Maryamsadat Tahavori

Frequency-Interval Model Reduction of Bilinear Systems

In this technical note, a new method for model reduction of bilinear systems is presented. The method is developed in particular for many applications in which one is interested to approximate a system in a given frequency-interval. To this end, new generalized frequency-interval gramians are introduced for bilinear systems. It is shown that these gramians are the solutions to the so-called frequency-interval generalized Lyapunov equations. The conditions for these equations to be solvable are derived and an algorithm is proposed to solve such equations iteratively. The method is further illustrated with the help of an illustrative example. The numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians.

Optimal Sensor and Actuator Location for Unstable Systems

Accurate and reliable process measurements play a key role in the success of process control. In addition to selecting the instrumentation hardware, it is important to determine which variables should be measured in a process. Therefore it is important to study techniques for placing sensors on the processes. Dually the problem of placing actuators on the processes is equally important. In this paper, the problem of determining optimal sensor and actuator locations for the linear systems is addressed. The problem of the sensor locations is viewed as the problem of maximizing the output energy generated by a given state and for the actuator locations is viewed as the problem of minimizing the input energy required to reach a given state. Such design problems occur in many applications, and therefore have been studied extensively. Unfortunately, the results in this context, which have been proposed so far, only support stable systems. However, in industrial practices it is often the case that the system, which is needed to be controlled, is not stable. The method which is proposed in this paper is a general method in the sense that it supports both stable and unstable systems. The technique is successfully used for determining the optimal sensor locations of the linearized Continuous Stirred-Tank Reactor unstable model.

Control configuration selection for bilinear systems via generalised Hankel interaction index array

Decentralised and partially decentralised control strategies are very popular in practice. To come up with a suitable decentralised or partially decentralised control structure, it is important to select the appropriate input and output pairs for control design. This procedure is called control configuration selection. It is well known that a suitable control configuration selection is an important prerequisite for a successful industrial control. In this paper the problem of control configuration selection for multiple-input and multiple-output (MIMO) bilinear processes is addressed. First, the concept of the cross-gramian is developed for bilinear systems. The conditions for the existence of generalised cross-gramian are derived. It is shown that if the cross-gramian exists it is the solution to the generalised Sylvester equation. To obtain the cross-gramian in a more computationally efficient way, an iterative method for solving the generalised Sylvester equation is proposed. The generalised cross-gramian is used to form the generalised Hankel interaction index array. The generalised Hankel interaction index array is used for control configuration selection of MIMO bilinear processes. Most of the results on control configuration selection, which have been proposed so far, can only support linear systems. The proposed method supports bilinear processes, takes the effects of dynamics of the process into account and can be used to propose a richer (sparse or block diagonal) controller structure. More importantly, since for each element of generalised Hankel interaction index array just one generalised Sylvester equation is needed to be solved, the proposed control configuration selection method is computationally more efficient than its gramian-based counterparts.

Time-interval model reduction of bilinear systems

In this paper, a new method for model reduction of bilinear systems is presented. The proposed technique is from the family of gramian-based model reduction methods. The method uses time-interval generalised gramians in the reduction procedure rather than the ordinary generalised gramians and in such a way that it improves the accuracy of the approximation within the time-interval for which the method is applied. The time-interval generalised gramians are the solutions to the generalised time-interval Lyapunov equations. The conditions for these equations to be solvable are derived and an algorithm is proposed to solve these equations iteratively. The method is further illustrated with the help of two illustrative examples. The numerical results show that the method is more accurate than its previous counterpart which is based on the ordinary gramians.

