this paper from the earlier publications. The behavior of both approaches is tested on a set of numerical experiments with the focus on performance under constrained computational resources. The obtained results demonstrate the superiority of shrinking horizon approach and its strong computational restriction resistance.

This paper presents continuous-time optimal control of penicillin production maximizing penicillin concentration at chosen final time. First, control input parametrization is performed and the input is expressed as a piece-wise affine function of continuous time reducing the number of optimizable parameters. Then, iterative numerical gradient optimization with the initial conditions corresponding to the optimal input obtained by projected gradient optimization is used to find the optimal values of the optimized parameters. The proposed approach is compared to both the original gradient method and the traditionally used classical nonlinear feedback controller. All these strategies are tested on a set of numerical experiments for various cultivation lengths and the results are evaluated and discussed. The comparison reveals significant superiority of the proposed algorithm. Together with impressive reduction of memory space needed to store the solution, the results improvement makes the proposed algorithm very attractive from the industrial point of view.

This paper studies synchronization of a dynamical complex network consisting of nodes being generalized Lorenz chaotic systems and connections created with transmitted synchronizing signals. The focus is on the robustness of the network synchronization with respect to its topology. The robustness is analyzed theoretically for the case of two nodes with two-sided (bidirectional) connections, and numerically for various cases with large numbers of nodes. It is shown that, unless a certain minimal coherent topology is present in the network, synchronization is always preserved. While for a minimal network where synchronization is global, the resulting synchrony reduces to semi-global if redundant connections are added.

The virtual constraints method is used here to design and control the walking-like trajectory of the 4-link having 4 degrees of freedom: stance angle, 2 knees angles and 1 hip angle and 3 actuators only as the stance leg angle is not actuated. Two different approaches are compared. First, the wellknown approach consists in setting virtual constraints as the dependencies of knees and hip angles on the stance leg angle. Therefore there are 3 virtual constraints enforced by all 3 available inputs and reducing thereby overall 4 degrees of freedom to a single degree of freedom unactuated system. Selecting suitable constraints functions, various walking-like trajectories can be designed. Secondly, this three constraints approach is compared with the alternative one developed very recently. Here, only two constraints are imposed being dependencies of knees angles on the hip angle thereby reducing the 4-link to 2 degrees of freedom system with a single actuator at the hip angle.

This article deals with online adaptation of control strategy for nonlinear tracking of a walking like motion of bipedal robot. Adaptation of the control rule is done according to results of online parameter estimation. Parameter estimation was realized by an extended Kalman filter due to recursive nature of the estimation problem and abundant a priori information. Proposed estimation strategy yields at least three advantages. By utilization of extensive knowledge about the system in consideration a multi-variable estimation problem was reduced to estimation problem involving one parameter only. A heavy computation burden required for recomputation of reference trajectory and feed-forward controller is removed. This approach can also be used to eliminate the modeling mismatch. A practical situation when a robot has to carry a load of an unknown weight is demonstrated.

The purpose of this paper is to provide theoretical framework enabling to design tracking feedback for a general Acrobot trajectory which allows rigorous convergence proof. It is based on the partial exact feedback linearization of the Acrobot model followed by further approximate feedback linearization of the tracking error dynamics for arbitrary target trajectory. The approach presented here enables to prove the convergence in a rigorous way at least for small initial tracking errors. The slight novelty here is that neglecting is made with respect to tracking error along any general trajectory to be tracked, not just in some neighborhood of fixed working point. To demonstrate viability of this approach, simulations of a tracking of a walking-like cyclic trajectory are presented. The walking includes several steps including impacts between them. As a matter of fact, the exponentially stable tracking during the swing phase only is capable to stabilize overall walking, including the effect of the impacts.

