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An optimal PID controller via LQR for standard second order plus time delay systems

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An improved tuning methodology of PID controller for standard second order plus time delay systems (SOPTD) is developed using the approach of Linear Quadratic Regulator (LQR) and pole placement technique to obtain the desired performance measures. The pole placement method together with LQR is ingeniously used for SOPTD systems where the time delay part is handled in the controller output equation instead of characteristic equation. The effectiveness of the proposed methodology has been demonstrated via simulation of stable open loop oscillatory, over damped, critical damped and unstable open loop systems. Results show improved closed loop time response over the existing LQR based PI/PID tuning methods with less control effort. The effect of non-dominant pole on the stability and robustness of the controller has also been discussed.

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An optimal PID controller via LQR for standard second order plus time delay systems

  1. 1. Research Article An optimal PID controller via LQR for standard second order plus time delay systems Saurabh Srivastava, Anuraag Misra, S.K. Thakur, V.S. Pandit n Variable Energy Cyclotron Center, 1/AF, Bidhan Nagar, Kolkata 700064, India a r t i c l e i n f o Article history: Received 17 July 2013 Received in revised form 21 September 2015 Accepted 19 November 2015 Available online 4 December 2015 This paper was recommended for publica- tion by Dr. Ahmad B. Rad. Keywords: Linear system PID controller System matrix Linear Quadratic Regulator (LQR) Time delay Closed-loop a b s t r a c t An improved tuning methodology of PID controller for standard second order plus time delay systems (SOPTD) is developed using the approach of Linear Quadratic Regulator (LQR) and pole placement technique to obtain the desired performance measures. The pole placement method together with LQR is ingeniously used for SOPTD systems where the time delay part is handled in the controller output equation instead of characteristic equation. The effectiveness of the proposed methodology has been demonstrated via simulation of stable open loop oscillatory, over damped, critical damped and unstable open loop systems. Results show improved closed loop time response over the existing LQR based PI/PID tuning methods with less control effort. The effect of non-dominant pole on the stability and robustness of the controller has also been discussed. & 2015 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction Proportional-Integral-Derivative (PID) controller, though very old design, is still one of the favorite and most widely used con- troller for many industrial process control applications. This is due to its simple structure, satisfactory control effect and acceptable robustness [1–3]. The PID controller is easier to understand due to intuitive simplicity of the algorithm and simple meaning of its tuning parameters proportional (Kp), integral (Ki) and derivative (Kd). In order to provide good and robust performance these PID parameters are required to be tuned individually to match the process dynamics. An improper PID setting results in sluggish, oscillatory time response and poor robustness. The PID controller tuning first proposed by Ziegler–Nichols [4] has been improved by several researchers [5]. As the high performance is always desired from the controller and due to the availability of fast computa- tional power, tuning methods based on optimization approach have received more attention in the recent years [6,7]. Many techniques have been developed and still research is going on for better tuning of the PID controller using complex numerical optimization procedures [8,9]. The design techniques based on linear Quadratic Regulator (LQR) are well known in modern control theory and have been widely used in many applications [10–12]. In a recent article Saha et al. [11] have obtained the PID parameters for second order systems via LQR using the dominant pole placement technique. However, their approach is applicable only for systems having no time delay. Most of the real industrial plants have time delay in their transfer function. Since the presence of time delay in a control loop is a source of instability and performance degrada- tion, it is, therefore, necessary to design the PID controller opti- mally to achieve good stability. Many researchers have worked on the tuning of controller for the systems having time-delay [13–20] with pole placement and mentioned the challenges due to the presence of exponential term in the characteristics equation which leads to the infinite roots. They have used different approaches to design the controller with some limitations. He et al. [10] have proposed an analytical method to tune the PI/PID parameters in an optimal way using LQR techniques with user specified closed loop damping ratio and natural frequency for the first order plus time delay (FOPTD) model. His method is based on the decomposition of state equation in two parts one for toL and another for tZL in such a way that the state equation for tZL become independent of L and then applied the usual LQR approach for obtaining the PID parameters for FOPTD. They have compared simulation results of their method with the gain-phase margin method [21] and pre- sented much improved results. