Time-delay estimation for nonlinear systems with piecewise-constant input

We consider a general nonlinear time-delay system in which the input signal is piecewise-constant. Such systems arise in a wide range of industrial applications, including evaporation and purification processes and chromatography. We assume that the time-delays—one involving the state variables and the other involving the input variables—are unknown and need to be estimated using experimental data. We formulate the problem of estimating the unknown delays as a nonlinear optimization problem in which the cost function measures the least-squares error between predicted and measured system output. The main difficulty with this problem is that the delays are decision variables to be optimized, rather than fixed values. Thus, conventional optimization techniques are not directly applicable. We propose a new computational approach based on a novel algorithm for computing the cost function’s gradient. We then apply this approach to estimate the time-delays in two industrial chemical processes: a zinc sulphate purification process and a sodium aluminate evaporation process

Computational methods for solving optimal industrial process control problems

In this thesis, we develop new computational methods for three classes of dynamic optimization problems: (i) A parameter identification problem for a general nonlinear time-delay system; (ii) an optimal control problem involving systems with both input and output delays, and subject to continuous inequality state constraints; and (iii) a max-min optimal control problem arising in gradient elution chromatography.In the first problem, we consider a parameter identification problem involving a general nonlinear time-delay system, where the unknown time delays and system parameters are to be identified. This problem is posed as a dynamic optimization problem, where its cost function is to measure the discrepancy between predicted output and observed system output. The aim is to find unknown time-delays and system parameters such that the cost function is minimized. We develop a gradient-based computational method for solving this dynamic optimization problem. We show that the gradients of the cost function with respect to these unknown parameters can be obtained via solving a set of auxiliary time-delay differential systems from t = 0 to t = T. On this basis, the parameter identification problem can be solved as a nonlinear optimization problem and existing optimization techniques can be used. Two numerical examples are solved using the proposed computational method. Simulation results show that the proposed computational method is highly effective. In particular, the convergence is very fast even when the initial guess of the parameter values is far away from the optimal values.Unlike the first problem, in the second problem, we consider a time delay identification problem, where the input function for the nonlinear time-delay system is piecewise-constant. We assume that the time-delays—one involving the state variables and the other involving the input variables—are unknown and need to be estimated using experimental data. We also formulate the problem of estimating the unknown delays as a nonlinear optimization problem in which the cost function measures the least-squares error between predicted output and measured system output. This estimation problem can be viewed as a switched system optimal control problem with time-delays. We show that the gradient of the cost function with respect to the unknown state delay can be obtained via solving a auxiliary time-delay differential system. Furthermore, the gradient of the cost function with respect to the unknown input delay can be obtained via solving an auxiliary time-delay differential system with jump conditions at the delayed control switching time points. On this basis, we develop a heuristic computational algorithm for solving this problem using gradient based optimization algorithms. Time-delays in two industrial processes are estimated using the proposed computational method. Simulation results show that the proposed computational method is highly effective.For the third problem, we consider a general optimal control problem governed by a system with input and output delays, and subject to continuous inequality constraints on the state and control. We focus on developing an effective computational method for solving this constrained time delay optimal control problem. For this, the control parameterization technique is used to approximate the time planning horizon [0, T] into N subintervals. Then, the control is approximated by a piecewise constant function with possible discontinuities at the pre-assigned partition points, which are also called the switching time points. The heights of the piecewise constant function are decision variables which are to be chosen such that a given cost function is minimized. For the continuous inequality constraints on the state, we construct approximating smooth functions in integral form. Then, the summation of these approximating smooth functions in integral form, which is called the constraint violation, is appended to the cost function to form a new augmented cost function. In this way, we obtain a sequence of approximate optimization problems subject to only boundedness constraints on the decision variables. Then, the gradient of the augmented cost function is derived. On this basis, we develop an effective computational method for solving the time-delay optimal control problem with continuous inequality constraints on the state and control via solving a sequence of approximate optimization problems, each of which can be solved as a nonlinear optimization problem by using existing gradient-based optimization techniques. This proposed method is then used to solve a practical optimal control problem arising in the study of a real evaporation process. The results obtained are highly satisfactory, showing that the proposed method is highly effective.The fourth problem that we consider is a max-min optimal control problem arising in the study of gradient elution chromatography, where the manipulative variables in the chromatographic process are to be chosen such that the separation efficiency is maximized. This problem has three non-standard characteristics: (i) The objective function is nonsmooth; (ii) each state variable is defined over a different time horizon; and (iii) the order of the final times for the state variable, the so-called retention times, are not fixed. To solve this problem, we first introduce a set of auxiliary decision variables to govern the ordering of the retention times. The integer constraints on these auxiliary decision variables are approximated by continuous boundedness constraints. Then, we approximate the control by a piecewise constant function, and apply a novel time-scaling transformation to map the retention times and control switching times to fixed points in a new time horizon. The retention times and control switching times become decision variables in the new time horizon. In addition, the max-min objective function is approximated by a minimization problem subject to an additional constraint. On this basis, the optimal control problem is reduced to an approximate nonlinear optimization problem subject to smooth constraints, which is then solved using a recently developed exact penalty function method. Numerical results obtained show that this approach is highly effective.Finally, some concluding remarks and suggestions for further study are made in the conclusion chapter

