Set Membership Identification Research Articles

Stability Guaranteed Model Predictive Control With Adaptive Lyapunov Constraint

This paper aims to design a stabilized Adaptive Model Predictive Control (MPC) for nonlinear continuous systems with unknown dynamics at the start time. The proposed method is based on the Lyapunov MPC (LMPC) where the derivative of the Lyapunov function of the closed-loop system is limited to be less than that under an auxiliary Lyapunov controller. Different from the existing LMPC methods, this paper provides an online constraint updating strategy, which gradually reduces the conservation of the constraint. Specifically, the piece-wise linear (PWL) model updated by the Set-Membership (SM) identification method is adopted in the constraint and the auxiliary controller design, which can result in a non-increasing boundary of the model error. Then, a sufficient condition that guarantees the stabilization of the closed-loop system is deduced, which takes the effect of sample-and-hold implementation and the error boundary of the PWL model into account. By this condition, an optimization problem to update the Lyapunov controller for relaxing the constraints of MPC is proposed. We prove that, by the proposed method, the states will eventually converge to a small region around the equilibrium, both this small region and the conservatism of MPC decrease with the increasing of the accuracy of the PWL model. The application of the proposed method to a chemical process demonstrates the effectiveness of the proposed method. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The control problem of nonlinear continuous systems with unknown dynamics has attracted the attention of researchers. This article develops an Adaptive MPC for this kind of system, which updates the boundary of uncertainties according to the measurement, and updates the constraints of MPC according to the updated uncertainty boundary. It provides a guaranteed stabilization and continuously relaxed constraint design for MPC where machine learning-based models are employed to predict the future trajectory. In the process of implementation, the method solves two optimization problems: one is to get the parameters of constraints which is based on the PWL model, and the other is used to optimize the future trajectory with the updated constraints. To proceed, the initial boundary of the uncertainty should be given at the initial instant, the initialization of PWL models and constraints can be configured based on it. The proposed method can be used for tracking control of chemical processes, power systems, robots, etc., where the accurate dynamics are not provided in advance and an improved performance is required.

Virtual reference feedback tuning for linear discrete-time systems with robust stability guarantees based on set membership

In this paper we propose a novel methodology that allows us to design, in a purely data-based fashion and for linear single-input and single-output systems, both robustly stable and performing control systems for tracking piecewise constant reference signals. The approach uses both (i) virtual reference feedback tuning for enforcing suitable performance and (ii) the set membership framework for providing a-priori robust stability guarantees. Indeed, an uncertainty set for the system parameters is obtained through set membership identification, where an algorithm based on the scenario approach is proposed to estimate the inflation parameter in a probabilistic way. Based on this set, robust stability conditions are enforced as linear matrix inequality constraints within an optimization problem whose linear cost function relies on virtual reference feedback tuning. To show the generality and effectiveness of our approach, we apply it to two of the most widely used yet simple control schemes, i.e., where tracking is achieved thanks to (i) a static feedforward action and (ii) an integrator in closed-loop.The proposed method is not direct due to the set membership identification. However, the uncertainty set is used with the objective of providing robust stability guarantees for the closed-loop system and it is not used for the definition of the cost function, which instead is based on data. The effectiveness of the method is shown with reference to three simulation examples. A comparison with other data-driven methods is carried out.

Computation of Input Disturbance Sets for Constrained Output Reachability

Linear models with additive unknown-but-bounded input disturbances are extensively used to model uncertainty in robust control systems design. Typically, the disturbance set is either assumed to be known a priori or estimated from data through set-membership identification. However, the problem of computing a suitable input disturbance set in case the set of possible output values is assigned a priori has received relatively little attention. This problem arises in many contexts, such as in supervisory control, actuator design, decentralized control, and others. In this paper, we propose a method to compute input disturbance sets (and the corresponding set of states) such that the resulting set of outputs matches as closely as possible a given set of outputs, while additionally satisfying strict (inner or outer) inclusion constraints. We formulate the problem as an optimization problem by relying on the concept of robust invariance. The effectiveness of the approach is demonstrated in numerical examples that illustrate how to solve safe reference set and input-constraint set computation problems.

An Explicit Dual Control Approach for Constrained Reference Tracking of Uncertain Linear Systems

A finite horizon optimal tracking problem is considered for linear dynamical systems subject to parametric uncertainties in the state-space matrices and exogenous disturbances. A suboptimal solution is proposed using a model predictive control (MPC) based explicit dual control approach which enables active uncertainty learning. A novel algorithm for the design of robustly invariant online terminal sets and terminal controllers is presented. Set membership identification is used to update the parameter uncertainty online. A predicted worst-case cost is used in the MPC optimization problem to model the dual effect of the control input. The cost-to-go is estimated using contractivity of the proposed terminal set and the remaining time horizon, so that the optimizer can estimate future benefits of exploration. The proposed dual control algorithm ensures robust constraint satisfaction and recursive feasibility, and navigates the exploration-exploitation trade-off using a robust performance metric.