This paper describes the chaos shift keying method based on the second derivative desynchronization error and provides its security analysis, especially against the attacks by power and return map analysis. Desynchronization chaos shift keying method (DECSK) uses methods to detect the correct bit by detecting the wrong bit. Various modifications are possible, here the method using sharp increase of error in the second derivative of synchronizing signal is used. The proposed method requires very reasonable amount of data to encrypt and time to decrypt one bit. Basically, to encrypt one bit, only one iteration (i.e. only one real number of six valid digits) is needed. At the same time, thanks to the desynchronization detection based on the synchronization error second derivative, almost 100% of the carrying chaotic signal can be used. The security of the proposed method can be systematically investigated showing its good resistance against typical decryption attacks. More detailed analysis is devoted to its analysis via power and return map analysis. Conclusion is that the DECSK method cannot be broken by the above two methods which together with other arguments developed there serves as a good basis for the DECSK security.

A fermentation process is generally defined as a biological process containing the growth of the biomass (bacteria, yeasts) resulting from the consumption of essential substrate supplies (source of carbon, nitrogen, oxygen, etc.). The biomass growth is usually followed by the production of various products from which especially the variety of antibiotics makes the fermentation processes attractive for the industrial utilization. However, the complicated dynamics, high level of uncertainty and nonlinearity and difficult online measurement of the process variables come hand in hand with the attractivity and turn the attempts on optimal control of the fermentation process into a very delicate challenge. To tackle it, the theory of the gradient projection method has been partially adapted and fully implemented by the authors of this paper. Numerical experiments show its significantly better performance than for other known methods. Moreover, these experiments reveal an interesting "superprofile" visible for long cultivation times.

Since their discovery, fermentation processes have gone along not only with the industrial beverages production and breweries but since the times of Alexander Fleming they have become a crucial part of the health care due to antibiotics production (from which the overwhelming majority of 90% is produced during a fermentation process). However, complicated dynamics and strong nonlinearities cause that the production with the use of linear control methods achieves only suboptimal yields. From the variety of nonlinear approaches, gradient method has proved the ability to handle these issues - nevertheless, its potential in the field of fermentation processes has not been revealed completely. In this paper, two fresh control strategies are introduced and compared - both of them are based on a double-input optimization approach, yet a successful reduction to a single-input optimization task is proposed. To accomplish this, model structure used in the previous work has been modified so that it corresponds with the new optimization strategies which together with the model stands for the main contribution of this paper.

This article deals with state estimation and filtering in nonlinear systems. More precisely the design of approximate nonlinear Kalman filter for nonlinear systems linearizable by a nonlinear coordinate transformation with possible application of nonlinear output injection is studied. The main idea is to reduce the errors introduced by the linearization used by the approximate filter using exact full or partial linearization of the system. Underlying transformations of both deterministic and stochastic signals are studied. An example of designing such an approximate nonlinear filter for nonlinear tracking of walking like motion of bipedal robot is given.

Unlike prevailing contemporary approach considering the influx feed flow as the only crucial input influencing the final product concentration in bioprocess cultivation, the current paper considers other (non-nutritional) inputs as well. Among them, the volume withdrawal is an important option. Two types of control strategies employing this second input are considered here, each of them using it in a different way. First, the so-called quasi-double-input strategies exploit the broth withdrawal to follow a pre-determined volume profile. Second, the true-double-input strategies consider the second input as a completely independent optimization variable. Main analysis tool here is the gradient optimization numerical algorithm and its adaptations. All resulting strategies are summarized and formulated in the context of optimization, those introduced earlier (i.e., the quasi-double-input) are unified into a common general strategy. Practical complication (inadmissible volume decrease) related to withdrawal introduction is successfully addressed and the summary results show indisputable improvement gained by the non-nutrient input introduction.

This contribution addresses a possible description of the chaotic behavior in multi-valued dynamical systems. An important area leading to description via the multi-valued dynamical systems is the non-smooth dynamical systems theory and their applications. Examples of such applications are mechanics with dry friction, electric circuits with small conductivity, systems with small inertia, economy, biology, control theory, game theory, optimization, etc.

This paper deals with estimation of coefficients of viscous friction for a model of an acrobot. The acrobot represents an underactuated nonlinear dynamic system, where typically not all states are measurable. Moreover effect of noise corruption on remaining measured states is often non negligible. However, except for friction coefficient, all remaining parameters of the model can usually be measured directly. To overcome mentioned difficulties and to take advantage of abundant prior knowledge, we applied hybrid extended Kalman filter to this task. Using Monte Carlo (MC) simulations we approximated probability density functions of friction coefficients estimate and showed that the bias and variance of the estimate can be controlled by properly designed experiment.

