**1. Introduction**

30 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

Zhuang, M. & Atherton, D.P. (1993). Automatic Tuning of Optimum PID Controllers, *IEE* 

ISSN 0143-7054, 1993

*Proceedings, Part D: Control Theory and Applications,* Vol. 140, No. 3, pp. 216-224,

The PID controllers (P, PD, PI, PID) are very widely used, very well and successfully applied controllers to many applications, for many years, almost from the beginning of controls applications (D'Azzo & Houpis, 1988)(Franklin et al., 1994). (The facts of their successful application, good performance, easiness of tuning are speaking for themselves and are sufficient rational for their use, although their structure is justified by heuristics: "These ... controls - called proportional-integral-derivative (PID) control - constitute the heuristic approach to controller design that has found wide acceptance in the process industries." (Franklin et al., 1994, pp. 168)).

 In this chapter we state a problem whose solution leads to the PID controller architecture and structure, thus avoiding heuristics, giving a systematic approach for explanation of the excellent performance of the PID controllers and gives insight why there are cases the PID controllers do not work well. Namely, by the use of Linear Quadratic Tracking (LQT) theory (Kwakernaak & Sivan, 1972)(Anderson & Moore, 1989) control-tracking problems are formulated and those cases when their solution gives the PID controllers are shown.

Further, problem of controlling-tracking high order polynomial inputs and rejecting high order polynomial disturbances is formulated. By applying the LQT theory extended family of PID controllers – the family of generalized PID controllers denoted PImDn-1 is derived. This provides tool for application of optimal controllers for those systems that the conventional PID controllers are not satisfactory, for generalization and derivation of further results. The notation of generalized PID controllers, PImDn-1, is consistent with the notation of controllers for fractional order systems (Podlubny, 1999).

The present work is strongly motivated by problems-question tackled by the author during a continuous work on high performance servo and motion control applications. Some of the theoretical results that have had motivated and led to the present work have been documented in (Rusnak, 1998, 1999, 2000a, 2000b). Some of the presented architectures appear and are recommended for use in (Leonhard, 1996, pp. 80, 347) without rigorous rationale and were partial trigger for the presented approach.

By Architecture we mean, loosely, the connections between the outputs/sensors and the inputs/actuators; Structure deals with the specific realization of the controllers' blocks; and Configuration is a specific combination of architecture and structure. These issues fall within the control and feedback organization theory that have been reviewed and presented in a concise form in (Rusnak, 2002, 2005) and in a widened form in (Rusnak, 2006, 2008). It is beyond the scope of this chapter. It is used here as a basis at a system theoretic level to

Family of the PID Controllers 33

Throughout this chapter the same notation for time domain and Laplace domain is used,

The optimal tracking problem is introduced in (Kwakernaak & Sivan, 1972) (Anderson &

; () , *o o x Ax Bu x t x*

where *x* is the state; *u* is the input and *y* is the measured output, *xo* is a zero mean random

; () , *r r r r r r o ro*

where *xr* is the state; *wr* is the input and *yr* is the reference output; *wr* is a zero mean stochastic process, *xro* is zero mean random vector. Further it is assumed that n=. The case

The integral action is introduced into the control in order to "force" zero tracking errors for polynomial inputs, and to attenuate disturbances (Kwakernaak & Sivan, 1972)(Anderson & Moore, 1989). This is done by introducing the auxiliary variables, integrals of the tracking error. This way the generalized PID controller, denoted PImDn-1, is created. That is, the state

11 1

*e r ex*

*C x x Ce*

;

1

<sup>2</sup> ; () , *o o*

() () () () () ()

*y t yt G y t yt t G t*

*<sup>T</sup> <sup>T</sup> rf f rf f f f*

2 () () () () () () () ()

*t*

1 2

*T T T*

1 2

*y t y t Q y t y t t Q t u t Ru t*

 

(4)

 

 

2 21

*C*

*e*

 

> 

*m em m*

1

*m*

where (m) is the number of integrators that are introduced on the tracking error.

*r r*

*C*

*x r*

 

*e xx*

*x Ax Bw x t x*

(1)

(2)

(3)

(5)

*y Cx*

*r rr*

*y Cx*

and the explicit Laplace variable (s) is stated to avoid confusion wherever necessary.