Model reduction via time-interval balanced stochastic truncation for linear time invariant systems

In this article, a new method for model reduction of linear dynamical systems is presented. The proposed technique is from the family of gramian-based relative error model reduction methods. The method uses time-interval gramians in the reduction procedure rather than ordinary gramians and in such a way it improves the accuracy of the approximation within the time interval which is applied. It is proven that the reduced order model is stable when the proposed method applies to a stable system. The method uses a recently proposed inner–outer factorisation algorithm which enhances the numerical accuracy and efficiency. In order to avoid numerical instability and also to further increase the numerical efficiency, projector matrices are constructed instead of the similarity transform approach for reduction. The method is illustrated by a numerical example and finally it is applied to a practical CD player example. The numerical results show that the method is more accurate than ordinary balanced stochastic truncation.

Stability analysis for a class of discrete-time two-dimensional nonlinear systems

Stability analysis for a class of discrete-time two-dimensional nonlinear systems is addressed in this paper. A linear matrix inequality based sufficient condition for asymptotic stability is proposed. This condition is an analogous counterpart for two-dimensional linear systems which is shown to be easily verifiable and suitable for design problems. The result is illustrated by a numerical example.

Time-weighted balanced stochastic model reduction

A new relative error model reduction technique for linear time invariant (LTI) systems is proposed in this paper. Both continuous and discrete time systems can be reduced within this framework. The proposed model reduction method is mainly based upon time-weighted balanced truncation and a recently developed inner-outer factorization technique. Compared to the other analogous counterparts, the proposed method shows to provide more accurate results in terms of time weighted norms, when applied to different practical examples. The results are further illustrated by a numerical example.

Generalized Hankel Interaction Index Array for control structure selection for discrete-time MIMO bilinear processes and plants

The control technology has been orientated towards decentralized and partially decentralized control strategies. To ensure the success of a decentralized or a partially decentralized control in practice, the first necessary step is to determine a suitable control structure. The control structure selection which is the task of selecting suitable input and output pairs for control design is therefore very important. This paper focuses on control structure selection for discrete-time MIMO bilinear systems. The generalized cross-gramian is introduced in this paper for discrete-time bilinear processes and plants. The existence of the generalized cross-gramian is studied and it is shown that if the cross-gramian exists, it can be obtained by solving a generalized Sylvester equation. To solve the generalized Sylvester equation in a computationally efficient way, an iterative method is developed and presented. The generalized cross-gramians are computed for all SISO subsystems of the discrete-time MIMO bilinear systems. These gramians are used to build the generalized Hankel Interaction Index Array which is used for control structure selection. The proposed method for control structure selection is among the few methods supporting bilinear processes and plants, enjoys the advantages of gramian based methods and it is more efficient in terms of computations compared to its counterparts.

Toward model-based control of non-linear hydraulic networks

Water leakage is an important component of water loss. Many methods have emerged from urban water supply systems (WSSs) for leakage control, but it still remains a challenge in many countries. Pressure management is an effective way to reduce the leakage in a system. It can also reduce the power consumption. To have a better understanding of leakage in WSSs, to control pressure and leakage effectively, and for optimal design of WSSs, suitable modeling is an important prerequisite. In this paper a model with the main objective of pressure control and consequently leakage reduction is presented. Following an analogy to electric circuits, first the mathematical expression for pressure drop over each component of the pipe network (WSS) such as pipes, pumps, valves and water towers is presented. Then the network model is derived based on the circuit theory and subsequently used for pressure management in the system. A suitable projection is used to reduce the state vector and to express the model in standard state-space form.

Relative Error Model Reduction via Time-Weighted Balanced Stochastic Singular Perturbation

A new mixed method for relative error model reduction of linear time-invariant systems is proposed in this paper. This order reduction technique is mainly based upon the time-weighted balanced stochastic model reduction method and singular perturbation model reduction technique. Compared to the other analogous counterparts, the proposed method provides more accurate results in terms of time-weighted norms when applied to the practical examples. It is shown that important properties of the time-weighted stochastic balanced reduction technique are extended to the mixed reduction method by using the concept and properties of the reciprocal systems. The results are further illustrated by two practical numerical examples: a model of compact disc player and a model of the atmospheric storm track.