Vector Relative Degree Research Articles

Adaptive controller design and disturbance attenuation for minimum phase multiple‐input multiple‐output linear systems with noisy output measurements and with measured disturbances

SummaryIn this paper, we present a systematic procedure for robust adaptive control design for minimum phase uncertain multiple‐input multiple‐output linear systems that are right invertible and can be dynamically extended to a linear system with vector relative degree using a dynamic compensator that is known. For this class of systems, it is always possible to dynamically extend them, and/or integrate a select set of output channels, and/or padding dummy state variables to arrive at a system model that admits uniform vector relative degree and uniform observability indices that is further minimum phase according to Reference 13. We assume that the uniform vector relative degree is known and an upper bound for the uniform observability indices is known. We also assume that the unknown parameter vector lies in a convex compact set such that the high frequency gain matrix remains invertible for any parameter vector value in the set. These are the assumptions that allow for a successful design of a robust adaptive controller. A numerical example is included to fully illustrate the controller design and the effectiveness of the controller.

Singularity-free adaptive control of MIMO discrete-time nonlinear systems with general vector relative degrees

This paper develops a singularity-free adaptive tracking control scheme for a general class of multi-input and multi-output uncertain discrete-time nonlinear systems with non-canonical control gain matrices. The estimation of the control gain matrices, especially in some non-canonical forms, may be singular during parameter adaptation, which leads to the singularity problems of the adaptive control laws. This paper employs the matrix decomposition technique to solve the problem under a linearly parameterized adaptive control framework. The state and output feedback cases are addressed, respectively, to ensure closed-loop stability and asymptotic output tracking. Compared with the existing results, the features of the proposed adaptive control scheme include: (i) the proposed control laws do not involve the high-gain issue commonly encountered in robust control methods; (ii) two different filtered tracking error signals are introduced for the state and output feedback cases, respectively. These filters are crucial to avoid causality contradiction of the adaptive control laws commonly encountered in adaptive control of discrete-time systems; and (iii) a future time signal estimation-based adaptive control law is developed to ensure asymptotic output tracking for the output feedback case without requiring the high-gain observer. Finally, an illustrative example is given to verify the validity of the proposed control scheme.

Iteration dependent interval based open‐closed‐loop iterative learning control for time varying systems with vector relative degree

AbstractFor linear time varying (LTV) multiple input multiple output (MIMO) systems with vector relative degree, an open‐closed‐loop iterative learning control (ILC) strategy is developed in this article, where the time interval of operation is iteration dependent. To compensate the missing tracking signal caused by iteration dependent interval, the feedback control is introduced in ILC design. As the tracking signal of many continuous iterations is lost in a certain interval, the feedback control part can employ the tracking signal of current iteration for compensation. Under the assumption that the initial state vibrates around the desired initial state uniformly in mathematical expectation sense, the expectation of ILC tracking error can converge to zero as the number of iteration tends to infinity. Under the circumstance that the initial state varies around the desired initial state with a bound, as the number of iteration tends to infinity, the expectation of ILC tracking error can be driven to a bounded range, whose upper bound is proportional to the fluctuation. It is revealed that the convergence condition is dependent on the feedforward control gains, while the feedback control can accelerate convergence speed by selecting appropriate feedback control gains. As a special case, the controlled system with integrated high relative degree is also addressed by proposing a simplified iteration dependent interval based open‐closed‐loop ILC method. Finally, the effectiveness of the developed iteration dependent interval based open‐closed‐loop ILC is illustrated by a simulation example with two cases on initial state.

An Output Feedback Stabilizer for Uncertain MIMO Nonlinear Systems: Nonlinear Nominal Input Gain Function

This article presents a robust output feedback stabilizer for multi-input–multi-output nonlinear systems subject to external disturbances and system uncertainties. We assume that the system has a well-defined vector relative degree and the zero dynamics is input-to-state stable. Based on the assumption that there exists a state feedback controller that makes the origin of the nominal closed-loop system asymptotically stable, we present an output feedback stabilizer that recovers the stability of the nominal closed-loop system in the practical sense. We allow that the nominal system can have a nonlinear input gain matrix that is a function of system state, and this is done by modifying the structure of the disturbance observer-based robust outputfeedback controller. Simulation results on a practical system are included to validate the proposed controller.

Matrix Decomposition-Based Adaptive Control of Noncanonical Form MIMO DT Nonlinear Systems

This article presents a new study on adaptive control of multi-input and multi-output (MIMO) discrete-time nonlinear systems with a noncanonical form involving parametric uncertainties. The adaptive control scheme employs a vector relative degree formulation to reconstruct the noncanonical system dynamics and derives a normal form. Then, a new matrix decomposition-based adaptive control scheme is proposed for the controlled plant with a vector relative degree <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[1, 1,{\ldots }, 1]$</tex-math></inline-formula> under some relaxed design conditions. In particular, the matrix decomposition technique is adopted to overcome the singularity problem during the adaptive estimation of an uncertain high-frequency gain matrix. The adaptive control scheme ensures closed-loop stability and asymptotic output tracking. An extension to the adaptive control of general canonical-form MIMO discrete-time nonlinear systems is also presented. Finally, through simulations, the effectiveness of the proposed control scheme is verified.