This paper aims to the further improve of the previously developed design for the Acrobot walking based on the partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4 dimensional linear time varying system having 3 time varying entries only. Unlike previous approaches treating time varying entries as uncertainties with various extent of conservatism, the present paper takes into the account an information about these time varying functions including their derivatives up to order 4. Using that, the time varying state and the feedback transformation enable to design a fundamental matrix of the error dynamics in an explicit form and pre-designed stability properties.

This work is concerned with the design of an observer for Switched Linear Systems (SLS) subject to an unknown switching signal. The convergence of the proposed SLS observer is proved, based on observability results for SLS, for the following two cases. First, it is shown that if the discrete state of the SLS is observable, then the observer estimates the switching signal, even though the individual Linear System (LS) may not be observable. And second, it is shown how the observer may provide an estimate of the continuous variables, even though the discrete state is unknown.

This paper aims to extend of the previously developed analytical design for the Acrobot walking. The corresponding state feedback controller is completed in this paper by an observer to estimate some states of the Acrobot. Both the controller and the observer are based on the partial exact feedback linearization of order 3. The feedback controller and the observer are extended for the tracking of the cyclic walking-like trajectory in order to demonstrate the cyclic Acrobot walking. The cyclic walking-like trajectory starts continuous phase from certain initial conditions, that at the end of the step makes an impact and after the impact it reaches the same initial conditions as at the beginning of the step. This cyclic motion of the Acrobot enable us to prove the stability of the feedback tracking with the observer numericaly by the method of Poincare mappings.

This article aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearisation of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4-dimensional linear time-varying system having three time-varying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilising a linear system with polytopic uncertainty. The current improvement is based on applying linear matrix inequalities (LMI) methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above-mentioned LMI design are demonstrated as well.

This chapter introduces the notion of chaos synthesis by means of evolutionary algorithms and develops a new method for chaotic systems synthesis. This method is similar to genetic programming and grammatical evolution and is applied alongside evolutionary algorithms: differential evolution, self organizingmigrating, genetic algorithm, simulated annealing and evolutionary strategies. The aim of this investigation is to synthesize new and "simple" chaotic systems based on some elements contained in a pre-chosen existing chaotic system and a properly defined cost function. The investigation consists of two case studies based on the aforementioned evolutionary algorithms in various versions. For all algorithms, 100 simulations of chaos synthesis were repeated and then averaged to guarantee the reliability and robustness of the proposed method. The most significant results are carefully selected, visualized and commented in this chapter.

This chapter deals with chaotic systems. Based on the characterization of deterministic chaos, universal features of that kind of behavior are explained. It is shown that despite the deterministic nature of chaos, long term behavior is unpredictable. This is called sensitivity to initial conditions.We further give a concept of quantifying chaotic dynamics: the Lyapunov exponent. Moreover, we explain how chaos can originate from order by period doubling, intermittence, chaotic transients and crises. In the second part of the chapter we discuss different examples of systems showing chaos, for instance mechanical, electronic, biological,meteorological, algorithmical and astronomical systems.

This paper provides the comparison of nonlinear observers for underactuated mechanical systems with aplication to Acrobot walking. Namely, it aims to compare a reduced observer for the Acrobot angular velocities based on the respective angles measurements with a observer for the Acrobot angular velocity and angle based on the other angular velocity and angle measurement and then to combine this observers with the earlier developed full state feedback controllers. For the Acrobot it is not possible to measure the angle between its stance leg and the surface directly because this angle is underactuated. One possibility of measuring this angle is using laser beam sensor and then computation the stance angle from that distance. Main disadvantage of this approach is the high price of the laser beam sensor. The other observer is based on angular velocity of stance leg measurement using the digital gyroscope.