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions http://dx.doi.org/10.1016/j.isatra.2015.11.020 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: saurabh@vecc.gov.in (S. Srivastava), pandit@vecc.gov.in (V.S. Pandit). ISA Transactions 60 (2016) 244–253
  2. 2. Most of the real plants can be more closely approximated using second order plus time delay (SOPTD) model compared to FOPTD model. The SOPTD processes are very rich in dynamics as they include under damped, critically damped and over damped sys- tems. Very few tuning rules are available for such processes. He et al. [10] have also extended their approach for SOPTD systems by equating the larger process pole with the derivative term of the PID controller and then applied the PI tuning approach using LQR to obtain other two parameters. This approach works satisfactory for SOPTD model if the system poles have real roots, but does not provide the optimum parameters of the PID controller as one of the PID parameters is prefixed. This technique cannot be applied for SOPTD systems with complex poles (such as highly oscillatory processes) of the system as they are always in pairs and cannot be eliminated with single complex zero of the controller. In the present work we have combined the concept of LQR based PI/PID tuning method together with the dominant pole placement approach to derive the PID parameters analytically for SOPTD systems. It is shown that the present technique gives a good closed loop time response for various processes as compared with the existing PI/PID tuning methods using LQR. In order to illustrate the utility of the present technique, simulations per- formed in MATLAB [22] have been presented for different types of SOPTD models such as critically damped and over-damped pro- cesses as well as processes having complex poles. The effect of non-dominant pole on the control signal and on the stability of the closed loop system has also been discussed. 2. LQR based PID controller design for SOPTD processes In this section we briefly outline the LQR solution for time delay systems formulated by He et al. [10] for the FOPDT models where the motion equation is reformulated into a first order differential equation that contains no time delay and then the optimal con- troller is designed according to the classical control theory. We then extend this approach utilizing the dominant pole placement technique to find the optimal PID parameters for SOPTD systems. 2.1. LQR solution for SOPTD systems A linear plant with time delay can be represented as _XðtÞ ¼ AXðtÞþBuðtÀLÞ tZ0 ; ð1Þ where A, B, X and L are the state transition matrix, control matrix, state matrix and the time delay term respectively. For toL, no control signal will be effective and thus we have Eq. (1) as control free equation. Control signal will be effective only for tZL. So by decomposing Eq. (1) into two components, one for toL and other for tZL, we have _XðtÞ ¼ AXðtÞ ; 0rtoL ; ð2Þ _XðtÞ ¼ AXðtÞþBum ðtÞ ; tZL ; ð3Þ where um ðtÞ ¼ uðtÀLÞ. Since Eqs. (2) and (3) are now delay free, one can easily apply the standard LQR approach [12] for delay free processes to find the optimum control vector um ðtÞ subjected to the minimization of the cost-function defined by J ¼ Z 1 0 XT ðtÞ Q XðtÞþumT ðtÞ R um ðtÞ dt ; ð4Þ where Q is the semi positive definite state weighting matrix and R is the positive definite control weighting matrix. The LQR solution gives the optimal control vector um ðtÞ as um ðtÞ ¼ ÀRÀ 1 BT PXðtÞ ; ð5Þ where P is the symmetric positive definite Riccati coefficient matrix which can be obtained by solving continuous algebraic Riccati equation AT PþPAþQ ÀPBRÀ 1 BT P ¼ 0 : ð6Þ From (5) we can write uðtÞ ¼ um ðtþLÞ ¼ ÀRÀ1 BT PXðtþLÞ : ð7Þ Here we see that uðtÞ gives the control signal in the whole time horizon of tZ0, however X(tþL) is not directly known at time t. With the use of Eqs. (2), (3) and (5), X(tþL) can be expressed in terms of X(t) [10]. The optimal control vector uðtÞ for the present case, thus can be written as uðtÞ ¼ ÀRÀ 1 BT PeðAcÞt eAðL ÀtÞ XðtÞ ; 0rtoL ; ð8Þ uðtÞ ¼ ÀRÀ 1 BT PeðAcÞL XðtÞ ; tZL ; ð9Þ where Ac ¼ AÀBRÀ 1 BT P : ð10Þ The beauty of above mathematical formulation lies in the fact that the optimal control vector u(t) handles the delay part as given by Eqs. (8) and (9). As the system matrix Ac given by Eq. (10) does not contain any time delay for tZL, one can easily apply the approach of direct pole placement to get `the desired closed loop time performance measures. In order to obtain the optimal feed- back gain uðtÞ we need to calculate eðAcÞt and eAðLÀ tÞ . By sub- stituting um ðtÞ from Eq.