Robust iterative learning control for linear continuous systems with vector relative degree under varying input trail lengths and random initial state shifts

SummaryFor linear continuous multiple‐input multiple‐output systems with vector relative degree, which operate repetitively with an input‐driven motion and a following autonomous motion over a finite time interval, this article proposes an iteratively moving average operator (IMAO) based D‐type iterative learning control (ILC) method to address the iteratively varying input trail lengths and random initial state shifts. The proposed ILC algorithm has a rectifying action to random initial state shifts at the designed initial time interval. From the terminal time of rectifying action to the end of control input, it makes the ILC tracking error convergent into a bounded range, the bound of which is proportional to the initial state shifts. Specifically, as the initial state shifts remain fixed, the ILC tracking error can be driven to zero at the relevant time interval along the iteration axis. An example manifests the validity of the proposed IMAO based D‐type ILC algorithm.

Zero dynamics assignment and its applications to the stabilization of linear time-varying systems

An algorithm to select an output for desired vector relative degree, which is not restricted to be uniform (the same for all outputs), and zero dynamics in linear time-varying (LTV) multi-input multi-output (MIMO) systems is proposed. The proposed algorithm is developed under reasonable assumptions of uniform (with respect to time) and lexicography-fixed controllability (with some choice of fixed controllability indices) and is inspired by a generalization of the Ackermann’s formula. One application of the proposed algorithm is the context of higher-order sliding-mode control for perturbed LTV MIMO systems, achieving finite-time stability. The developed algorithm also provides an extension of the classical Ackermann–Utkin formula for outputs of uniform relative degree one and includes the design of flat outputs for LTV MIMO systems. Hence other application is the feedback controller design for trajectory tracking task. Finally, a complete HOSM methodology is provided and tested in an LTV model of the non-holonomic car.

A Generalized Minimum Phase Property for Finite-Dimensional Continuous-Time MIMO LTI Systems with Additive Disturbances

In this paper, we further generalize the definition of the extended zero dynamics to finite-dimensional continuous-time MIMO LTI systems with additive disturbances; and then introduce the concept of minimum phase for this class of systems. We show that the extended zero dynamics is invariant under the application of dynamic extension to its input, and therefore, a minimum phase system remains minimum phase after a finite number of steps of dynamic extension. We further introduce the extended zero dynamics canonical form for square MIMO LTI systems with uniform vector relative degree. We prove that a system is minimum phase according to our definition if its zero dynamics is asymptotically stable. The converse of the statement holds if the system is stabilizable from the control input.The objective of this research is to solve the model reference robust adaptive control problem for finite-dimensional continuous-time square MIMO LTI systems that is minimum phase according to the generalized definition using an appropriately vectorized version of Pan and Başar (2000), and results here constitute important building blocks. In a subsequent paper, Başar and Pan (2019), we further connect the dots: starting with a square MIMO LTI system that is minimum phase with respect to admissible initial conditions and admissible disturbance waveforms, we must obtain a true system representation that admits both the extended zero dynamics canonical form representation and the strict observer canonical form representation. Toward this end, we need to be able to extend the given system to one with uniform vector relative degree and with uniform observability indices without changing its minimum phase property, and this has been done in Başar and Pan (2019), fully resolving this issue.

Application of a nonovershooting tracking control method for the Double Buck Converter

The problem of avoiding overshoot in tracking control problems has an extensive history, but only a few works have offered methods that are applicable to a nonlinear plant. The recent paper (Schmid, 2019) adapted some methods from the linear control systems literature to offer state feedback laws to deliver a nonovershooting response in all outputs of a multi-input multi-output feedback linearisable plant.A double-buck converter consists of cascaded buck converters connected to a single voltage supply. Taking the system outputs as the voltages across the two load resistors, we use dynamic averaging to obtain a nonlinear state model for the converter. We introduce suitable coordinates to show it has a well-defined vector relative degree and show the system is feedback linearisable. The methods of (Schmid, 2019) are then employed to obtain a state feedback law that ensures both outputs track arbitrary time-varying reference trajectories without overshoot.

