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LQR based improved discrete PID controller design via optimum selection of weighting matrices using fractional order integral performance index

Saptarshi Das, Indranil Pan, Kaushik Halder, Shantanu Das, Amitava Gupta

arXiv:1301.0931v1math.OC

TL;DR

LQR-based PID design links controller gains to optimal state feedback, but arbitrary LQR weights can yield poor time-domain tracking, and discretizing a continuous-time design may not preserve optimality. The paper uses genetic search to select weights for analog and digital PID controllers and extends the objective to fractional orders. It reports process-dependent preferred fractional orders and lower cost of control than the integer-order case in its examples.

  • Problem

    Arbitrary LQR weighting matrices can leave time-domain performance poor, and an analog LQR-PID controller may not remain optimal after discretization at an arbitrary sampling time.

    Source: §2.2 · p. 6, §3.2 · p. 9

  • Method

    The paper uses a Genetic Algorithm to select LQR weighting matrices for analog and digital PID controllers while minimizing a time-domain integral performance index, including fractional-order forms.

    Source: §1 · p. 3, §2.1 · p. 5, §3.2 · p. 9

  • Results

    For both illustrative processes, fractional-order performance-index designs outperform the integer-order case; order 1.5 gives the lowest cost of control for sluggish processes, while order 0.5 does so for oscillatory processes.

    Source: §4 · p. 20

  • Takeaways & Limitations

    The authors recommend fractional orders above 1 for sluggish processes and below 1 for oscillatory processes within the studied arbitrary-order cost function.

    Source: §4 · p. 20, §5 · p. 20

    The fractional-order integrands can oscillate and vary over time rather than increase monotonically or reach a steady-state value as the integer-order integral does.

    Source: §3.4 · p. 14

Abstract

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The continuous and discrete time Linear Quadratic Regulator (LQR) theory has been used in this paper for the design of optimal analog and discrete PID controllers respectively. The PID controller gains are formulated as the optimal state-feedback gains, corresponding to the standard quadratic cost function involving the state variables and the controller effort. A real coded Genetic Algorithm (GA) has been used next to optimally find out the weighting matrices, associated with the respective optimal state-feedback regulator design while minimizing another time domain integral performance index, comprising of a weighted sum of Integral of Time multiplied Squared Error (ITSE) and the controller effort. The proposed methodology is extended for a new kind of fractional order (FO) integral performance indices. The impact of fractional order (as any arbitrary real order) cost function on the LQR tuned PID control loops is highlighted in the present work, along with the achievable cost of control. Guidelines for the choice of integral order of the performance index are given depending on the characteristics of the process, to be controlled.

1. Introduction

LQR tuning offers a way to combine PID set-point tracking with the regulator’s trade-off between state deviation and controller effort. This paper uses genetic search over LQR weights and fractional-order time-domain indices to tune analog and digital PID controllers, illustrating the method on lightly and heavily damped second-order systems.

1. Introduction

  • LQR-based tuning can combine PID set-point tracking with the LQR trade-off between state-trajectory deviation and controller effort.The authors motivate the combination by noting that arbitrary LQR weights alone may not provide good set-point tracking because the regulator lacks an integral term.Source: §1 · pp. 1–2
  • Earlier optimal PID approaches include inverse-optimal, custom-cost, state-space, and LMI formulations, motivating the paper’s LQR-based treatment.Source: §1 · p. 2
  • The method uses a Genetic Algorithm to select LQR weighting matrices while minimizing a time-domain performance index, then examines arbitrary fractional-order indices for PID design.The paper studies the resulting closed-loop performance and cost of control for continuous and discrete LQR-based PID controllers.Source: §1 · pp. 2–3
  • Prior work includes GA tuning of PID controllers and LQR weights, while this paper applies an arbitrary fractional-order integral performance index to the weight-selection objective.The introduction situates the method alongside earlier GA-based time-domain and frequency-domain PID tuning approaches.Source: §1 · p. 2
  • The proposed method tunes both analog and digital PID controllers for second-order systems with very low and high damping.Source: §1 · p. 3

2. Formulation of LQR Based Optimal PID Controller for Second Order Systems

The paper formulates PID gains as LQR state-feedback gains, using error, its rate, and its integral as the controller states. For digital PID design, it discretizes the augmented system and uses discrete-time LQR because sampling can shift poles and zeros and alter the analog design’s behavior.

2.1. Tuning of PID Controllers as Continuous Time Linear Quadratic Regulators

  • The continuous-time formulation takes error, its rate, and its integral as states, making the optimal state-feedback gains the PID controller parameters.The approach is extended to lightly damped processes, where the earlier pole-cancellation reduction to a PI design does not apply and simultaneous selection of all three PID gains is lost.Source: §2.1 · p. 3
  • The continuous-time LQR minimizes a quadratic cost on state trajectories and control effort; its feedback gain follows from the CARE solution.The designer chooses diagonal Q weights to penalize states, while R regulates the penalty on control effort and helps prevent actuator saturation.Source: §2.1 · pp. 4–5

3.2. Optimum Selection of Weighting Matrices Using Genetic Algorithm Minimizing Time Domain Integral Performance Index

  • Choosing Q and R optimally is important because they determine the state-feedback gains and indirectly affect closed-loop time-domain performance.The paper motivates a GA search over the weights against a separate time-domain performance index rather than relying on arbitrary weights.Source: §2.1 · p. 5