(5) into Eq. (3) we have for tZL, _XðtÞ ¼ AcXðtÞ : ð11Þ The matrix Ac can be determined by setting the characteristic equation of the closed loop system ΔðsÞ ¼ sIÀAcj equal to the desired closed loop equation. For example, in the case of FOPTD process, where the matrix Ac is a 2  2 matrix, we have ΔðsÞ ¼ sIÀAc ¼ ðsþp1Þðsþp2Þ ¼ ðs2 þ2ςclωclsþωcl 2 Þ ; ð12Þ where p1 ¼ ζclωcl þiωcl ffiffiffiffiffiffiffiffiffiffiffiffiffi 1Àζ2 cl q ; p2 ¼ ζclωcl Àiωcl ffiffiffiffiffiffiffiffiffiffiffiffiffi 1Àζ2 cl q ; with ζcl and ωcl as the desired closed loop damping ratio and natural frequency. For the SOPTD process the dimension of matrix Ac will be 3  3. Utilizing the help of dominant pole placement technique matrix Ac can be evaluated in terms of known parameters ζcl and ωcl from the equation sIÀAc ¼ ðsþp1Þðsþp2Þðsþp3Þ ¼ ðsþmζclωclÞðs2 þ2sζclωcl þω2 clÞ ; ð13Þ where the location of non-dominant pole p3 ¼ mζclωcl is placed m times away from the real part of the dominant closed loop poles. We call this m as the relative dominance and as per the literature its value should be chosen around 3 or more [2]. 2.2. Determination of state weighting and Riccati coefficient matrices In the case of second order process matrices Q, R and P are generally taken as Q ¼ q1 0 0 0 q2 0 0 0 q3 2 6 4 3 7 5; R ¼ r½ Š; P ¼ p11 p12 p13 p12 p22 p23 p13 p23 p33 2 6 4 3 7 5 : ð14Þ In the optimal control it is a standard practice to design reg- ulator by varying Q and keeping R fixed [10,11]. A schematic of S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 245
  3. 3. closed loop system with PID controller for SOPTD process is shown in Fig. 1. The state variables for the present case are XðtÞ ¼ ½x1ðtÞ x2ðtÞ x3ðtފT ; ð15Þ where x1ðtÞ ¼ Z eðtÞ dt; x2ðtÞ ¼ eðtÞ; x3ðtÞ ¼ deðtÞ dt ; ð16Þ with error eðtÞ ¼ rðtÞÀyðtÞ. Here r(t) and y(t) are the reference and output signals respectively. From Fig. 1, the control signal can be expressed in terms of the state variable as uðtÞ ¼ Kpx2ðtÞþKix1ðtÞþKdx3ðtÞ : ð17Þ The transfer function of the PID controller can be express in s domain as CðsÞ ¼ Kp þ Ki s þKds : ð18Þ In the case of unity output feedback system such as shown in Fig. 1, if we put the reference signal rðtÞ ¼ 0 ; we have eðtÞ ¼ ÀyðtÞ : With this condition, the second order transfer function with time delay can be written as GðsÞ ¼ yðsÞ uðsÞ ¼ K eÀ sL s2 þasþb ¼ ÀeðsÞ uðsÞ ; ð19Þ in which a ¼ 2ζolωol and b ¼ ω2 ol, where ζol and ωol are the damping ratio and natural frequency of the open loop plant respectively. Using Eq. (16) we can express Eq. (19) in terms of state variables as _x3ðtÞ ¼ Àax3ðtÞÀbx2ðtÞÀKuðtÀLÞ : In terms of state-space formulation the derivative of the state variables can be written as _x1ðtÞ _x2ðtÞ _x3ðtÞ 2 6 4 3 7 5 ¼ 0 1 0 0 0 1 0 Àb Àa 2 6 4 3 7 5 x1ðtÞ x2ðtÞ x3ðtÞ 2 6 4 3 7 5þ 0 0 ÀK 2 6 4 3 7 5 uðtÀLÞ : ð20Þ Comparing Eq. (20) with Eq. (1), it is straightforward to α ¼ rÀ1 K2 obtain matrices A and B as A ¼ 0 1 0 0 0 1 0 Àb Àa 2 6 4 3 7 5; B ¼ 0 0 ÀK 2 6 4 3 7 5 : ð21Þ Using Eqs. (10), (14) and (21) we have sIÀAc ¼ s À1 0 0 s À1 η p13 bþη p23 sþaþη p33 : ð22Þ where η ¼ rÀ 1 K2 : Now from Eqs. (22) and (13) we have s3 þðaþη p33Þs2 þðbþη p23Þsþη p13 ¼ s3 þðð2þmÞζclωclÞs2 þðωcl 2 þ2mζcl 2 ωcl 2 Þsþmζclωcl 3 : ð23Þ By comparing the coefficients of powers of s from both sides of Eq. (23), the elements p13, p23 and p33 can be obtained as p13 ¼ mζclω3 cl η ; p23 ¼ ω2 cl þ2mζ2 clω2 cl Àb η ; p33 ¼ ð2þmÞζclωcl Àa η : ð24Þ The remaining three elements of the matrix P and three ele- ments of the matrix Q can be obtain by solving Riccati equation Eq. (6), which gives six equations for six variables in terms of known parameters. With some algebraic manipulations we obtain, p11 ¼ mζcl ω5 cl ð1þ2m ζ2 clÞ η ; p12 ¼ ð2þmÞ m ζ2 cl ω4 cl η ; p22 ¼ 2 ω3 cl ðζcl þ2m ζ3 cl þm2 ζ3 cl ÞÀab η ; q1 ¼ m2 ζ2 cl ω6 cl η ; q2 ¼ ω4 clð1þ4m2 ζ4 cl À2m2 ζ2 cl ÞÀb 2 η ; q3 ¼ ω2 clð4ζ2 cl þm2 ζ2 cl À2Þþ2bÀa2 η : ð25Þ 2.3. Evaluation of eAðLÀ tÞ In order to obtain the value of eAðL ÀtÞ we proceed as follows. eAðL ÀtÞ ¼ ℓÀ1 ðsIÀAÞÀ 1 h i t ¼ LÀ t ¼ ℓÀ 1 1 s ðsþ aÞ sðsþ p01Þðsþ p02Þ 1 sðsþ p01Þðsþ p02Þ 0 sþ a ðs þp01Þðsþp02Þ 1 ðs þp01Þðs þp02Þ 0 Àb ðs þp01Þðsþp02Þ s ðs þp01Þðs þp02Þ 2 6 6 6 4 3 7 7 7 5 0 B B B @ 1 C C C A ¼ f 0 11ðtÞ f 0 12ðtÞ f 0 13ðtÞ f 0 21ðtÞ f 0 22ðtÞ f 0 23ðtÞ f 0 31ðtÞ f 0 32ðtÞ f 0 33ðtÞ 2 6 4 3 7 5 t ¼ LÀ t : ð26Þ Here p01 and p02 are the poles of the open loop system (see Eq. (19)) given by p01 ¼ aÀ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 À4b p 2 ; p02 ¼ aþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 À4b p 2 ; ð27Þ Using partial fraction approach f 0 11,f 0 12,f 0 13, f 0 21,f 0 22,f 0 23,f 0 31,f 0 32 and f 0 33 can be evaluated as f 0 11ðLÀtÞ ¼ 1 ; f 0 12ðLÀtÞ ¼ p02 eÀp01ðLÀ tÞ p01ðp01 Àp02Þ À p01 eÀp02ðL ÀtÞ p02ðp01 Àp02Þ þ a b ; f 0 13ðLÀtÞ ¼ eÀ p01ðLÀ tÞ p01ðp01 Àp02Þ À eÀp02ðLÀ tÞ p02ðp01 Àp02Þ þ 1 b ; f 0 21ðLÀtÞ ¼ 0 ; f 0 22ðLÀtÞ ¼ Àp02 eÀ p01ðL ÀtÞ ðp01 Àp02Þ þ p01 eÀ p02ðLÀ tÞ ðp01 Àp02Þ ; f 0 23ðLÀtÞ ¼ À eÀ p01ðL ÀtÞ ðp01 Àp02Þ þ eÀ p02ðLÀ tÞ ðp01 Àp02Þ ; f 0 31ðLÀtÞ ¼ 0 ; Kp Ki Kd x3(t) + (t) PIDControllerC(s) sL e bass K − ++2 x2(t) x1(t) s 1 s + + + - (t) y(t)(t) Plant G(s) Fig. 1. Schematic of closed loop system with PID controller. S. Srivastava et al. / ISA Transactions 60 (2016) 244–253246
  4. 4. f 0 32ðLÀtÞ ¼ beÀ p01ðLÀ tÞ ðp01 Àp02Þ À beÀp02ðL À tÞ ðp01 Àp02Þ ; f 0 33ðLÀtÞ ¼ p01eÀp01ðL À tÞ ðp01 Àp02Þ À p02eÀp02ðLÀ tÞ ðp01 Àp02Þ : ð28Þ 2.4. Evaluation of eðAcÞt Using Eqs. (10), (14) and (21) we have Ac ¼ ðAÀBRÀ 1 BT PÞ ¼ 0 1 0 0 0 1 Àγ Àβ Àα 2 6 4 3 7 5: where γ ¼ η p13, α ¼ aþη p33 and β ¼ bþη p23. Now eðAcÞt ¼ ℓÀ 1 ðsIÀAcÞÀ1 h i ¼ ℓÀ 1 1 sIÀAc s2 þαsþβ sþα 1 Àγ s2 þαs s Àsγ ÀsβÀγ s2 2 6 4 3 7 5 0 B @ 1 C A ¼ f 11ðtÞ f 12ðtÞ f 13ðtÞ f 21ðtÞ f 22ðtÞ f 23ðtÞ f 31ðtÞ f 32ðtÞ f 33ðtÞ 2 6 4 3 7 5 : ð29Þ Using Eq. (13) in Eq. (29) and with some algebraic manipula- tions, it is straightforward to get f11, f12, f13, f21, f22, f23, f31, f32 and f33 as f 11ðtÞ ¼ X3 i ¼ 1 p2 i Àα pi þβ Di eÀpi t ; f 12ðtÞ ¼ X3 i ¼ 1 Àpi þα Di eÀpi t ; f 13ðtÞ ¼ X3 i ¼ 1 1 Di eÀpi t ; f 21ðtÞ ¼ X3 i ¼ 1 Àγ Di eÀpi t ; f 22ðtÞ ¼ X3 i ¼ 1 pi 2 Àα pi Di eÀ pi t ; f 23ðtÞ ¼ X3 i ¼ 1 Àpi Di eÀ pi t ; f 31ðtÞ ¼ X3 i ¼ 1 γ pi Di eÀ pi t ; f 32ðtÞ ¼ X3 i ¼ 1 β pi Àγ Di eÀ pi t ; f 33ðtÞ ¼ X3 i ¼ 1 pi 2 Di eÀ pi t ; ð30Þ where D1 ¼ Àðp1 Àp2Þðp3 Àp1Þ ; D2 ¼ Àðp1 Àp2Þðp2 Àp3Þ ; D3 ¼ Àðp3 Àp1Þðp2 Àp3Þ : 2.5. Evaluation of PID parameters for 0rtoL ; Using Eqs. (28) and (30) in Eq. (8) the optimal value of control u (t) for 0rtoL ; can be expressed as uðtÞ ¼ ÀRÀ 1 BT PeAc t eAðLÀ tÞ XðtÞ ; ¼ rÀ1 K p13 p23 p33 2 6 4 3 7 5 T f 11ðtÞ f 12ðtÞ f 13ðtÞ f 21ðtÞ f 22ðtÞ f 23ðtÞ f 31ðtÞ f 32ðtÞ f 33ðtÞ 2 6 4 3 7 5  f 0 11ðLÀtÞ f 0 12ðLÀtÞ f 0 13ðLÀtÞ f 0 21ðLÀtÞ f 0 22ðLÀtÞ f 0 23ðLÀtÞ f 0 31ðLÀtÞ f 0 32ðLÀtÞ f 0 33ðLÀtÞ 2 6 4 3 7 5 x1ðtÞ x2ðtÞ x3ðtÞ 2 6 4 3 7 5 : ð31Þ By comparing the coefficients of x1ðtÞ, x2ðtÞ and x3ðtÞ in Eqs. (17) and (31), one can easily obtain the PID parameters for 0rtoL ;as KiðtÞ ¼ rÀ1 K p13 X3 i ¼ 1 f 1iðtÞf 0 i1ðLÀtÞþp23 X3 i ¼ 1 f 2iðtÞf 0 i1ðLÀtÞ þp33 X3 i ¼ 1 f 3iðtÞf 0 i1ðLÀtÞ ! ; KpðtÞ ¼ rÀ 1 K p13 X3 i ¼ 1 f 1iðtÞf 0 i2ðLÀtÞþp23 X3 i ¼ 1 f 2iðtÞf 0 i2ðLÀtÞ þp33 X3 i ¼ 1 f 3iðtÞf 0 i2ðLÀtÞ ! ; KdðtÞ ¼ rÀ 1 K p13 X3 i ¼ 1 f 1iðtÞf 0 i3ðLÀtÞþp23 X3 i ¼ 1 f 2iðtÞf 0 i3ðLÀtÞ þp33 X3 i ¼ 1 f 3iðtÞf 0 i3ðLÀtÞ ! ; ð32Þ 2.6. Evaluation of PID parameters for tZL Similarly using Eqs. (9) and (30) the optimal control u(t) for tZL can be evaluated as uðtÞ ¼ ÀRÀ 1 BT PeAcL XðtÞ ; ¼ rÀ1 K p13 p23 p33 2 6 4 3 7 5 T f 11ðLÞ f 12ðLÞ f 13ðLÞ f 21ðLÞ f 22ðLÞ f 23ðLÞ f 31ðLÞ f 32ðLÞ f 33ðLÞ 2 6 4 3 7 5 x1ðtÞ x2ðtÞ x3ðtÞ 2 6 4 3 7 5 : ð33Þ A comparison of coefficients of x1ðtÞ, x2ðtÞ and x3ðtÞ in Eqs. (33) and (17) gives the PID parameters for tZL as Ki ¼ rÀ1 K p13f 11ðLÞþp23f 21ðLÞþp33f 31ðLÞ À Á ; Kp ¼ rÀ 1 K p13f 12ðLÞþp23f 22ðLÞþp33f 32ðLÞ À Á ; Kd ¼ rÀ 1 K p13f 13ðLÞþp23f 23ðLÞþp33f 33ðLÞ À Á : ð34Þ Note that for L¼0, the matrix elements fij ¼1 when i¼j and fij ¼0 for iaj and Eq. (34) leads to the optimal PID parameters for systems having no time delay. 3. Results and discussion In order to demonstrate the application of the PID tuning methodology proposed in this paper, we now present simulation results for different processes performed using MATLAB. We have considered the examples of under damped, critically damped and over damped SOPTD processes. The present day control challenge is to design a controller to tune unstable and highly oscillatory processes [3]. Therefore, in Examples 4 and 5 we have discussed two plants; one with unstable open loop time response and other with highly oscillatory behavior. 3.1. Example 1: non-minimum phase process In this example we will consider an over damped SOPTD model of a non-minimum phase process and evaluate the PID parameters for 0rtoL and tZL. The closed loop time response is compared S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 247
