Cleveland State University EngagedScholarship@CSU Electrical Engineering & Computer Science Electrical Engineering & Computer Science Faculty Publications Department 4-2002 From Linear to Nonlinear Control Means: A Practical Progression Zhiqiang Gao Cleveland State University, [email protected] Follow this and additional works at: https://engagedscholarship.csuohio.edu/enece_facpub Part of the Controls and Control Theory Commons How does access to this work benefit ou?y Let us know! Publisher's Statement NOTICE: this is the author’s version of a work that was accepted for publication in ISA Transactions. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in ISA Transactions, 41, 2, (04-01-2002); 10.1016/S0019-0578(07)60077-9 Original Citation Gao, Z. (2002). From linear to nonlinear control means: A practical progression. ISA Transactions, 41(2), 177-189. doi:10.1016/S0019-0578(07)60077-9 Repository Citation Gao, Zhiqiang, "From Linear to Nonlinear Control Means: A Practical Progression" (2002). Electrical Engineering & Computer Science Faculty Publications. 61. https://engagedscholarship.csuohio.edu/enece_facpub/61 This Article is brought to you for free and open access by the Electrical Engineering & Computer Science Department at EngagedScholarship@CSU. It has been accepted for inclusion in Electrical Engineering & Computer Science Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information, please contact [email protected]. From linear to nonlinear control means: A practical progression Zhiqiang Gao* DeP.1l'lll Ie/!/ of EJoclI"icai ami Computer Eng ineering. Cle l'f~/a /l(1 Slate UlliI'ersily. Clel'claml, Ohio 44115, USA 1. Introduction made in the last 40 years in mathematical control theory. But the level of mathematics required 10 The proportional-integral-derivative (PID) based understand it, the assumptions of linearity and existing control technology has been in existence time invariance, and its dependency on the math- for close to 80 years. N. Minorsky first invented it ematical model of the physical plant have limited in 1922 [I). Ziegler and Nichols provided a set of its appeal for most control practitioners. tuning tables that has been used to this day [2]. Judging from the state of control technology, it Various control fonns and tuning methods related seems that the development in control theory has \0 PID have continued \0 appear in literature, see, not been translated to a breakthrough technology. for example [3 - 7]. Nevertheless, the PID is still In the meanwhile, the control industry has com- the tool of choice in over 90% of current industrial pleted the transition from analog to digital and is control applications [8]. capable of implementing much more sophisticated Classical control theory, particularly the fre- algorithms than PID. But where is our next gen- quency response method, helps control engineers eration control technique that can tap into the 10 get helpful insight on how and why feedback power of the new digital control hardware? control system works, as well as how to improve This paper proposes a new vision on control il. Modem control theory, from Kalman filters to technology progression. Keep in mind that the H", control, represents the tremendous progress new technology should not be overly dependent on the math- ·Tel.: + 1-216-681-3528; fax: + 1-216-681-5405. E-mail ematical model since they are often unavail - address: [email protected] able in practice • needs to provide much better performance at equation is written in terms of the error, e=r low cost -y, where r is the desired output (constant) • must be intuitive and simple to implement e˙ =-w-u. (2) It is in this background that several novel design methods are presented here. Some of these ideas, The objective is to design a control law such that such as the use of nonlinear gains, have been uti- the closed-loop system satisfies e(t)!0, as t lized, in one form or the other, by practicing engi- !0. In practice, in the presence of the distur- neers as ad hoc solutions but were not systemati- bance w, a proportional-integral (PI) control is cally studied. The pioneering work of Professor usually employed: Jingqing Han’s in nonlinear control [9–15] pro- vided a much needed framework for the study of u=K e+K f edt. (3) such techniques. p I In the following sections, several novel and practical design techniques, including nonlinear It is well known that the integrator control helps PID mechanisms and an innovative active distur- to eliminate steady-state error. Unfortunately, it of- ( bance rejection scheme, are presented. Practical ten causes saturation problems known as integra- ) applications are used to