Dual adaptive MPC using an exact set-membership reformulation

Adaptive model predictive control (MPC) methods using set-membership Identification to reduce parameter uncertainty are considered in this work. Strong duality is used to reformulate the set-membership equations exactly within the MPC optimization. A predicted worst-case cost is then used to enable performance-oriented exploration. The proposed approach guarantees robust constraint satisfaction and recursive feasibility. It is shown that method can be implemented using homothetic tube and flexible tube parameterizations of state tubes, and a simulation study demonstrates performance improvement over state-of-the-art controllers.

Adaptive dual model predictive control for linear systems with parametric uncertainties

Adaptive dual model predictive control (DMPC) for linear systems with constant parametric uncertainties is investigated in this paper. In particular, chance constraints on output and input are considered. The online set-membership identification and recursive least squares are utilized for shrinking the uncertain parameter set and getting the estimation point of the parameters. The dual effect represented by the parameters' prediction error is considered in the receding horizon optimization problem for easing the impact of the uncertainties and improving the performance simultaneously. Chance constraints on output are tackled by converting into convex alternatives. The output tracking problem is transformed into a quadratically constrained quadratic-programming (QCQP) problem, which is computationally tractable. A numerical example is provided to illustrate the effectiveness of the proposed method.

Adaptive Data-Driven Control for Linear Time Varying Systems

In this paper, we propose an adaptive data-driven control approach for linear time varying systems, affected by bounded measurement noise. The plant to be controlled is assumed to be unknown, and no information in regard to its time varying behaviour is exploited. First, using set-membership identification techniques, we formulate the controller design problem through a model-matching scheme, i.e., designing a controller such that the closed-loop behaviour matches that of a given reference model. The problem is then reformulated as to derive a controller that corresponds to the minimum variation bounding its parameters. Finally, a convex relaxation approach is proposed to solve the formulated controller design problem by means of linear programming. The effectiveness of the proposed scheme is demonstrated by means of two simulation examples.

A unified framework for the identification of a general class of multivariable nonlinear block‐structured systems

AbstractThis article addresses the parametric identification of block‐structured nonlinear systems in a general form, characterized by the feedback interconnection of a multivariable linear system and a static multivariate nonlinear map. We assume that both the input and the output collected data are affected by bounded noise, which casts the problem in the context of set‐membership (SM) errors‐in‐variables identification. We introduce a single‐stage SM identification algorithm for the computation of the parameter uncertainty intervals. The proposed solution exploits the formulation of a suitable optimization problem solved through convex relaxation techniques. Numerical simulations and an experimental test show the effectiveness of the proposed approach.

3D Model Identification of a Soft Robotic Neck

Soft robotics is becoming an emerging solution to many of the problems in robotics, such as weight, cost and human interaction. In order to overcome such problems, bio-inspired designs have introduced new actuators, links and architectures. However, the complexity of the required models for control has increased dramatically and geometrical model approaches, widely used to model rigid dynamics, are not enough to model these new hardware types. In this paper, different linear and non-linear models will be used to model a soft neck consisting of a central soft link actuated by three motor-driven tendons. By combining the force on the different tendons, the neck is able to perform a motion similar to that of a human neck. In order to simplify the modeling, first a system input–output redefinition is proposed, considering the neck pitch and roll angles as outputs and the tendon lengths as inputs. Later, two identification strategies are selected and adapted to our case: set membership, a data-driven, nonlinear and non-parametric identification strategy which needs no input redefinition; and Recursive least-squares (RLS), a widely recognized identification technique. The first method offers the possibility of modeling complex dynamics without specific knowledge of its mathematical representation. The selection of this method was done considering its possible extension to more complex dynamics and the fact that its impact in soft robotics is yet to be studied according to the current literature. On the other hand, RLS shows the implication of using a parametric and linear identification in a nonlinear plant, and also helps to evaluate the degree of nonlinearity of the system by comparing the different performances. In addition to these methods, a neural network identification is used for comparison purposes. The obtained results validate the modeling approaches proposed.