In this paper, a modified version of the Chaos Shift Keying (CSK) scheme for secure encryption and decryption of data will be discussed. The classical CSK method determines the correct value of binary signal through checking which initially unsynchronized sys- tem is getting synchronized. On the contrary, the new anti-synchronization CSK (ACSK) scheme determines the wrong value of binary signal through checking which already syn- chronized system is loosing synchronization. The ACSK scheme is implemented and tested using the so-called generalized Lorenz system (GLS) family making advantage of its special parametrization. Such an implementation relies on the parameter dependent synchroniza- tion of several identical copies of the GLS obtained through the observer-based design for nonlinear systems. The purpose of this paper is to study and compare two different meth- ods for the anti-synchronization detection, including further underlying theoretical study of the GLS.

This paper aims to compare the performance of various techniques for the stabilization of the error dynamics of the Acrobot's walking like reference trajectory. Both the walking reference planning and the tracking feedback design are based on the Acrobot's model partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4 dimensional linear time varying system having 3 time varying entries only, the remaining entries are either zero or equal to one.

In nonlinear output regulation problems, it is necessary to solve the so-called regulator equations consisting of a partial differential equation and an algebraic equation. It is known that, for the hyperbolic zero dynamics case, solving the regulator equations is equivalent to calculating a center manifold for zero dynamics of the system. The present paper proposes a successive approximation method for obtaining center manifolds and shows its effectiveness by applying it for an inverted pendulum example.

This paper aims to further improve previously developed design for the Acrobot walking based on the partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4 dimensional linear time varying system having 3 time varying entries only, the remaining entries are either zero or equal to one. Previously, the exponentially stable tracking was obtained by solving quadratic stability of a linear system with polytopic uncertainty applying LMI methods to solve this problem numerically. Here, the new approach is presented allowing to design the tracking feedback and to prove the corresponding stability completely analytically. Moreover, this approach gives even better results than the LMI based one in the sense of the convergence speed.

This paper studies a new modication of the anti-synchronization chaos shift keying scheme for the secure encryption and decryption of data. A new concept of the detection of the correct/incorect binary value in the receiver is used. The method proposed here requires very reasonable amount of data to encrypt and time to decrypt a single bit. Basically, to encrypt a single bit, only one iteration is needed. Moreover, the security of this method is systematically investigated showing its good resistance to typical decryption attacks. Theoretical results are supported by the numerical simulations.

This paper studies yet another improvement of the anti-synchronization chaos shift keying scheme for the secure encryption and decryption of the digital data. A new concept of the detection of the correct binary value in the receiver is introduced here. The proposed method requires very reasonable amount of data to encrypt and time to decrypt one bit. Basically, to encrypt one bit, only one iteration (i.e. only one real number of 6 valid digits) is needed. At the same time, thanks to the anti-synchronization detection based on the synchronization error second derivative, almost 100% of the carrying chaotic signal can be used. The security of the proposed method can be systematically investigated showing its good resistance against typical decryption attacks. The theoretical analysis of the introduced method is supported by the numerical experiments with digital data encryption.

This paper aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4-dimensional linear time-varying system having 3 time-varying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilizing a linear system with polytopic uncertainty. The current improvement is based on applying LMI methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above mentioned LMI design are demonstrated as well.

It aims to compare the performance of various techniques fo the stabilization of the error dynamics of the Acrobot's walking like reference trajectory.

A functional version of the LaSalle invariance principle is introduced. Rather than the usual pointwise Lyapunov-like functions, this extended version of the principle uses specially constructed functionals along system trajectories. This modification enables the original principle to handle not only autonomous, but also some nonautonomous systems. The new theoretical result is used to study robust synchronization of general Li´enard-type nonlinear systems. The new technique is finally applied to coupled chaotic van der Pol oscillators to achieve synchronization. Numerical simulation is included to demonstrate the effectiveness of the proposed methodology.

A numerical method to solve the so-called regulator equation is presented here. This equation consists of partial differential equations combined with the algebraic ones and arises when solving the output-regulation problem. Output regulation problem aims to find feedback compensator to ensure tracking a given reference and/or rejecting unknown disturbance where both reference and disturbances are generated by a finite dimensional autonomous exogenous system. Solving the regulator equation is becoming difficult especially for the non-minimum phase systems where reducing variables against algebraic part leads to a potentially unsolvable differential part.