2.2. Discrete Time Quadratic Regulator Theory Applied to Optimal Digital PID Controller Design

  • A bilinear transformation relates the discrete-time controller poles and zeros to their continuous-time counterparts.Source: §2.2 · p. 6
  • The discrete-time design discretizes the augmented system at the specified sampling time and obtains the optimal feedback gain from the DARE solution.The discrete controller’s performance depends heavily on sampling time, and discretizing an analog PID does not generally preserve optimality.Source: §2.2 · p. 6
  • Discretizing the analog LQR-PID controller can shift dominant poles and controller zeros toward higher frequencies, so the paper derives the digital controller with discrete-time LQR at the specified sampling time.The comparison uses a second-order system with gain 1, damping ratio 0.2, and natural frequency 1; sampling times range from 0.1 to 1 s.Source: §2.2 · pp. 6–8
  • The figure-based sampling-time comparison shows that discretization changes the open-loop root locus and closed-loop pole locations, motivating a discrete-time design.Source: Figure 2 · p. 7, Figure 3 · p. 7, §2.2 · p. 8

3. LQR Based PID Controller Design with Optimum Selection of Weighting Matrices

The section motivates optimizing LQR weighting matrices because their choice affects PID time response, then extends the performance index to fractional integration orders. It describes a genetic algorithm search over the weights and examines how fractional order changes the cost behavior.

3.1. Effect of Weighting Matrices on the Control System Performance

3.2. Optimum Selection of Weighting Matrices Using Genetic Algorithm Minimizing Time Domain Integral Performance Index

  • A GA selects Q1,Q2,Q3,R by minimizing a weighted ITSE+ISCO index, since LQR optimality at fixed weights need not yield good time-domain response.ITSE and ISCO receive equal weights to penalize both loop error and control signal.Source: §3.2 · p. 9

3.3. Details of the Genetic Algorithm for Optimal Controller Design

  • The real-coded GA searches Q1,Q2,Q3,R in [0,100], using population 20, elite count 2, crossover 0.8, mutation 0.2, and instability penalties.The authors report repeating runs and selecting the lowest-cost results.Source: §3.3 · p. 10

3.4. Fractional Order Integral Performance Indices and Their Impact on the LQR Based PID Design

  • The cost index generalizes integration order to arbitrary Λ, making order an additional design tool for obtaining superior or inferior control performance.Source: §3.4 · p. 13
  • At Λ=1, ITSE increases monotonically toward a steady value, whereas lower-order fractional indices are nonmonotonic and do not reach a steady final value.The authors caution that finite-horizon final values may therefore be inappropriate for controller design or comparison.Source: §3.4 · p. 14

4. Simulations and Results

The simulations compare continuous- and discrete-time LQR-based PID controllers on oscillatory and sluggish second-order systems, and assess fractional-order choices using controller cost and time responses. The preferred fractional order depends on process characteristics: lower orders suit oscillatory systems, while higher orders suit sluggish systems.

4. Simulations and Results

  • The simulations use a heavily oscillatory and a sluggish system as illustrative second-order cases.The stated oscillatory-system parameters include damping ratio ξ = 0.2; the excerpt also specifies n = 1 and ω_ol = 5, excluding time delay.Source: §4 · p. 15
  • For each integer- or fractional-order case, the CARE-based design has a lower minimum J than the DARE-based design, with the sampled-data system showing slightly worse tracking.Simulations use a finite 100-second horizon; the reported minimum-J values across different fractional orders are not directly comparable because their performance indices differ.Source: §4 · p. 15
  • GA selection of weighting matrices yields the PID gains as optimal state-feedback gains for CARE- and DARE-based designs.Tables I and II report the corresponding controller parameters and minimum fractional-order cost-function values; their captions identify CARE and DARE, respectively.Source: §4 · p. 15, Table I · p. 15, Table II · p. 15
  • As the fractional order increases, minimum J increases; orders below one yield lower values than the integer-order index, while orders above one yield higher values.The paper attributes the higher values for Λ > 1 to memory effects in fractional integration and cautions that different-order minimum-J values are not directly comparable.Source: §4 · pp. 15–16
  • Discrete-time controllers give satisfactory set-point tracking for the oscillatory plant but weaker load-disturbance rejection; for the sluggish plant, they give dead-beat tracking and comparable disturbance response.The discrete controllers also produce small, smooth control signals for both plants; for the sluggish plant, this may reduce actuator size and associated costs.Source: §4 · p. 16, Figure 9 · p. 16, Figure 10 · p. 17
  • The authors compare control costs through Riccati solutions because step-input initial state values cannot be directly used to calculate the optimal cost for a PID controller.The initial error rate tends to infinity and the integral error tends to zero, so the comparison uses eigenvalues of differences between the GA-based Riccati matrices.Source: §4 · p. 19
  • The fractional-order performance-index approach gives better results than the integer-order case in both illustrative systems.The integer-order case lies between the fractional-order cases in control cost.Source: §4 · p. 20
  • For oscillatory systems, Λ = 0.5 gives the lowest control cost; for sluggish systems, Λ = 1.5 gives the lowest, supporting low and high fractional orders, respectively.In both cases the integer-order result lies between the two fractional-order cases.Source: §4 · pp. 19–20

5. Conclusion

The conclusion presents GA-based selection of LQR weighting matrices for PID design and reports that control cost depends on process characteristics, damping, and fractional order. It recommends choosing the fractional order according to process type and identifies LQG with noisy measurements and disturbances as future work.

5. Conclusion

  • GA-based weighting-matrix selection designs a discrete-time LQR for PID control, and the reported control cost depends on process characteristics, damping, and fractional order.The conclusion also describes extending the formulation to continuous-time LQR and second-order sluggish and oscillatory systems.Source: §5 · p. 20
  • The conclusion recommends choosing the fractional order according to process characteristics and proposes extending the method to LQG problems with noisy measurements and disturbances.It links the fractional differ-integrals' long-memory behavior to potentially useful controller designs.Source: §5 · p. 20
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