Adaptive interpolating control for constrained systems with parametric uncertainty and disturbances

SummaryAn adaptive interpolating control (AIC) algorithm is proposed for constrained linear systems with parametric uncertainty and additive disturbance. This adaptive algorithm consists of an iterative set membership identification algorithm, which updates the uncertain parameter set at each time step, and an interpolating controller, which robustly stabilizes the uncertain system with state and input constraints. We prove that the AIC algorithm is recursively feasible and guarantees robust constraint satisfaction and robust asymptotic stability of the closed‐loop system in the presence of uncertainties. Moreover, we detail two possible extensions of the AIC algorithm: (a) persistent excitation conditions can be embedded into the AIC algorithm to accelerate the convergence of system parameters and (b) the combination of the AIC algorithm and aggressive learning is able to enlarge the size of the feasible region with every iteration by exploiting information from previous iterations. We illustrate the effectiveness of the proposed algorithms through comparisons with adaptive model predictive control and one example of mobile carrier robot.

Design of experiments for nonlinear system identification: A set membership approach

Design of Experiments (DoE) is an important step in system identification. Regardless of the chosen model structure and identification method, the DoE quality determines an upper bound on the accuracy of the identified models. One of the greatest challenges in this context is to design an experiment which gives the maximum information about the dynamics of the system of interest. In this paper, a novel DoE algorithm for input-constrained MISO nonlinear systems, based on set membership identification, is proposed. The DoE algorithm is aimed to minimize the so-called radius of information, a quantity giving the worst-case model error. Two numerical examples are presented, showing the effectiveness of the approach and its potential in view of real-world applications.

One-shot set-membership identification of Generalized Hammerstein–Wiener systems

In this paper we consider the identification of Generalized Hammerstein–Wiener systems in the presence of bounded noise affecting both the input and the output sequences. The considered class of systems is characterized by the cascade connection of Hammerstein building blocks, where the nonlinear static input map is a polynomial, which is not assumed to be invertible. The problem of computing the parameter uncertainty intervals is formulated in terms of a set of suitable semialgebraic optimization problems, where both the parameter to be estimated and the inner unmeasurable signals connecting all the subsystems are considered as decision variables. The effectiveness of the proposed algorithm is shown by means of a numerical simulation example. The presented approach is also applied to the problem of identifying the mathematical model of a JFET amplifier from a set of experimentally collected data.

Set Membership Identification of Linear Systems With Guaranteed Simulation Accuracy

The problem of model identification for linear systems is considered, using a finite set of sampled data affected by a bounded measurement noise, with unknown bound. The objective is to identify one-step-ahead models and their accuracy in terms of worst-case simulation error bounds. To do so, the Set Membership identification framework is exploited. Theoretical results are derived, allowing one to estimate the noise bound and system decay rate. Then, these quantities and the data are employed to define the Feasible Parameter Set (FPS), which contains all possible models compatible with the available information. Here, the estimated decay rate is used to refine the standard FPS formulation, by adding constraints that enforce the desired converging behavior of the models' impulse response. Moreover, guaranteed simulation error bounds for an infinite future horizon are derived, improving over recent results pertaining to finite simulation horizon only. These bounds are the basis for a result and method to guarantee asymptotic stability of the identified model. Finally, the desired one-step-ahead model is identified by means of numerical optimization, and the related simulation error bounds are evaluated. Both input-output and state-space model structures are addressed. The approach is showcased on a numerical example and on real-world experimental data of the roll rate dynamics of an autonomous glider.

A novel set membership fast NLMS algorithm for acoustic echo cancellation

In order to improve the performances of the Fast Normalized Least Mean Square type (FNLMS) adaptive filtering algorithm in the context of Acoustic Echo Cancellation (AEC), this work proposes an Improved Set Membership version by exploiting, firstly, the theory of set membership identification (SMI) to the FNLMS algorithm to get more complexity reduction, then by using the estimated output error to update the step size which results to better convergence speed and tracking ability. The obtained algorithm is called the Improved Set Membership FNLMS (ISM-FNLMS) algorithm. Simulation results have demonstrated the superiority and the good performances of the proposed algorithm compared with NLMS, SM-NLMS, FNLMS, SM-FNLMS and SM-NLMS with Robust Error Bound (SMREB-NLMS) algorithms in terms of convergence speed, steady-state mean square filtering error (MSE), tracking capability and computational complexity performances.

Robust Adaptive Model Predictive Control with Worst-Case Cost

Abstract A robust adaptive model predictive control (MPC) algorithm is presented for linear, time invariant systems with unknown dynamics and subject to bounded measurement noise. The system is characterized by an impulse response model, which is assumed to lie within a bounded set called the feasible system set. Online set-membership identification is used to reduce uncertainty in the impulse response. In the MPC scheme, robust constraints are enforced to ensure constraint satisfaction for all the models in the feasible set. The performance objective is formulated as a worst-case cost with respect to the modeling uncertainties. That is, at each time step an optimization problem is solved in which the control input is optimized for the worst-case plant in the uncertainty set. The performance of the proposed algorithm is compared to an adaptive MPC algorithm from the literature using Monte-Carlo simulations.