  5. 5. with the previously developed LQR-based PI/PID tuning method [10], where the derivative term of the PID controller for SOPTD process was set equal to one of the process pole and thus is not obtained in an optimum way. For fair comparison, the desired closed loop damping ratio ζcl ¼0.8 and natural frequency ωcl ¼0.793 rad/s are taken same in the simulation. The non- dominant pole is placed 6 times away from the desired domi- nant real poles i.e. m¼6. The transfer function of the non-minimum phase process is P1 ¼ 1Às ð1þsÞ2 ð2þsÞ ð35Þ and its corresponding over damped SOPTD model is P1 ¼ 1 s2 þ3sþ2 eÀ 1:64s ð36Þ Matrices A and B can be obtained from Eqs. (36) and (21) as A ¼ 0 1 0 0 0 1 0 À2 À3 2 6 4 3 7 5 and B ¼ 0 0 À1 2 6 4 3 7 5 : ð37Þ Using Eqs. (24) and (25), matrices P and Q with R¼[1] can be evaluated as Q ¼ 5:7158 0 0 0 1:4889 0 0 0 9:8290 2 6 4 3 7 5; P ¼ 13:0394 12:1288 2:3908 12:1288 19:2785 3:4540 2:3908 3:4540 2:0732 2 6 4 3 7 5 : ð38Þ The eigen values of matrices P and Q are eig P½ Š ¼ 1:4050 3:6579 29:3282 2 6 4 3 7 5 and eig Q½ Š ¼ 5:7158 1:4889 9:8290 2 6 4 3 7 5 : ð39Þ The positive eigen values of matrices P and Q indicate that the positive definite condition of LQR is satisfied. Finally, the PID parameters for 0rto1:64 s can be obtain using Eq. (32). The time varying PID parameters are plotted in Fig. 2. The PID para- meters for tZ1:64 s can be calculated using Eq. (34) as ½Kp Ki KdŠ ¼ 0:6984 0:4602 0:1543½ Š : ð40Þ Fig. 2 shows the variation of PID parameters used in the simulation at m ¼ 6. For comparison, PID parameters evaluated at m ¼ 3 and m ¼ 10 are also shown. It is clear that values of all the PID parameters are very high at the beginning (t ¼ 0s), followed by a decrease with t up to t ¼ 1:64 s and then remain constant thereafter. Note that design of PID controller at higher value of m leads to lower values for all the PID parameters for tZ1:64 s whereas the situation is completely reverse in the case of to1:64 s. The time response of the step input for process P1 with 20% disturbance at t ¼ 40 s is shown in Fig. 3 by solid black line. The observed behavior of the closed loop time response during the initial period is due to the high values of initial PID parameters, which are responsible for the decrease in the system rise time and hence enhancement in the overshoot. Note that PID parameters between 0rtoL are time varying and large initially. This leads to a comparatively larger control efforts and may cause the actuator saturation in some cases. It is also difficult to implement them practically, particularly in analog domain. It is obvious that a choice of constant PID parameters throughout eases the practical implementation, needs low control effort and maintains the state optimality for all values of tZL. The plots of time response using only constant PID parameters throughout (i.e. for tZ0) obtained for tZ1:64 s using Eq. (34) for various values of relative dom- inance m are also shown in Fig. 3 for comparison. In the simulation all other parameters are kept constant. It can be seen from Fig. 3 that as the value of m is decreased from 6 to 3, the PID controller based on constant parameters tries to cope with the actual time varying PID controller and produces overshoot with improved rise time. An increase in the value of m reduces the overshoot but at the same time increases the rise time. At higher values of m, say around m¼50, this effect saturates and further increase in m has no significant effect on the time response. Thus the choice of m depends upon a particular requirement whether one needs fast rise time or less overshoot. In our experience a good choice for m is between 3 and 10. It is interesting to point out here that increase in the rise time with m in the delayed processes is just opposite to the LQR based PID tuning with no delay [11], where an increase in m decreases the rise time of the closed loop time response. For a given process, our simulation results indicate that an increase in the value of m results in the lower values of PID parameters as shown in Fig. 2 and thus a reduction in the control effort. This fact can also be explained using Eqs. (9), (24) and (25) where an increase in m increases the value of matrix elements of P. This finally causes reduction in the control effort u(t) due to the presence of the term PeðAcÞL which decreases with increase in the value of elements of matrix P. Note that in the case of delay free process eðAcÞL ¼ 1 and u (t) is proportional to matrix P. 0.511.522.533.544.5 0.0 3.0 6.0 Kp 0.51.01.52.02.53.03.54.5 0.0 1.0 2.0 3.0 Ki 0.0 1.0 2.0 3.0 4.0 5.0 0.0 0.5 1.0 1.5 Time (s) Kd m = 10 m = 6 m = 3 t = 1.64 s Fig. 2. PID parameters Kp, Ki and Kd as a function of time. 0 10 20 30 40 50 60 70 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time(s) y(t) Time varying PID with m=6 Constant PID with m=6 Constant PID with m=5 Constant PID with m=4 Constant PID with m=3 Constant PID with m=10 Constant PID with m=50 Constant PID with m=200 Fig. 3. Comparison of time responses of time varying PID with constant PID parameters for process P1 at ζcl ¼ 0:8 and ωcl ¼ 0:793 rad=s at different m. S. Srivastava et al. / ISA Transactions 60 (2016) 244–253248