NGMV control of delayed piecewise affine systems

A Nonlinear Generalized Minimum Variance (NGMV) control algorithm is introduced for the control of piecewise affine (PWA) systems. Under some conditions, discrete-time PWA systems can be transferred into an equivalent state-dependent nonlinear system form. The equivalent state-dependent systems maintain the hybrid nature of the original PWA systems and include both the discrete and continuous signals in one general description. In a more general way, the process is assumed to include common delays in input or output channels of magnitude k. Then the NGMV control strategy [1] can be applied. The NGMV controller is related to a well-known and accepted solution for time delay systems (Smith Predictor) but has the advantage that it may stabilize open-loop unstable processes [2]

Reliable H∞ control for discrete-time piecewise linear systems with infinite distributed delays

In this paper, the reliable H∞ control problem is investigated for discrete-time piecewise linear systems with time delays and actuator failures. The time delays are assumed to be infinitely distributed in the discrete-time domain, and the possible failure of each actuator is described by a variable varying in a given interval. The aim of the addressed reliable H∞ control problem is to design a controller such that, for the admissible infinite distributed delays and possible actuator failures, the closed-loop system is exponentially stable with a given disturbance attenuation level γ. The controller gain is characterized in terms of the solution to a linear matrix inequality that can be easily solved by using standard software packages. A simulation example is exploited in order to illustrate the effectiveness of the proposed design procedures

Modeling of complex-valued Wiener systems using B-spline neural network

In this brief, a new complex-valued B-spline neural network is introduced in order to model the complex-valued Wiener system using observational input/output data. The complex-valued nonlinear static function in the Wiener system is represented using the tensor product from two univariate Bspline neural networks, using the real and imaginary parts of the system input. Following the use of a simple least squares parameter initialization scheme, the Gauss–Newton algorithm is applied for the parameter estimation, which incorporates the De Boor algorithm, including both the B-spline curve and the first-order derivatives recursion. Numerical examples, including a nonlinear high-power amplifier model in communication systems, are used to demonstrate the efficacy of the proposed approaches

Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument

We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.Comment: 24 pages, 1 figur

Constrained Nonlinear Model Predictive Control of an MMA Polymerization Process via Evolutionary Optimization

In this work, a nonlinear model predictive controller is developed for a batch polymerization process. The physical model of the process is parameterized along a desired trajectory resulting in a trajectory linearized piecewise model (a multiple linear model bank) and the parameters are identified for an experimental polymerization reactor. Then, a multiple model adaptive predictive controller is designed for thermal trajectory tracking of the MMA polymerization. The input control signal to the process is constrained by the maximum thermal power provided by the heaters. The constrained optimization in the model predictive controller is solved via genetic algorithms to minimize a DMC cost function in each sampling interval.Comment: 12 pages, 9 figures, 28 reference