Adaptive MPC for constrained systems with parameter uncertainty and additive disturbance

In this study, the authors propose an adaptive model predictive control (MPC) algorithm for constrained linear systems in state space subject to uncertain model parameters and disturbances. An iterative set membership identification algorithm is first presented to update the uncertain parameter set at each time step. Based on the shrunken uncertain parameter set, an MPC controller is then designed to robustly stabilise the uncertain systems subject to state and input constraints. The algorithm can efficiently reduce the size of the uncertain parameter set in min-max MPC setting, and therefore improve the control performance. The algorithm is proved to ensure constraint satisfaction, recursive feasibility and input-to-state practical stability of the closed-loop system even in the presence of system uncertainties. A numerical example and a brief comparison with traditional min-max MPC are provided to demonstrate the efficiency of the proposed algorithm.

Adaptive model predictive control for linear time varying MIMO systems

A robust, adaptive Model Predictive Control (MPC) approach for asymptotically stable, constrained linear time-varying (LTV) systems with multiple inputs and outputs is proposed. The approach consists of two-steps, carried out on-line with a receding horizon strategy. In the first one, a real-time Set Membership identification algorithm exploits the measured input–output data and the available prior knowledge to build and refine a set of admissible models of the plant (Feasible Parameter Set, FPS). This set is guaranteed to contain also the true system dynamics under the considered working assumptions. In the second step, a robust finite-horizon optimal control problem is formulated and solved. The variation of system dynamics is taken into account by inflating the FPS over the prediction horizon, according to worst-case bounds, assumed a priori, on the parameters’ rate of change. The resulting optimal control sequence guarantees that the outputs of all possible plants inside the FPS satisfy the operational constraints, also considering all possible future parameter changes. The main theoretical properties of the proposed approach are demonstrated and the method is showcased in numerical simulations, highlighting the fundamental improvement over previous approaches not designed for LTV systems.

A set-membership algorithm based parameter identification method for lithium-ion batteries

The accuracy of lithium-ion battery model is one of the most important factors that affects the applicability of power battery in electrical vehicles. Based on the traditional forgetting factor recursive least square (FFRLS) method, the random noise should be subjected to the normal distribution of zero mean and zero covariance, which, however, is very difficult to be satisfied in practical application. In this paper, based on the first-order RC equivalent circuit model, the identification of lithium-ion battery model parameters is performed by using the set-membership identification algorithm with unknown but bounded noise. The model parameters are identified by the set-membership algorithm with the experimental data of UDDS test on the NCM battery module. Experiments and simulation results show that the new method can simulate the dynamics of battery well, it can keep terminal voltage error within 1%, alongside with the root mean square error(RMSE) improved up to 8% compared with the FFRLS, which verifies the feasibility and the effectiveness of the new method, as well as providing data support for accurate estimation of battery state.

Parameter identification for structural dynamics based on interval analysis algorithm

A parameter identification method using interval analysis algorithm for structural dynamics is presented in this paper. The proposed uncertain identification method is investigated by using central difference method and ARMA system. With the help of the fixed memory least square method and matrix inverse lemma, a set-membership identification technology is applied to obtain the best estimation of the identified parameters in a tight and accurate region. To overcome the lack of insufficient statistical description of the uncertain parameters, this paper treats uncertainties as non-probabilistic intervals. As long as we know the bounds of uncertainties, this algorithm can obtain not only the center estimations of parameters, but also the bounds of errors. To improve the efficiency of the proposed method, a time-saving algorithm is presented by recursive formula. At last, to verify the accuracy of the proposed method, two numerical examples are applied and evaluated by three identification criteria respectively.

Set-membership errors-in-variables identification of MIMO linear systems

In this paper, we consider the problem of set-membership identification of multiple-input multiple-output (MIMO) linear models when both input and output measurements are affected by bounded additive noise. Firstly, we propose a general formulation that allows the user to take into account possible a-priori information on the structure of the MIMO model to be identified. Then, we formulate the problem in terms of a suitable polynomial optimization problem that is solved by means of a convex relaxation approach. To show the effectiveness of the proposed approach, we test the original MIMO identification algorithm on a simulation example, as well as on a set of input–output experimental data, collected on a multiple-input multiple-output electronic process simulator.