  6. 6. Observing the simulation results shown in Fig. 3, it appears that the time varying part of PID parameters though, improves the rise time of the closed loop time response, but at the same time it produces substantial overshoot as compared to the cases where only constant PID parameters are used. As it will be easy to implement practically, we therefore, in subsequent examples consider only constant value of PID parameters evaluated for tZL using Eq. (34). To show the effectiveness of the present method, we now compare our results with the previously developed LQR based PI/PID tuning method at same values of closed loop damping ratio and natural frequency. The optimal PID controller for process P1 with ζcl ¼ 0:8 ; ωcl ¼ 0:793 rad=s and m ¼ 6 obtained for tZ1:64 s is C1½presentŠ ¼ 0:6984þ 0:4602 s þ0:1543 s ; and the PID controller used in Ref.[10] is C1½He et al:Š ¼ 0:6138þ 0:5561 s þ1:0 s : Fig. 4(a) compares the step responses with 20% disturbance at t ¼ 40 s. It is easy to observe that the present method gives very less overshoot, only 4% as compared to the 14% of the earlier method. Note that the value of derivative gain in controller C1½He et al:Š is larger than the controller C1½presentŠ. From the simula- tion it is clear that a choice of larger derivative gain not necessarily reduced the overshoot. The main reason for the reduction in overshoot is the optimal tuning of derivative parameter Kd in the present case, which was taken as one of the real pole of the open loop system in earlier case and thus, was not optimum one. Due to the optimal design of all the three parameters in the present method, there is almost 46% reduction in the settling time toge- ther with a substantial reduction in the overshoot. Fig. 4(b) compares the control energy required to achieve good closed loop time response. Since the cost-function is optimized properly in the present method, the required control energy is also less. PID parameters and closed loop performance measures such as percentage overshoot (%OS), settling time (Ts) and rise time (Tr) are presented in Table 1 for comparison. 3.2. Example 2: higher order process Now we consider a higher order process [1] given by P2 ¼ 1 ð1þsÞ8 : ð41Þ The corresponding over damped SOPTD model of this process is P2 ¼ 0:3360 s2 þ1:3878sþ0:3360 eÀ 4:3s : ð42Þ The controller parameters for this model calculated using the method presented by [10] where one of the pole is taken equal to the Kd are given in Table 1. The optimal PID controller for the above process using present method for m ¼ 4 and tZ4:3 s is C2 ¼ 0:3919þ 0:0912 s þ0:2834s : ð43Þ In both the cases same values for desired closed loop para- meters ζcl ¼ 0:9 ; ωcl ¼ 0:3 rad=s are used. The eigen values of matrices P and Q for this model are also positive and therefore satisfying the condition of LQR. Fig. 5(a) compares of the step response with 20% disturbance at t ¼ 70 s. Due to optimal design of all the three parameters, overshoot is almost negligible with an improvement in rise time and disturbance rejection. The control effort required for desired time response, plotted in Fig. 5(b), is also slightly less in the present optimization method. In order to test the present method with large time delay, we have varied the time delay of process P2 from 4.3 s to 44.3 s in steps of 10 s and performed simulations. In all the cases fixed value of ωclL ¼ 1:3 is used. Simulation results indicate that the satisfactory closed loop time response. As usual, we found the response time to become slow as the time delay L increases. 0.0 0.5 1.0 y(t) 0 10 20 30 40 50 60 70 80 0.5 1.0 1.5 2.0 2.5 Time (s) u(t) Present He et. al. a b Fig. 4. Time response and controller response for process P1 with 20% disturbance at t ¼ 40 s. Table 1 Closed loop performance measures. Processes Kp Ki Kd ζcl ωclL m %OS Tr(s) Ts(s) P1 (He et al.) P1 (Present) 0.6138 0.5561 1 0.8 1.3 14 4.5 15 0.6984 0.4602 0.1543 0.8 1.3 6 4 4.5 8 P2 (He et al.) P2 (Present) 0.2873 0.0851 1.0753 0.9 1.3 4 16 23 0.3919 0.0912 0.2834 0.9 1.3 4 0 12 20 P3 (He et al.) P3 (Present) 1.7342 2.1759 1 0.98 0.4 35 1.2 8 3.7238 1.9858 1.6867 0.98 0.4 4 15 1.1 5 0.0 0.5 1.0 y(t) 0 20 40 60 80 100 120 140 0.0 0.5 1.0 Time (s) u(t) Present He et. al. Fig. 5. Time response and controller response for higher order process P2 with 20% disturbance at t ¼ 70 s. S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 249
