Inverse optimal control formulation for guaranteed dominant pole placement with PI/PID controllers
01 Mar 2012-pp 1-6
TL;DR: Inverse optimal control formulation has been done, considering that the PID controller gains are equivalent to optimal state feedback gains of a quadratic regulator and a brief comparison of the achievable cost of control for various specified open loop and closed loop conditions are reported.
Abstract: This paper presents an analytical approach of guaranteed dominant pole placement tuning for PID controllers to handle second order systems. Analytical expression of PID controller gains are reported in terms of the system's open loop characteristics and desired closed loop damping ratio, natural frequency and relative dominance of pole placement. Inverse optimal control formulation has been done, considering that the PID controller gains are equivalent to optimal state feedback gains of a quadratic regulator and a brief comparison of the achievable cost of control for various specified open loop and closed loop conditions are reported. The idea has been extended for pole placement tuning of PI controllers as well to handle first order systems along with discussion about the corresponding inverse optimality.
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24 citations
TL;DR: New formulations for designing dominant pole placement based proportional-integral-derivative (PID) controllers to handle second order processes with time delays (SOPTD) are derived and tuning rules for the robust stable solutions using the test-bench processes are reported.
Abstract: This paper derives new formulations for designing dominant pole placement based proportional-integral-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 non-dominant 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.
18 citations
TL;DR: This paper proposes a new formulation of proportional–integral–derivative (PID) controller design using the dominant pole placement method for handling second-order-plus-time-delay (SOPTD) systems that transforms the transcendental exponential delay term of the plant into finite number of discrete-time poles by a suitable choice of the sampling time.
Abstract: This paper proposes a new formulation of proportional–integral–derivative (PID) controller design using the dominant pole placement method for handling second-order-plus-time-delay (SOPTD) systems....
9 citations
Cites methods from "Inverse optimal control formulation..."
...This paper proposes a novel dominant pole placement based PID controller design method to control SOPTD plants which is an extension of the previous works reported in (Das et al., 2012; Das et al., 2018; Saha et al., 2012)....
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TL;DR: Linear quadratic regulator (LQR), a popular technique for designing optimal state feedback controller, is used to derive a mapping between continuous and discrete time inverse optimal equivalence of proportional integral derivative (PID) control problem via dominant pole placement.
Abstract: Linear quadratic regulator (LQR), a popular technique for designing optimal state feedback controller, is used to derive a mapping between continuous and discrete time inverse optimal equivalence of proportional integral derivative (PID) control problem via dominant pole placement. The aim is to derive transformation of the LQR weighting matrix for fixed weighting factor, using the discrete algebraic Riccati equation (DARE) to design a discrete time optimal PID controller producing similar time response to its continuous time counterpart. Continuous time LQR-based PID controller can be transformed to discrete time by establishing a relation between the respective LQR weighting matrices that will produce similar closed loop response, independent of the chosen sampling time. Simulation examples of first/second order and first-order integrating processes exhibiting stable/unstable and marginally stable open loop dynamics are provided, using the transformation of LQR weights. Time responses for set-point and disturbance inputs are compared for different sampling times as fraction of the desired closed loop time constant.
8 citations
References
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01 Jan 1995
4,110 citations
"Inverse optimal control formulation..." refers background in this paper
...These weighting matrices regulate the penalties on the excursion of state variables ( x ) and control signal ( u )....
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Book•
01 Jun 1979TL;DR: In this article, an augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems, with step-by-step explanations that show clearly how to make practical use of the material.
Abstract: This augmented edition of a respected text teaches the reader how to use linear quadratic Gaussian methods effectively for the design of control systems. It explores linear optimal control theory from an engineering viewpoint, with step-by-step explanations that show clearly how to make practical use of the material. The three-part treatment begins with the basic theory of the linear regulator/tracker for time-invariant and time-varying systems. The Hamilton-Jacobi equation is introduced using the Principle of Optimality, and the infinite-time problem is considered. The second part outlines the engineering properties of the regulator. Topics include degree of stability, phase and gain margin, tolerance of time delay, effect of nonlinearities, asymptotic properties, and various sensitivity problems. The third section explores state estimation and robust controller design using state-estimate feedback. Numerous examples emphasize the issues related to consistent and accurate system design. Key topics include loop-recovery techniques, frequency shaping, and controller reduction, for both scalar and multivariable systems. Self-contained appendixes cover matrix theory, linear systems, the Pontryagin minimum principle, Lyapunov stability, and the Riccati equation. Newly added to this Dover edition is a complete solutions manual for the problems appearing at the conclusion of each section.
3,254 citations
"Inverse optimal control formulation..." refers background in this paper
...Analytical expression of PID controller gains are reported in terms of the system’s open loop characteristics and desired closed loop damping ratio, natural frequency and relative dominance of pole placement....
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19 Dec 2002
TL;DR: This chapter discusses Dynamical Systems and Modeling, a Treatise on Modeling Equations and its Applications to Neural Networks, and its applications to Genetic and Evolutionary Algorithms.
Abstract: 1. Dynamical Systems and Modeling 2. Analysis of Modeling Equations 3. Linear Systems 4. Stability 5. Optimal Control 6. Sliding Modes 7. Vector Field Methods 8. Fuzzy Systems 9. Neural Networks 10. Genetic and Evolutionary Algorithms 11. Chaotic Systems and Fractals Index
362 citations
TL;DR: In this paper, an inverse problem of nonlinear regulator design is posed and solved, and some useful properties are isolated for nonlinear optimal regulators are discussed, and an inverse solution to the problem is presented.
Abstract: Nonlinear optimal regulators are discussed, and some useful properties are isolated. An inverse problem of nonlinear regulator design is posed and solved.
276 citations
TL;DR: In this article, the authors proposed two simple and easy methods which can guarantee the dominance of the two assigned poles for PID control systems They are based on root locus and Nyquist plot respectively if a solution exists, the parametrization of all the solutions is explicitly given.
145 citations
"Inverse optimal control formulation..." refers background in this paper
...Clearly, with this principle of inverse optimal control, the cost of control ( J ) can be compared for arbitrary set of state feedback controller gains....
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