Reduced modelling and fixed‐order control of delay systems applied to a heat exchanger

Abstract

This paper presents an integrated approach for designing low-order multi-objective controllers for linear time-delay systems, combining recently developed methods for reduction of delay systems and fixed-order control design, respectively. First, as a benchmark problem for process control applications, a model of an experimental heat transfer setup is discussed in detail, which serves to motivate the adopted combination of model reduction and control technique. The former corresponds to a Krylov based reduction procedure, which is generalized to multiple-input-multiple-output systems with state, input and output delays, and allows the adaptive construction of an accurate low-order linear time-invariant approximation of the original linear time-delay system. Concerning the latter, a fixed-order controller is synthesized for the delay-free approximation, exploiting a recently proposed linear matrix inequality based framework for fixed-order controller design, and validated on the original linear time-delay system. In this way, the systematic overall design procedure, which is grounded in convex optimization, complements approaches based on directly optimizing stability and performance measures as a function of the controller parameters, which may lead to highly nonconvex and even non-smooth optimization problems. The successful design of a fixed-order multi-H 2 controller is validated on the benchmark problem, confirming the potential of the adopted approach for realistic industrial applications. IntroductionWe consider continuous-time models for control systems, described in terms of delay differential equations of the form(1) i = 1, . . . , my, where x(t) ∈ R n is the state, u i (t) ∈ R, i = 1, . . . , mu are the inputs, and y i (t) ∈ R, i = 1, . . . , my correspond to the outputs at time t. The quantities τ i , i = 1, . . . , mx, µ i , i = 1, . . . , mu and ν i , i = 1, . . . , my represent time-delays in the system's state, inputs and outputs, respectively, where the largest state delay is denoted by τmax = max i∈{1,...,mx} τ i . The inputs u i and outputs y i are grouped in vectors as follows:where we assume that the number of scalar inputs does not exceed the dimension of the system state, i.e., mu ≤ n. Without loss of generality, the input u and output y are subdivided asto distinguish between exogenous inputs u 1 and control inputs u 2 on the one hand, and regulated outputs y 1 and measured outputs y 2 on the other hand. The targeted class of delay systems described by (1) results from the physics based modeling of complex interconnections of components, where time-delay elements naturally appear in modeling propagation phenomena. The latter are due to the fact that the transfer of material, energy and information is mostly not instantaneous. For this class of systems, the experimental heat-transfer setup described in Section 3 serves as a benchmark problem for the control design. An analogous model has been derived in [25] for a greenhouse. More applications include chemical reactors [26], plate...

Further experimental results on modelling and algebraic control of a delayed looped heating-cooling process under uncertainties

“…[ 27 ]. A similar modelling idea was applied to a looped experimental heat transfer setup followed by a Krylov-based model order reduction procedure [ 28 ]. A very close appliance model was also used in Ref.…”

“…A fixed-order controller for a reduced-order delayed model of a looped HX laboratory process was derived and tested in Ref. [ 28 ]. The controller was tuned via the solution of a specific H 2 -norm optimization problem.…”

Parametric system identification and robust controller design for liquid–liquid heat exchanger system

“…Considering the industrial heat exchanger system as an infinite‐dimension system, proper model reduction technique is used to obtain a fixed‐dimensional linear time invariant(LTI) system. A fixed‐order controller is designed for a fixed‐dimensional LTI model of the heat exchanger system in [14]. Some standard textbooks of the control system have provided a lumped parameter model of the heat exchanger system and approximated the model to a first‐order process with dead time with acceptable assumptions [15, 16].…”

Design of robust decentralised controllers for MIMO plants with delays through network structure exploitation

“…Time-delays are present in the system model as the transfer of energy, material or information is usually not instantaneous. They appear, for instance, as computation and communication lags, they model the transport phenomena and heredity, and arise as feedback delays in control loops (Michiels, Hilhorst, Pipeleers, Vyhlídal, & Swevers, 2017). For large-scale multiple-input, MIMO systems, it is often infeasible or costly to implement centralised controllers (see (Siljak, 1991), (Lunze, 1992) and references within).…”

“…First, with the aforementioned approaches imposing constraints on the structure or order of the controller gives rise to non-convex bi-linear matrix inequalities, which are difficult to solve. Second, by including delays the system models become infinite-dimensional (Michiels et al, 2017), hence, any controller design problem involving tuning of finitely many controller parameters can be considered as a reduced-order controller design problem.…”