This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON CYBERNETICS
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Adaptive GSA-Based Optimal Tuning of PI Controlled Servo Systems With Reduced Process Parametric Sensitivity, Robust Stability and Controller Robustness Radu-Emil Precup, Senior Member, IEEE, Radu-Codrut David, Emil M. Petriu, Fellow, IEEE, Mircea-Bogdan Radac, Member, IEEE, and Stefan Preitl, Senior Member, IEEE
Abstract—This paper suggests a new generation of optimal PI controllers for a class of servo systems characterized by saturation and dead zone static nonlinearities and second-order models with an integral component. The objective functions are expressed as the integral of time multiplied by absolute error plus the weighted sum of the integrals of output sensitivity functions of the state sensitivity models with respect to two process parametric variations. The PI controller tuning conditions applied to a simplified linear process model involve a single design parameter specific to the extended symmetrical optimum (ESO) method which offers the desired tradeoff to several control system performance indices. An original back-calculation and tracking anti-windup scheme is proposed in order to prevent the integrator wind-up and to compensate for the dead zone nonlinearity of the process. The minimization of the objective functions is carried out in the framework of optimization problems with inequality constraints which guarantee the robust stability with respect to the process parametric variations and the controller robustness. An adaptive gravitational search algorithm (GSA) solves the optimization problems focused on the optimal tuning of the design parameter specific to the ESO method and of the anti-windup tracking gain. A tuning method for PI controllers is proposed as an efficient approach to the design of resilient control systems. The tuning method and the PI controllers are experimentally validated by the adaptive GSA-based tuning of PI controllers for the angular position control of a laboratory servo system. Index Terms—Anti-windup tracking gain, controller robustness, objective functions, PI controllers, process parametric sensitivity, servo systems.
I. Introduction ERVO systems are important in many applications which belong to complex control systems in infrastructures for energy, transportation, sustenance, medical care, emergency
S
Manuscript received March 9, 2013; revised September 29, 2013; accepted February 11, 2014. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0109, and by a grant from the NSERC of Canada. This paper was recommended by Associate Editor W. Smart. R.-E. Precup, R.-C. David, M.-B. Radac, and S. Preitl are with Department of Automation and Applied Informatics, Politehnica University of Timisoara, Timisoara 300223, Romania (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). E. M. Petriu is with the School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON K1N 6N5, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCYB.2014.2307257
response, and physical and cyber security [1]–[9]. With this regard the optimal control of servo systems is important to ensure very good control system (CS) performance by the minimization of objective functions expressed as the integral quadratic performance indices where the variables are the controller tuning parameters. As shown in [10]–[14], resilience is the capacity of a CS to maintain state awareness and to proactively maintain a safe level of operational normalcy in response to anomalies, including threats of a malicious and unexpected nature. The resilience is especially important in the control of servo systems in the context of complex and critical infrastructures. This paper will propose a new approach to the design of resilient CSs based on the fulfillment of three performance specifications. First, the controller tuning ensures a reduced sensitivity with respect to the parametric variations of the process. Second, the robust stability of the CSs is guaranteed accepting that the parameters of the CSs vary within certain intervals set by the levels of operational normalcy. Third, as mentioned in [15]–[18], for practical implementations any controller must be nonfragile or controller robust so that the CS is not destabilized by round-off errors during the implementations, and the parameters are allowed to vary around the nominal design values. Building upon [19], this paper considers that the controller robustness is guaranteed if it fulfils the robust stability of the CS with respect to the parametric variations of the controller. The main contributions of this paper are the following. 1) An original tuning method which offers resilient CSs by the incorporation of the three performance specifications in terms of the definition of the optimization problems where the variables are the tuning parameters of the controllers. The objective functions are expressed as the integral of time multiplied by absolute error (ITAE) plus the weighted sum of the integrals of output sensitivity functions of the state sensitivity models with respect to the parametric variations of the process. The inequality constraints are defined such that to guarantee the robust stability with respect to the parametric variations of the process and the controller robustness, and the output
c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2