Robust precision attitude tracking of an uncertain rigid spacecraft based on regulation theory

SummaryThis paper explores regulation theory for the design of robust precision attitude tracking of an uncertain rigid spacecraft with external disturbances. Focusing on the attitude system in terms of unit quaternions with an unknown inertia matrix and unmodeled input disturbances, we first introduce specific nonlinear logic such that the resultant error‐quaternion system has an input‐to‐state stability property. Then, we establish an attitude deviation system with an output feedback normal form that has a strict vector relative degree of unity. This enables us to achieve robust output regulation based on an internal model. In particular, we can construct a high‐gain stabilizer for the relevant augmented system. As a major consequence, our study achieves not only precision attitude tracking with exponential convergence but also the input‐to‐state stability disturbance attenuation for the closed‐loop system. Finally, we show extensive simulation and experimental results to illustrate the approach.

A matrix decomposition based adaptive control scheme for a class of MIMO non-canonical approximation systems

This paper presents a unified study on adaptive control of multi-input multi-output (MIMO) uncertain approximation systems in non-canonical forms. For adaptive control of such systems, this paper develops a new state feedback parametrization-based adaptive control scheme, by specifying relative degrees and establishing a normal form, deriving new system reparametrizations needed for adaptive control designs, and constructing adaptive state feedback parametrized controllers for the controlled plant with different vector relative degrees, where the uncertain control gain matrices are handled using a matrix decomposition technique. The main features of such a new control framework are demonstrated by new results for non-canonical MIMO recurrent high-order neural network systems with closed-loop stability and output tracking performance analyzed. An illustrative example with simulation results is presented to show the control design procedure and desired system performance.

Cooperative output regulation for linear uncertain MIMO multi-agent systems by output feedback

In this paper, we study the cooperative output regulation problem for a class of general linear uncertain multi-input multi-output (MIMO) multi-agent systems subject toa well-defined vector relative degree. By proposing some suitable internal model, the problem is first converted into the auxiliary cooperative stabilization problem of the augmented system in the so-called strict feedback normal form. This auxiliary problem is then solved by some developed robust control techniques, such as multiple high-gain feedback and mixed distributed observer, leading to an effective distributed output feedback regulator synthesis for the original cooperative output regulation problem.

Improved nonlinear dynamic inversion control for a flexible air-breathing hypersonic vehicle

This paper presents an improved nonlinear dynamic inversion control approach for the longitudinal dynamics of a flexible air-breathing hypersonic vehicle. The control design of the approach is based on a control-oriented model that represents the nominal state. By establishing a three inputs and three outputs control system, the control-oriented model in this study has full vector relative degree, without dynamic extension. To maintain tracking performance in the presence of disturbances, a nonlinear disturbance observer is adopted to estimate the disturbances. To achieve good transient performance, an adaptive damping term is proposed to the pitch dynamics. Based on approximate input–output linearization, linear control theory is applied to design a pole placement controller for the equivalent linear system. The damping ratio, natural frequency, and simple pole of the pole placement controller are optimized by the genetic algorithm along with the full nonlinear model of the vehicle. During the optimization, 11 uncertain parameters are introduced to the nonlinear model. Monte Carlo evaluation shows that the optimized pole placement controller provides robust tracking of reference trajectories. Simulation results indicate the effectiveness of the proposed control approach.

Data-Efficient Multirobot, Multitask Transfer Learning for Trajectory Tracking

Transfer learning has the potential to reduce the burden of data collection and to decrease the unavoidable risks of the training phase. In this letter, we introduce a multirobot, multitask transfer learning framework that allows a system to complete a task by learning from a few demonstrations of another task executed on another system. We focus on the trajectory tracking problem where each trajectory represents a different task, since many robotic tasks can be described as a trajectory tracking problem. The proposed multirobot transfer learning framework is based on a combined L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> adaptive control and an iterative learning control approach. The key idea is that the adaptive controller forces dynamically different systems to behave as a specified reference model. The proposed multitask transfer learning framework uses theoretical control results (e.g., the concept of vector relative degree) to learn a map from desired trajectories to the inputs that make the system track these trajectories with high accuracy. This map is used to calculate the inputs for a new, unseen trajectory. Experimental results using two different quadrotor platforms and six different trajectories show that, on average, the proposed framework reduces the firstiteration tracking error by 74% when information from tracking a different single trajectory on a different quadrotor is utilized.

Minimum-Rank Dynamic Output Consensus Design for Heterogeneous Nonlinear Multi-Agent Systems

In this paper, we propose a new and systematic design framework for output consensus in heterogeneous Multi-Input Multi-Output (MIMO) general nonlinear Multi-Agent Systems (MASs) subjected to directed communication topology. First, the input-output feedback linearization method is utilized assuming that the internal dynamics is Input-to-State Stable (ISS) to obtain linearized subsystems of agents. Consequently, we propose local dynamic controllers for agents such that the linearized subsystems have an identical closed-loop dynamics which has a single pole at the origin whereas other poles are on the open left half complex plane. This allows us to deal with distinct agents having arbitrarily vector relative degrees and to derive rank-$1$ cooperative control inputs for those homogeneous linearized dynamics which results in a minimum rank distributed dynamic consensus controller for the initial nonlinear MAS. Moreover, we prove that the coupling strength in the consensus protocol can be arbitrarily small but positive and hence our consensus design is non-conservative. Next, our design approach is further strengthened by tackling the problem of randomly switching communication topologies among agents where we relax the assumption on the balance of each switched graph and derive a distributed rank-$1$ dynamic consensus controller. Lastly, a numerical example is introduced to illustrate the effectiveness of our proposed framework.