A state feedback law is proposed for on-line optimization of microalgal growth in a photobioreactor in the presence of non-measurable disturbances. The objective is to maximize a photosynthetic production rate (specific growth rate of microalgae) by manipulating the irradiance. The reduction to a slow dynamics is used to derive analytically the approximation of the optimal feedback control. The analysis of the obtained explicit formula shows that the optimal feedback control actually performs optimal transfer to a constant optimal irradiance developed earlier but does not achieve principally better performance than the optimal control within the class of constant irradiance. Therefore, by reducing fast dynamics one can not reveal possible more complex optimal solutions. At the same time, this contradicts to a common belief in biotechnological community that the fast phenomena may be neglected. Illustrative simulations are included.

A new control concept for a class of simple underactuated mechanical system, the so-called Acrobot, is presented here. Despite being seemingly a simple system, the acrobot comprises many important difficulties when controlling the most challenging underactuated system - the walking robot. This paper presents the design of the asymptotical tracking of the prescribed trajectory generated by a suitable openloop input of the acrobot. Such a design is based on the partial exact linearization of the third order combined with a certain robust stabilization technique. The proposed control is then demonstrated by the exponential tracking of the walkinglike trajectory of the acrobot. Besides theoretical proofs, our approach is supported by numerical simulations and illustrated by acrobot movement animations.

A new control concept for a class of an underactuated mechanical system is presented here. More precisely, the system in question is the chain of n links having n − 1 actuators between them and attached by one its ends to the horizontal plane. This paper presents its stabilization from a wide range of initial positions, much wider than typical local results in the current literature.

A class of mechanical systems is studied and a notion of equivalence is introduced. More generally, one identifies whenever a given system is state equivalent to some subdynamics of an other system. This property plays a crucial role to control large complex mechanical systems as walking robots.

The regulator equation is the fundamental equation whose solution must be found in order to solve the output regulation problem. It is a system of first-order partial differential equations combined with an algebraic equation. The classical approach to its solution is to use the Taylor series. In this contribution, another path is followed: the equation is solved using the finite-element method. There are two methods developed to satisfy the algebraic condition: the first one is based on iterative minimization of a cost functional (the square of the error made in the algebraic condition), the main idea of the other one is to convert the algebro-differential equation into a singularly perturbed system of partial differential equations. Both methods are compared and results of simulations are presented.

In this paper, a modified version of the Chaos Shift Keying (CSK) scheme for secure encryption and decryption of data is proposed. The proposed scheme uses the effect of anti-synchronization, rather than synchronization. More specifically, the classical CSK method determines the correct value of binary signal through checking which unsynchronized system is getting synchronized. On the contrary, our novel method determines wrong value of binary signal through checking which already synchronized system is loosing synchronization. Our method is implemented and thoroughly tested on the recently introduced generalized Lorenz system (GLS) family making advantage of its special parametrization. The proposed scheme relies on parameter depended synchronization obtained through observer-based design for nonlinear systems.

A real time implementation of an error feedback output regulation problem for the gyroscopical platform is presented here. It is based on a numerical method for the solution of the so-called regulator equation. The regulator equation consists of partial differential equations combined with algebraic ones and arises when solving the output-regulation problem. Error-feedback output regulation problem aims to find a dynamic feedback compensator using only tracking error measurements to ensure tracking given reference and/or rejecting unknown disturbance. Solving the regulator equation is becoming difficult especially for the non-minimum phase systems where reducing variables against algebraic part leads to possible unsolvable differential part.

The aim of this paper is to study synchronization of dynamical complex network consisting of nodes being generalized Lorenz chaotic systems and connections are created with transmitted synchronizing signals. Focus is on the robustness of the network synchronization with respect to its connectional structure.

The regulator equation arises when solving the output regulation problem. This equation consists of partial differential equations combined with algebraic ones. Error-feedback output regulation problem aims to find a dynamic feedback compensator to ensure tracking of a given reference and/or rejecting unknown disturbances. Both reference and disturbances are generated by a finite- dimensional autonomous exogenous system. Solving the regulator equation is becoming difficult especially for the non-minimum phase systems where reducing variables against algebraic part leads to possibly unsolvable differential part. The proposed numerical method is based on the approximation of the algebraic part of the regulator equation by a singularly perturbed differential equation.