sensitivity functions are obtained from the sensitivity models with respect to the parametric variations of the process. 2) A new generation of optimal PI controllers for a class of servo systems characterized by saturation and dead zone static nonlinearities and second-order models with an integral component. The optimization problems are solved by the modification of the adaptive Gravitational Search Algorithm (GSA) proposed in [20] and proved to be an efficient evolutionary nature-inspired optimization algorithm in the optimal tuning of the parameters of Takagi-Sugeno PI-fuzzy controllers. These modifications concern the reduction of two random variables (which appear in the formula of total force acting on an agent and in the velocity update formula) in two steps of the algorithm. 3) A new back-calculation and tracking anti-windup scheme to prevent the integrator wind-up and to compensate for the dead zone nonlinearity of the process, i.e., of the actuator. This scheme includes a dead zone inverse and saturation nonlinearity that replaces the saturation nonlinearity in the classical back-calculation and tracking anti-windup scheme with actuator model. 4) The tuning of two parameters of the PI controller. One of the parameters is the design parameter specific to the extended symmetrical optimum (ESO) method [21], [22], which sets the desired tradeoff to several CS performance indices (overshoot, settling time, rise time, phase margin, etc.) accepting a simplified linear process model. The second parameter is the anti-windup tracking gain. Our new approach and tuning method to the design of resilient CSs is motivated by the fact that the robust stability and the controller robustness are treated in the linear case for the simplified linear process model and the state sensitivity models are derived on the basis of the nonlinear process model. The sensitivity analysis is required by the unavoidable parametric variations of the process models which affect the resilience. The robust stability analysis is not conducted in the nonlinear case because it is complicated. In addition, the robust stability is guaranteed by inserting a validation condition for the new solution in the evaluation stage of the GSA; this condition guarantees the convergence of the objective functions. The new approach and tuning method proposed in this paper are important in the context of the state-of-the-art [10]–[14] because they represent an efficient, viz. optimal approach to the design of resilient CSs by the simultaneous fulfillment of three performance specifications: reduced process parametric sensitivity, robust stability, and controller robustness. The combination of resilience and robust stability has been investigated in [13], and the resilience and controller robustness have been discussed in [17]. The use of GSA is employed for this problem in order to overcome its difficulty as it has been already proven as a reliable solution in [20] and [23]. Other evolutionary algorithms might present as viable alternatives as shown in [24] and [25], but for this particular problem a novel variation of the GSA will be implemented: the adaptive GSA, as it
IEEE TRANSACTIONS ON CYBERNETICS
is based on the framework of the regular GSA but with an improved use of allocated resources. The adaptive GSA implemented in this paper is thus justified because it provides a more advantageous solution compared to the regular GSA as it will be established in the next sections. The application of the ESO method is motivated by the reduction of the design parameters of the PI controller to only two ones, the design parameter specific to the ESO method and the anti-windup tracking gain. This reduction of the number of variables of the objective functions to only two ones has benefic effects on the implementation of the adaptive GSA in terms of a search space of dimension two. That is the reason why our approach is attractive in cost-effective automation solutions for processes in many applications including CSs in complex systems and infrastructures [26]–[33]. The consideration of the anti-windup tracking gain in the tuning parameter of the PI controller is advantageous as it does not lead to discussions on the strong or weak impacts of the integrator wind-up and of saturation [34]–[38]. The anti-windup tracking gain is obtained as a solution to the optimization problem and the impacts are incorporated in the minimization of the objective functions performed by the adaptive GSA. Our back-calculation and tracking antiwindup scheme yields not only the prevention of integrator wind-up but also ensures the compensation of the dead zone nonlinearity of the process. This paper is organized as follows. Section II presents the problem which includes the optimization problem, the derivation of the state sensitivity models, and the robust stability and controller robustness analyses. The adaptive GSA and the new tuning method are discussed in Section III. Section IV deals with the case study of the optimal PI controller tuning for the angular position control of a laboratory DC servo system. A set of experimental results is given. The conclusions are pointed out in Section V. II. Optimization Problem, Sensitivity Models, Robust Stability, and Controller Robustness Analyses Let the process as part of servo systems be characterized by the following continuous-time nonlinear time-invariant single input-single output (SISO) state-space model:
m(t) =
⎧ −1, ⎪ ⎪ ⎪ ⎪ [u(t) + u c ]/(ub − uc ), ⎪ ⎪ ⎨ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