demonstrate their applica- tor windup and its 90° phase lag may adversely bility and effectiveness. Note that this research is affect system stability. For these reasons, it is not by no means complete. Instead, it points to a uncommon to see that the integral control is gain- promising new direction of control technology de- scheduled in practice. For example, a common velopment: break out from the constraints of the practice is to zero the integral gain during the tran- existing control practice and theory, and actively sient to prevent overshoot, saturation, and instabil- explore new control design strategies that are not ity. This and other variations of PID are the results overly dependent on mathematical models. Com- of ingenuities of practicing engineers. But, as con- puter aided design tools, such Matlab/Simulink, trol researchers, can we provide them with a more provide an ideal environment for such endeavors powerful set of tools that are less problematic? Let where new ideas can be quickly tested. In fact, this us start the investigation by examining a novel is how the proposed techniques presented here building block of a controller: a nonlinear gain. were first developed. 2.1.1. Nonlinear proportional control 2. Nonlinear PID control mechanisms Intuitively, the reason integral control helps to reduce the steady state error is that it makes the Engineers have long used various modifications controller more sensitive to small errors. This of PID, from gain scheduling to fuzzy logic, in compliments the proportional control actions. An practice. The nonlinear variations usually provide alternative is to use a nonlinear mechanism in the much needed improvements of performance and controller that yields higher gain for smaller er- flexibility of the controller. However, the design rors, such as the one below: process tends to be tedious and solutions problem u=K lela sgn(e), (4) dependent. In this section, a systematic nonlinear • • PID design approach is proposed. It leads to which was systematically used by Han [12–14] to simple, effective, and practical control solutions explore the advantage of nonlinear feedback. The that apply to a wide range of industrial control sgn(e) function is defined as problems. 1, e�0 2.1. Nonlinear proportional-integral control sgn(e)={ -1, e<0. Consider a single integrator plant Note that for a=0 and 1, (4) represents the y˙ =w+u, (1) well-known bang-bang control and the linear pro- portional control, respectively. It is in the case of with y and u as the output and input, respectively, 0<a<1 that an interesting phenomenon was ob- and w as input disturbance. For convenience, the served. The selection of the parameter a is stud- Fig. 2. Disturbance rejection comparisons. Fig. 1. Comparison of the linear and nonlinear controllers. control signal requires a different set of hardware. ied in terms of steady-state error, transient re- For example, in temperature applications, the posi- sponse, disturbance rejection, and robustness, as tive control signal means heating and the negative shown below. one cooling. The above results imply that, with a proper control strategy that produces fast response 2.1.1.1. Steady-state error. For the same gain of without overshoot, the cooling actuator may not K=10, when lw l<1, \t, the steady-state error is be needed even for an aggressive control design 0 < < less than 0.1 for the linear controller with a=1, with 0 a 1. In drive applications, it may result and 0.01 for the nonlinear controller with a in a motor drive that never applies the reversed =1/2. This bound is further reduced to 0.001 with torque, which not only helps to save energy, but a=1/3. In general, as a!0, this bound will ap- also prolongs the life of motor and other parts in proach zero. the drive train, such as coupling and transmission. 2.1.1.2. Transient response. With K=10, the tran- 2.1.1.5. Disturbance rejection. To examine the sient responses for a=0, 1/2, 1, are shown in Fig. disturbance rejection properties of linear and non- = 1. Note that the linear control has the slowest re- linear controllers, a disturbance of w(t) 2 + sponse because it has a long tail end. This is due to sin(10t) is introduced. The results are shown in = its inherent nature of not being sensitive to small Fig. 2 for a 1, 1/2, 1/5. Again, the linear control errors. On the other hand, the bang-bang control is the least desirable one. has the fastest response but the chattering in the control signal renders it impractical. The nonlinear 2.1.1.6. Robustness. Let us examine the robust- ( ) control with 0<a<1 seems to offer a good bal- ness of the control law in 4 with respect to the ( ) ance.