**2. The optimal tracking problem** 

Moore, 1989). The nth order system is

The th order reference trajectory generator is

n*≠* is beyond the scope of this chapter.

vector.

is augmented by

The control objective is

1

*J E*

*f*

*t*

*o*

*t*

enable formulation of the control-feedback loops organization problem that leads to the family of generalized PID controllers. This article does not deal with the numerical values of the controllers' filters coefficients/gains; rather it concentrates in organization of the control loop and structure of the filters. This is the way the optimal LQT theory is used.

The LQT theory requires a reference trajectory generator. The reference trajectory is generated by a system that reflects the nominal behavior of the plant. The differences are the initial conditions, the input to the reference trajectory generator and the deviation of the actual plant from the nominal one. The zero steady state error is imposed by integral action of a required order on the state tracking error.

The generalized controllers derived by the presented methodology have been applied to high performance motion control in (Nanomotion, 2009a, 2009b) and to high performance missile autopilot in (Rusnak and Weiss, 2011).

The novelty of the results in this approach is that it shows for what problems a controller from the family of PID controllers is the optimal controller and for which it is not.

The importance of this result is:


The results on the architecture and structure of the PID controllers' family for 1st and 2nd order are rederived in the article. Specifically, it is shown that the classical one block PID controller is optimal for Linear Quadratic Tracking problem of a 2nd order minimum phase plant. For plants with non-minimum phase zero the family of PID controllers is only suboptimal. Multi output single input architectures are proposed that are optimal.

enable formulation of the control-feedback loops organization problem that leads to the family of generalized PID controllers. This article does not deal with the numerical values of the controllers' filters coefficients/gains; rather it concentrates in organization of the control

The LQT theory requires a reference trajectory generator. The reference trajectory is generated by a system that reflects the nominal behavior of the plant. The differences are the initial conditions, the input to the reference trajectory generator and the deviation of the actual plant from the nominal one. The zero steady state error is imposed by integral action

The generalized controllers derived by the presented methodology have been applied to high performance motion control in (Nanomotion, 2009a, 2009b) and to high performance

The novelty of the results in this approach is that it shows for what problems a controller

1. From theoretical point of view it is important to know that widely used control

2. The solution shows for what kind of systems the PID controller is optimal and for which systems it is not, thus showing why a PID controller does not perform well for all systems. This will enable to forecast what control designs not to apply a PID controller. 3. For those systems that the PID is not the controller architecture derived from the optimal control approach shows what is the optimal controller architecture and

4. The present approach advises how to design PID controller on finite time interval, i.e.

5. The generalization can be used in deriving generalized PID controllers for high order SISO systems, for SIMO and MIMO systems (Rusnak, 1999, 2000a), for time–varying and non-linear systems; thus enabling a systematic generalization of the PID controller

6. The design procedures of PID controllers are assuming noise free environment. The presented approach advises how to generalize the PID controller configuration in presence of noises by the use of the Linear Quadratic Gaussian Tracking-LQGT theory

7. The conventional PID paradigm introduces integral action in order to drive the steady state tracking errors in presence of constant reference trajectory or disturbance. The present approach enables to systematically generalize the controller to drive the steady

state tracking errors to zero for high order polynomial inputs and disturbances. 8. Choosing the optimal generalized PID controller reduces the quantity of controller parameters-gains that are required for tuning, Thus saving time during the design process. 9. The LQT motivated architecture enables separate treatment of the transient, by the trajectory generator, and the steady state performance by introducing integrators into

The results on the architecture and structure of the PID controllers' family for 1st and 2nd order are rederived in the article. Specifically, it is shown that the classical one block PID controller is optimal for Linear Quadratic Tracking problem of a 2nd order minimum phase plant. For plants with non-minimum phase zero the family of PID controllers is only

suboptimal. Multi output single input architectures are proposed that are optimal.

from the family of PID controllers is the optimal controller and for which it is not.

architecture can be derived from an optimal control/tracking problem.

loop and structure of the filters. This is the way the optimal LQT theory is used.

of a required order on the state tracking error.

missile autopilot in (Rusnak and Weiss, 2011).

structure, thus achieving generalization.

the controllers (Rusnak and Weiss, 2011).

when the gains are time varying.

paradigm.

(Rusnak, 2000b).

The importance of this result is:

Throughout this chapter the same notation for time domain and Laplace domain is used, and the explicit Laplace variable (s) is stated to avoid confusion wherever necessary.