if u(t)≤ − ub , if − ub
1
Adaptive GSA-Based Optimal Tuning of PI Controlled Servo Systems With Reduced Process Parametric Sensitivity, Robust Stability and Controller Robustness Radu-Emil Precup, Senior Member, IEEE, Radu-Codrut David, Emil M. Petriu, Fellow, IEEE, Mircea-Bogdan Radac, Member, IEEE, and Stefan Preitl, Senior Member, IEEE
Abstract—This paper suggests a new generation of optimal PI controllers for a class of servo systems characterized by saturation and dead zone static nonlinearities and second-order models with an integral component. The objective functions are expressed as the integral of time multiplied by absolute error plus the weighted sum of the integrals of output sensitivity functions of the state sensitivity models with respect to two process parametric variations. The PI controller tuning conditions applied to a simplified linear process model involve a single design parameter specific to the extended symmetrical optimum (ESO) method which offers the desired tradeoff to several control system performance indices. An original back-calculation and tracking anti-windup scheme is proposed in order to prevent the integrator wind-up and to compensate for the dead zone nonlinearity of the process. The minimization of the objective functions is carried out in the framework of optimization problems with inequality constraints which guarantee the robust stability with respect to the process parametric variations and the controller robustness. An adaptive gravitational search algorithm (GSA) solves the optimization problems focused on the optimal tuning of the design parameter specific to the ESO method and of the anti-windup tracking gain. A tuning method for PI controllers is proposed as an efficient approach to the design of resilient control systems. The tuning method and the PI controllers are experimentally validated by the adaptive GSA-based tuning of PI controllers for the angular position control of a laboratory servo system. Index Terms—Anti-windup tracking gain, controller robustness, objective functions, PI controllers, process parametric sensitivity, servo systems.
I. Introduction ERVO systems are important in many applications which belong to complex control systems in infrastructures for energy, transportation, sustenance, medical care, emergency
S
Manuscript received March 9, 2013; revised September 29, 2013; accepted February 11, 2014. This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project number PN-II-ID-PCE-2011-3-0109, and by a grant from the NSERC of Canada. This paper was recommended by Associate Editor W. Smart. R.-E. Precup, R.-C. David, M.-B. Radac, and S. Preitl are with Department of Automation and Applied Informatics, Politehnica University of Timisoara, Timisoara 300223, Romania (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). E. M. Petriu is with the School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, ON K1N 6N5, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCYB.2014.2307257
response, and physical and cyber security [1]–[9]. With this regard the optimal control of servo systems is important to ensure very good control system (CS) performance by the minimization of objective functions expressed as the integral quadratic performance indices where the variables are the controller tuning parameters. As shown in [10]–[14], resilience is the capacity of a CS to maintain state awareness and to proactively maintain a safe level of operational normalcy in response to anomalies, including threats of a malicious and unexpected nature. The resilience is especially important in the control of servo systems in the context of complex and critical infrastructures. This paper will propose a new approach to the design of resilient CSs based on the fulfillment of three performance specifications. First, the controller tuning ensures a reduced sensitivity with respect to the parametric variations of the process. Second, the robust stability of the CSs is guaranteed accepting that the parameters of the CSs vary within certain intervals set by the levels of operational normalcy. Third, as mentioned in [15]–[18], for practical implementations any controller must be nonfragile or controller robust so that the CS is not destabilized by round-off errors during the implementations, and the parameters are allowed to vary around the nominal design values. Building upon [19], this paper considers that the controller robustness is guaranteed if it fulfils the robust stability of the CS with respect to the parametric variations of the controller. The main contributions of this paper are the following. 1) An original tuning method which offers resilient CSs by the incorporation of the three performance specifications in terms of the definition of the optimization problems where the variables are the tuning parameters of the controllers. The objective functions are expressed as the integral of time multiplied by absolute error (ITAE) plus the weighted sum of the integrals of output sensitivity functions of the state sensitivity models with respect to the parametric variations of the process. The inequality constraints are defined such that to guarantee the robust stability with respect to the parametric variations of the process and the controller robustness, and the output
c 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. 2168-2267 See http://www.ieee.org/publications standards/publications/rights/index.html for more information.