In this paper, a modified version of the Chaos Shift Keying scheme for secure encryption and decription of data is proposed. The proposed scheme uses the effect of anti-synchronization, rather than synchronization. More specifically, the classical CSk method determines the correct value of binary signal through checking which unsynchronized system is getting synchronized. On the contrary, our novel method determines wrong value of binary signal through checking which already synchronized system is loosing synchronization. The advantage of the proposed method is two-fold. First, it requires very reasonable amount of data to encrypt and time to decrypt a single bit. Secondly, its security can be investigated and estimated as practically unbreakable.

Final presentation of the project interested in design of an experimental device for study of a robotics walking. The aim was development and solution the simple benchmark model for an experimental part of work in this research area, with respect to be applicable for a research work and a system and control education as well.

This paper studies possible application of the observerbased chaos synchronization to secure encryption. Unlike common approaches that use continuous time synchronized chaos for the so-called chaotic masking, the chaos shift keying (CSK) is being used here. CSK uses piece of synchronized chaotic signal to encode directly a given symbol. This paper uses properties of observer-based synchronization to achieve reasonable amount of information per symbol, moreover, specific properties of dependence of this observer-based synchronization on parameters make possible direct encryption of multi-alphabet symbols. Such a breakthrough is enabled by using anti-synchronization effect, rather than the synchronization one. Our method, further referred as the anti-synchronization chaos shift keying (ACSK), is implemented and thoroughly tested on the recently introduced generalized Lorenz system (GLS) family making advantage of its special parametrization.

This paper studies possible application of the observerbased chaos synchronization to secure encryption. Unlike common approaches that use continuous time synchronized chaos for the so-called chaotic masking, the chaos shift keying (CSK) is being used here. CSK uses piece of synchronized chaotic signal to encode directly a given symbol. This paper uses properties of observer-based synchronization to achieve reasonable amount of information per symbol, moreover, specific properties of dependence of this observer-based synchronization on parameters make possible direct encryption of multi-alphabet symbols. Such a breakthrough is enabled by using anti-synchronization effect, rather than the synchronization one. Our method, further referred as the anti-synchronization chaos shift keying (ACSK), is implemented and thoroughly tested on the recently introduced generalized Lorenz system (GLS) family making advantage of its special parametrization.

In this paper, an algorithm for computing heteroclinic orbits of nonlinear systems, which can have several hyperbolic equilibria, is suggested and analyzed both analytically and numerically. The method is based on a representation of the invariant manifold of a hyperbolic equilibrium via a certain exponential series expansion. The algorithm for computing the series coefficients is derived and the uniform convergence of the series is theoretically proved. The algorithm is then applied to computing heteroclinic orbits numerically in the generated Lorenz system, thereby theoretically justifying the previously demonstrated existence of chaotic oscillations in this important class of dynamical systems.

The merit of the output-regulation problem is to find a feedforward control so that asymptotic tracking of a trajectory generated by an external system (exosystem) is achieved. Then the resulting control consists of the sum of this feedforward and a stabilizing feedback. The crucial point is solution of the regulator equation. It consists of a system of partial differential equations and an algebraic condition. In the proposed article the regulator equation is solved via the finiteelement method. To achieve solvability the system was stabilized at the first stage. Then the regulator equation was built for this stabilized system. The solution of the regulator equation was divided into two parts that were repeated iteratively. First the feedforward control was chosen and the partial differential equations was solved. Then the error was evaluated and a change of the proposed feedforward was carried out so that this error decreases. The results are demonstrated on an example.

The aim of this paper is twofold. It provides another option how to obtain a universal easily implementable method for the solution of the regulator equations using the FEMLAB package. The regulator equation originates from the output regulation problem. The main idea is making a slight change of the regulator equation. It is then solved using the finite-element method. Some theoretical aspects concerning solvability of the equations and convergence to the original problem are introduced. Secondly, to demonstrate viability of our approach, the results were applied to the real-time control of a gyroscope. Both simulations and real-time laboratory experiments are included.