  7. 7. 3.3. Example 3: critically damped SOPTD process Consider a critically damped SOPTD process [5] given by P3 ¼ eÀ0:2s ð1þsÞ2 : ð44Þ The optimal PID controller designed for ζcl ¼ 0:98 ; ωcl ¼ 2 rad=s and m ¼ 4 is C3 ¼ 3:7238þ 1:9858 s þ1:6867s : ð45Þ Fig. 6(a) shows the comparison of step responses of the criti- cally damped SOPTD process with 20% disturbance at t ¼ 20 s. Clearly, the present method gives an improved performance. Both the overshoot and settling time are improved by considerable amount (see Table 1) with slight improvement in the rise time and disturbance rejection time. Although the present tuning method takes slightly more control signal initially (Fig. 6(b)), but one can easily verify that the total control cost is almost identical in both the cases. 3.4. Example 4: unstable SOPTD process Now we consider an unstable plant [23] given by P4 ¼ 1:5 ð0:5sþ1ÞðsÀ1Þ eÀ 0:3s : ð46Þ With some algebraic manipulation we can easily write Eq. (46) in standard second order TF as given in Fig. 1 and get the value of a¼1, b¼ À2 and K¼3. The optimal LQR based PID controller obtained with ζcl ¼ 0:9 ; ωcl ¼ 0:8rad=s and m ¼ 4 is C4½presentŠ ¼ 1:2153þ 0:1688 s þ0:5682 s : ð47Þ The controller designed by adopting the method of He et al. [10] taking Kd ¼ 2, the larger real system pole with same ζcl and ωcl is given by C4½He et al:Š ¼ 0:5619þ 0:0824 s þ2:0 s : ð48Þ The time response plotted in Fig. 7 clearly shows the advantage of the proposed method for control of unstable plant dynamics. Expect for slightly higher percentage overshoot all other closed loop performance measures are quite reasonable (Tr ¼ 1:8 s, %OS ¼ 150 and Ts ¼ 5 s). Simulation results indicate that the range of ωclL is limited. In the case of stable system the appropriate range is ωclL A ð1:0; 1:5Þ and for the case of unstable systems it is ωclL A ð0:1; 0:4Þ. 3.5. Example 5: highly oscillatory SOPTD process Here we consider a SOPTD process with highly oscillatory open loop response [24] with transfer function given by P5 ¼ 1 s2 þsþ5 eÀ0:1s : ð49Þ Our aim is to design a controller with very small percentage overshoot and settling time. Since the roots of the process are complex, the method used by He et al. for SOPTD process cannot be applied here. The controller with ζcl ¼ 0:9 ; ωcl ¼ 1:5 rad=s and m ¼ 4 is C5 ¼ 3:9434þ 5:8325 s þ3:6339s : ð50Þ Simulation result presented in Fig. 8 shows a remarkable time response (Tr ¼ 1:5 s, %OS ¼ 2 and Ts ¼ 3:3 s) for process P5. 0.0 0.5 1.0 1.5 y(t) 0 5 10 15 20 25 30 35 40 1.0 2.0 3.0 4.0 Time (s) u(t) Present He et. al. Fig. 6. Time response and controller response for process P3 with 20% disturbance at t ¼ 20 s. -4.0 0.0 4.0 y(t) 0 10 20 30 40 50 -4.0 0.0 4.0 Time (s) u(t) Present He et. al. Fig. 7. Time response and controller response for process P4 with 20% disturbance at t ¼ 30 s. 0.0 0.4 0.8 1.2 y(t) 0 10 20 30 40 50 2.0 3.0 4.0 5.0 Time (s) u(t) Fig. 8. Time response and controller response for process P5 with 20% disturbance at t¼30 s. S. Srivastava et al. / ISA Transactions 60 (2016) 244–253250
  8. 8. 3.6. Comparison with other time domain tuning methods In order to check the relative merits and demerits of the pre- sent method, simulations have been performed for processes P1, P2, P3 and P5 using other time domain tuning methods [1,3] such as Integral of Square Error (ISE), Integral of Time Square Error (ITSE), Integral of Absolute Error (IAE), Integral of Time Absolute Error (ITAE). We have used the fmincon() function of the MATLAB [22] optimization toolbox for finding the sets of optimized PID controller parameters subject to a given time domain performance index based cost-function. In all the cases the optimization started with same initial value of PID parameters equal to 0.3. i.e. Kp ¼0.3, Ki ¼0.3, Kd ¼0.3. Results are compared in Fig. 9. It is easy to observe that present method gives overall satisfactory closed loop time response. In other cases the response time is fast but with substantial overshoot and oscillations. We have also calculated the control energy using the square of the MATLAB function norm (u(t),2). Except for process P3, where ITAE and ITSE require slightly less control energy, controllers designed with present method need comparatively less control energy for all other cases. X Data y(t) 0.0 0.4 0.8 1.2 IAE ISE ITAE ITSE Present Time (s) 0 10 20 30 40 50 60 70 80 u(t) 0.0 0.8 1.6 2.4 3.2 a b X Data y(t) 0.0 0.4 0.8 1.2 IAE ISE ITAE ITSE Present Time (s) 0 20 40 60 80 100 120 140 u(t) 0.0 0.4 0.8 1.2 1.6 a b X Data y(t) 0.0 0.4 0.8 1.2 IAE ISE ITAE ITSE Present Time (s) 0 10 20 30 40 u(t) 0.0 1.8 3.6 5.4 7.2 a b X Data y(t) 0.0 0.4 0.8 1.2 IAE ISE ITAE ITSE Present Time (s) 0 10 20 30 40 50 u(t) 0 2 4 6 8 10 a b P1 P2 P3 P5 Fig. 9. Time response and controller response for processes P1, P2, P3 and P5 obtained using methods based on different time domain performance measures. S. Srivastava et al. / ISA Transactions 60 (2016) 244–253 251