This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2
sensitivity functions are obtained from the sensitivity models with respect to the parametric variations of the process. 2) A new generation of optimal PI controllers for a class of servo systems characterized by saturation and dead zone static nonlinearities and second-order models with an integral component. The optimization problems are solved by the modification of the adaptive Gravitational Search Algorithm (GSA) proposed in [20] and proved to be an efficient evolutionary nature-inspired optimization algorithm in the optimal tuning of the parameters of Takagi-Sugeno PI-fuzzy controllers. These modifications concern the reduction of two random variables (which appear in the formula of total force acting on an agent and in the velocity update formula) in two steps of the algorithm. 3) A new back-calculation and tracking anti-windup scheme to prevent the integrator wind-up and to compensate for the dead zone nonlinearity of the process, i.e., of the actuator. This scheme includes a dead zone inverse and saturation nonlinearity that replaces the saturation nonlinearity in the classical back-calculation and tracking anti-windup scheme with actuator model. 4) The tuning of two parameters of the PI controller. One of the parameters is the design parameter specific to the extended symmetrical optimum (ESO) method [21], [22], which sets the desired tradeoff to several CS performance indices (overshoot, settling time, rise time, phase margin, etc.) accepting a simplified linear process model. The second parameter is the anti-windup tracking gain. Our new approach and tuning method to the design of resilient CSs is motivated by the fact that the robust stability and the controller robustness are treated in the linear case for the simplified linear process model and the state sensitivity models are derived on the basis of the nonlinear process model. The sensitivity analysis is required by the unavoidable parametric variations of the process models which affect the resilience. The robust stability analysis is not conducted in the nonlinear case because it is complicated. In addition, the robust stability is guaranteed by inserting a validation condition for the new solution in the evaluation stage of the GSA; this condition guarantees the convergence of the objective functions. The new approach and tuning method proposed in this paper are important in the context of the state-of-the-art [10]–[14] because they represent an efficient, viz. optimal approach to the design of resilient CSs by the simultaneous fulfillment of three performance specifications: reduced process parametric sensitivity, robust stability, and controller robustness. The combination of resilience and robust stability has been investigated in [13], and the resilience and controller robustness have been discussed in [17]. The use of GSA is employed for this problem in order to overcome its difficulty as it has been already proven as a reliable solution in [20] and [23]. Other evolutionary algorithms might present as viable alternatives as shown in [24] and [25], but for this particular problem a novel variation of the GSA will be implemented: the adaptive GSA, as it
IEEE TRANSACTIONS ON CYBERNETICS
is based on the framework of the regular GSA but with an improved use of allocated resources. The adaptive GSA implemented in this paper is thus justified because it provides a more advantageous solution compared to the regular GSA as it will be established in the next sections. The application of the ESO method is motivated by the reduction of the design parameters of the PI controller to only two ones, the design parameter specific to the ESO method and the anti-windup tracking gain. This reduction of the number of variables of the objective functions to only two ones has benefic effects on the implementation of the adaptive GSA in terms of a search space of dimension two. That is the reason why our approach is attractive in cost-effective automation solutions for processes in many applications including CSs in complex systems and infrastructures [26]–[33]. The consideration of the anti-windup tracking gain in the tuning parameter of the PI controller is advantageous as it does not lead to discussions on the strong or weak impacts of the integrator wind-up and of saturation [34]–[38]. The anti-windup tracking gain is obtained as a solution to the optimization problem and the impacts are incorporated in the minimization of the objective functions performed by the adaptive GSA. Our back-calculation and tracking antiwindup scheme yields not only the prevention of integrator wind-up but also ensures the compensation of the dead zone nonlinearity of the process. This paper is organized as follows. Section II presents the problem which includes the optimization problem, the derivation of the state sensitivity models, and the robust stability and controller robustness analyses. The adaptive GSA and the new tuning method are discussed in Section III. Section IV deals with the case study of the optimal PI controller tuning for the angular position control of a laboratory DC servo system. A set of experimental results is given. The conclusions are pointed out in Section V. II. Optimization Problem, Sensitivity Models, Robust Stability, and Controller Robustness Analyses Let the process as part of servo systems be characterized by the following continuous-time nonlinear time-invariant single input-single output (SISO) state-space model:
m(t) =
⎧ −1, ⎪ ⎪ ⎪ ⎪ [u(t) + u c ]/(ub − uc ), ⎪ ⎪ ⎨ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
if u(t)≤ − ub , if − ub
