Performance analysis of robust stable PID controllers using dominant pole placement for SOPTD process models

Abstract

Abstract:This paper derives new formulations for designing dominant pole placement based proportionalintegral-derivative (PID) controllers to handle second order processes with time delays (SOPTD). Previously, similar attempts have been made for pole placement in delay-free systems. The presence of the time delay term manifests itself as a higher order system with variable number of interlaced poles and zeros upon Pade approximation, which makes it difficult to achieve precise pole placement control. We here report the analytical expressions to constrain the closed loop dominant and nondominant poles at the desired locations in the complex s-plane, using a third order Pade approximation for the delay term. However, invariance of the closed loop performance with different time delay approximation has also been verified using increasing order of Pade, representing a closed to reality higher order delay dynamics. The choice of the nature of non-dominant poles e.g. all being complex, real or a combination of them modifies the characteristic equation and influences the achievable stability regions. The effect of different types of non-dominant poles and the corresponding stability regions are obtained for nine test-bench processes indicating different levels of open-loop damping and lag to delay ratio. Next, we investigate which expression yields a wider stability region in the design parameter space by using Monte Carlo simulations while uniformly sampling a chosen design parameter space. The accepted data-points from the stabilizing region in the design parameter space can then be mapped on to the PID controller parameter space, relating these two sets of parameters. The widest stability region is then used to find out the most robust solution which are investigated using an unsupervised data clustering algorithm yielding the optimal centroid location of the arbitrary shaped stability regions. Various time and frequency domain control performance parameters are investigated next, as well as their deviations with uncertain process parameters, using thousands of Monte Carlo simulations, around the robust stable solution for each of the nine test-bench processes. We also report, PID controller tuning rules for the robust stable solutions using the test-bench processes while also providing computational complexity analysis of the algorithm and carry out hypothesis testing for the distribution of sampled data-points for different classes of process dynamics and non-dominant pole types. Keywords:dominant pole placement, PID controller tuning, second order plus time delay (SOPTD), control performance, robust stable controller, stability region, signal/system norms, gain/phase margin 2 Nomenclature: m : Non-dominance parameter for pole placement,

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“…This technique does not need any approximation for the delay term e.g. 3 rd order Pade during the pole placement design as reported in Das et al [18]. We also compare the effect of different non-dominant pole types and stability regions obtained from a sampling approach using PSO optimizer with the ISE performance criteria.…”

“…( ) , , 1 LT = , which are further divided in three different damping scenarios for representing different open-loop oscillation levels i.e. under-damped, critically-damped, overdamped ( , , 1 ol  = ) respectively [18]. Table 1 describes these realistic process models along with their open-loop parameters e.g.…”

“… simultaneously using a random search and optimization method, in particular the particle swarm optimization (PSO) algorithm due to its well-known capability to quickly search a large parameter space even for complex cost function landscape. This can often be advantageous than employing a rejection sampling algorithm like [18] to filter out only the stabilizing set of specifications and the corresponding controller gains. We use here the integral of squared error (ISE) criteria (31) ( ) ( )   ( ) , , The simulations were run on a 64-bit Windows PC with 64 GB memory and an AMD Ryzen 7, 3.6 GHz processor where simulations on a single core takes 302 sec ≈ 5 min time on an average and the same setup running on 12 parallel cores takes an average run time of 64 sec ≈ 1 min.…”

Filtered PID Control Loop for Third Order Plants With Delay: Dominant Pole Placement Approach

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Quantum Algorithm of Imperfect KB Self-organization. Pt II: Robotic Control with Remote Knowledge Base Exchange

“…Different approaches to design of expanded conventional controller's structures in [4][5][6][7][8][9] described. In compar-Distributed under creative commons license 4.0 ison with the much simpler PI control, which still attracts attention of the contemporary research, the design is yet more complicated also due to the fact that an increased speed of transients exhibits all modeling and tuning imperfections.…”