  9. 9. 3.7. Example 6: robustness test Most of the real plants operate in a wide range of operating conditions and it is required that the controller must be able to stabilize the system with slight change in the operating conditions. In such situation, the robustness of the closed loop system is an important feature. The purpose of studying this process is to check the robustness property of the optimal LQR-PID controller when there is a mismatch between the delay time of the process and the delay time for which the PID controller is designed. We consider an under damped SOPTD process given by P6 ¼ 9 s2 þ1:2sþ9 eÀ 2s : ð51Þ It is easy to observe that the open loop system poles are complex. We have designed the PID controller with closed loop parametric demand of ζcl ¼ 0:98 ; ωcl ¼ 2 rad=s. The optimal LQR based PID controller obtained with m ¼ 3 is C6 ¼ 0:0979þ 0:1913 s þ0:0111s : ð52Þ Fig. 10(a) shows the closed loop step response of the under damped SOPTD process P4 with 20% disturbance at t ¼ 30 s and the corresponding control effort is plotted in Fig. 10(b). It can be readily seen that the stabilization of load disturbance by present controller is quite satisfactory. We have also studied the robustness of the present controller by varying the mismatched delay time Lm from 0.5 s to 4 s covering both sides of the actual delay L ¼ 2 s for which the controller is designed. Results of simulation are presented in Fig. 10(c) for comparison. The time response with designed parameters is shown by black solid line. It can be readily seen that an increase in the value of time Lm from the designed value of L ¼ 2 s, causes an overshoot in the time response and finally leads to the oscillation if the value of Lm becomes larger. In contrast, the mismatched value of Lm less than L is responsible for the increase in the rise time as well as in the settling time and thus making the response sluggish. Since in the present LQR based PID method, we have an extra tuning factor that is the value of relative dominance m which one can utilized to improve the robustness of the time response in the case of a mismatch in the delay time. To explain the effect of m on robustness of the controller we have designed another optimal LQR-PID controller keeping all the parameter same except the value of m. The optimal PID controller for m ¼ 10 is Cm10 ¼ 0:0658þ 0:1586 s þ0:0029s : ð53Þ A comparison of time response curves for cases Lm ¼ 4 s, m ¼ 3 and Lm ¼ 4 s, m ¼ 10 clearly indicates that a controller designed at higher m shows less overshoot and thus will be more robust in the case of mismatch between the process delay time and the delay time at which the controller is designed. However, the penalty one has to pay is the increase in the rise time. 4. Conclusion In this paper an improved design methodology of PID con- troller for standard SOPTD system has been developed by com- bining the optimal approach of LQR and the dominant pole pla- cement technique. The proposed tuning method allows more flexible pole placement, which results in better time response. The PID parameters have been calculated analytically using user defined closed loop damping ratio and natural frequency. It is demonstrated by simulation that present tuning methodology gives improved closed loop time response with less control effort as compared to the earlier developed LQR based PI/PID tuning method. Simulation results indicate that present method works well for most of the SOPTD models such as under-damped, criti- cally-damped, over-damped, unstable and highly oscillatory pro- cesses. It is observed from the simulation that most appropriate range of ωclL for stable SOPTD processes is ωclL A ð1:0; 1:5Þ and for the unstable SOPTD process is ωclL A ð0:1; 0:4Þ. A comparison of simulations results with other time domain performance indices indicates that the present methods gives an overall better closed loop time response with comparatively less control effort. It is observed that the location of non-dominant pole (value of m) affects the closed loop time response provided all others parameters are kept constant. An increase in the value of m, increases the rise time with a substantial control on the overshoot. This observed behavior of the closed loop time response with m in the case of processes with time delay is completely opposite to the cases of delay free processes. A slightly higher value of m adds an extra robustness to the closed loop time response in the case of mismatch between the process delay time and the delay time at which the controller is designed. The proposed analytical tuning method to obtain optimum PID parameters for SOPTD process will be helpful for the on-line applications. We like to point out here that the present approach cannot be applied to integrating pro- cesses because they cannot be represented in the form of standard second order transfer function. Fig. 10. The plots of (a) time response and (b) controller response for under damped SOPTD process P6 with 20% disturbance at t ¼ 30 s. (c) Time response of process P6 at various mismatched delay time. S. Srivastava et al. / ISA Transactions 60 (2016) 244–253252
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