We address the problem of output regulation for nonlinear systems driven by a partially unknown nonautonomous exosystem via error feedback. We generalize the classical output regulation problem in order to expand the class of reference or disturbance signals. Our study of the above problem, refereed to as the so-called generealized output regulation problem depends on the classical notion of the exact disturbance decoupling. A local necessary and sufficient conditions for the solvability of the problem are given.

A functional version of LaSalle's invariance principle is derived, i.e., rather than the usual pointwise Lyapunov-like functions it uses specially constructed functionals along system trajectories. This modification enables the principle to handle even nonautonomous systems to which the classical LaSalle's principle is not directly applicable. The new theoretical results are then used to study robust synchronization of general Liénard type of systems. The developed technique is finally applied to chaotic oscillators synchronization. Numerical simulation is included to demonstrate the effectiveness of the proposed methodology.

This short note is to briefly introduce the new notion of generalized Lorenz canonical form (GLCF), which contains the classical Lorenz system and the newly discovered Chen system as two extreme cases, along with infinitely many chaotic systems in between. It also points out that some recently reported chaotic systems are special cases of GLCF.

The aim of this note is two-fold. First, it discusses synchronization of chaotic systems from a control theoretic point of view and introduces the concept of secure synchronization, i.e., a communication scheme that resists possible intrusion based on either adaptive or robust control techniques. Second, for a large class of chaotic systems, i.e., the generalized Lorenz system, global exponential synchronization via a scalar communication signal is suggested and its security is analyzed from a control theoretic viewpoint. Both theoretical analysis and numerical simulations are provided, verifying the proposed chaos-synchronization-based secure communication design principle and methodology.

This paper studies the generalized Lorenz canonical form of dynamical systems introduced by Čelikovský and Chen [International Journal of Bifurcation and Chaos 12(8), 2002, 1789]. It proves the existence of a heteroclinic orbit of the canonical form and the convergence of the corresponding series expansion. The Ši'lnikov criterion along with some technical conditions guarantee that the canonical form has Smale horseshoes and horseshoe chaos. As a consequence, it also proves that both the classical Lorenz system and the Chen system have Ši'lnikov chaos. When the system is changed into another ordinary differential equation through a nonsingular one-parameter linear transformation, the exact range of existence of Ši'lnikov chaos with respect to the parameter can be specified. Numerical simulation verifies the theoretical results and analysis.

With the help of numerical examples, this paper describes new fixed-order robust controller design functions implemented in version 3.0 of the Polynomial Toolbox for Matlab. The functions use convex optimization over linear matrix inequalities (LMIs) solved with the SeDuMi solver.

This paper shows that a large class of chaotic systems introduced in Čelikovský and Chen [7], as the hyperbolic-type generalized Lorenz system, can be systematically used to generate sysnchronized chaotic oscillations. While the generalized Lorenz system unifies the famous Lorenz system and Chen's system, the hyperbolic-type generalized Lorenz system is in some way complementary to it. Synchronization of two such systems is made through a scalar coupling signal based on nonlinear observer design using special change of coordinates to the so-called observer canonical form of the hyperbolic-type generalized Lorenz system. The properties of the suggested synchronization that make it attractive for the the secure encrypted communication application are discussed in detail. Theoretical results are supported by the computer simulations.

The output regulation problem for the nonlinear systems with nonhyperbolic zero dynamics is considered. The usual approach uses the solution of the so-called regulator equation being rather non-standard first order PDE combined with an algebraic equation. Its solvability, as well as the corresponding numerical methods, remains to be an open question, especially in case of nonminimum phase systems having non-hyperbolic zero dynamics. Here, an approach based on FEMLAB software is presented. A key idea here is to avoid the usual reducing the overall regulator equation to pure PDE equation corresponding to a possibly unstable zero dynamics, but replace algebraic equation by a certain penalty functional for solvable PDE system. Such an approach appears as the promising one. An illustrative example of two cart with an inverted pendulum system is presented as well.