The Zero Dynamics Form for Nonlinear Differential-Algebraic Systems

We show that any nonlinear differential-algebraic system can be locally transformed into zero dynamics form, which is a normal form with respect to the input-output behavior. Only mild assumptions on the maximal output zeroing submanifold are required and thus the zero dynamics form even generalizes the Byrnes-Isidori form for nonlinear systems with existing vector relative degree. Left- and right-invertibility of the system can be studied in terms of the solution properties of a subsystem in the zero dynamics form. This is the basis for the investigation of various classical control problems, such as output regulation and trajectory tracking.

Feedback Linearization of Continuous and Discrete Multidimensional Systems

The problem of feedback linearization (FL) of continuous and discrete nonlinear MIMO systems is considered. The idea of FL method consists in converting the original nonlinear system into a linear one by means of feedback. Then, the methods of control theory for linear systems are used for system design. A widespread approach to FL design is based on the method of normal form, that uses a nonsingular transformation of system state variables z = T(x). In order to obtain a normal form of the nonlinear system in the neighbor of a some point, it is necessary to determine a special function - the system virtual output, for which a relative degree (in the case of single input) or a vector relative degree (in the case of multiple input) is determined. Applicability of the normal form method for FL is provided by the conditions of controllability and involutivity for the considered nonlinear system, which are not always true. Moreover, when developing a linearizing control law, the main difficulty lies in the transition from transformed variables z to state variables x of the original system. In this paper, we propose another approach, based on representing the original nonlinear system into a state-dependent coefficient form and applying the canonical similarity transformation z = T(x)x, that allow getting the system to canonical form, that considerably simplifies the FL problem. Such similarity transformation allow accomplishing linearization of system without determining of the virtual system output. Another advantage of the proposed method is that the technique of the transition from the transformed variables z to the state variables x of the original system is simpler. The results are illustrated by examples for continuous and discrete nonlinear systems.

Slewing and vibration control of a single-link flexible manipulator using filtered feedback linearization

This article presents the filtered feedback linearization controller for the simultaneous positioning and vibration suppression of a slewing flexible-link manipulator. Filtered feedback linearization requires only limited model information, specifically, knowledge of the vector relative degree and knowledge of the dynamic-inversion matrix, and it makes the [Formula: see text] command, following errors, arbitrarily small despite the presence of unknown disturbances. A closed-loop performance is determined by a parameter gain, which can be viewed as the cut-off frequency of a low-pass filter. For sufficiently large parameter gain, the closed-loop performance is arbitrarily small. Furthermore, we present an adaptive filtered feedback linearization controller, which uses a high-gain adaptive law to increase the controller parameter until the desired performance is achieved. Numerical and experimental results demonstrate the effectiveness of the proposed controller.

Varying trail lengths-based iterative learning control for linear discrete-time systems with vector relative degree

ABSTRACTIn this article, to tackle with the iteration-varying trail lengths and random initial state shifts, an average operator-based PD-type iterative learning control (ILC) law is firstly presented for linear discrete-time multiple-input multiple-output (MIMO) systems with vector relative degree. The proposed PD-type ILC law includes an initial rectifying action against initial state shifts, and pursues the reference trajectory tracking beyond the initial time points. As special cases of the PD-type ILC law, P-type and D-type ILC laws are then introduced. It is proved that for linear discrete-time MIMO systems with vector relative degree, the three proposed ILC laws can drive the varying trail lengths-based ILC tracking errors to zero in mathematical expectation beyond the initial time points. A numerical example is used to illustrate the effectiveness of the proposed ILC laws.

Adaptive robust fault-tolerant control for linear MIMO systems with unmatched uncertainties

ABSTRACTIn this paper, two novel fault-tolerant control design approaches are proposed for linear MIMO systems with actuator additive faults, multiplicative faults and unmatched uncertainties. For time-varying multiplicative and additive faults, new adaptive laws and additive compensation functions are proposed. A set of conditions is developed such that the unmatched uncertainties are compensated by actuators in control. On the other hand, for unmatched uncertainties with their projection in unmatched space being not zero, based on a (vector) relative degree condition, additive functions are designed to compensate for the uncertainties from output channels in the presence of actuator faults. The developed fault-tolerant control schemes are applied to two aircraft systems to demonstrate the efficiency of the proposed approaches.