Abstract
In previous papers by the use of the architecture generating property of the optimal linear quadratic tracking theory those problems that give the PI, PD and PID controllers had been presented, thus avoiding heuristics and giving a systematic approach to explanation for their excellent performance. This approach has been used also in derivation of the family of generalized PI m D n-1 controllers. In this paper by the combined use of the optimal linear quadratic tracking and system sensitivity theories a problem is formulated and solutions are shown that give a family of robust PID controllers. Examples of the control architectures and structures for first and second order systems are presented.
INTRODUCTION
The PD, PI and PID controllers are successfully applied controllers to many applications, almost from the beginning of controls applications; see D'Azzo and 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. Robustness issues are dealt with mainly by tuning algorithms of the PID controller's gains. Previous papers, Rusnak (1998 Rusnak ( , 1999 , have shown that the PID controllers can be derived by the use the optimal tracking control. Namely, problems had been stated whose solutions lead to the PD, PI and PID controllers. This enables avoiding heuristics and is giving a systematic approach to explanation of the good performance of the PID controllers. This approach enabled generalization and derivation of the family of the generalized PID controllers, the PI m D n-1 controllers; Rusnak (2000a Rusnak ( , 2000b . The classical one block PID controller has been generalized, the cascade architecture -the PIV controller, has been presented for PID controller with one integrator, Rusnak (2011) , and the PI2D, IPIV and PI2V controllers for controller with two integrators, Rusnak (2011) , have been derived. One of the main contributions of the results in Rusnak (2011) is that they show for what problems the PID controllers are the optimal controllers and for which they are not. The generalized PI 2 D controller had been implemented in real-time on Linear Motion Control System (Dvash et. all) . The use of the optimal linear quadratic tracking (LQT) theory gives the architecture as well as the structure of the controllers. 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 presented in Rusnak (2006 Rusnak ( , 2008 . This issue is beyond the scope of this paper. The optimal tracking theory is used here as a basis, at a system theoretic level, that enables formulation of the control-feedback loops organization problem that leads to the robust PID controllers. This paper does not deal with the numerical values of the controllers' filters coefficients/gains; rather it concentrates in organization of the control loops 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 behaviour 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 is imposed by integral action of the required order on the state tracking error. The issue of sensitivity of control systems, in general, and design of robust controllers, in particular, are widely treated issues. These are for example the H ∞ , QFT design techniques and more. These techniques, albeit their vitality, do not enable generalization. This is as they do not possess the architecture generating property. The author is unaware of specific treatment of the issue of the architecture and structure of robust PID and robust generalized PID controllers. The novelty and contribution of this paper is the simultaneous application of the LQT theory and sensitivity theory, Frank (1978) , in derivation of architectures and structures of robust PID controllers. It is shown that the presented approach: 1) Can systematically rationalize the existing practices. 2) Enables, due to the architecture generating property of the LQT criterion, systematic generalization of problems for which the generalized PID controllers are optimal. 3) Enables to derive controllers that are robust with respect to specific parameters. 4) Shows the exact structure of the robust controller. Examples of robust controllers for first and second order systems are derived. Throughout this paper the same notation for time domain and Laplace domain is used, and the explicit Laplace variable (s) is stated to avoid confusion wherever necessary.
PRELIMINARIES OF PROBLEM STATEMENT
In this section preliminary definitions essential for the problem statement are introduced. Let's consider the n th order continuous linear time-invariant single input single output system in the observer canonical form.
PRELIMINARIES OF PROBLEM STATEMENT
Cx y Further we assume that the reference system (2.3) is equal to the nominal-average system of (2.2), i.e. In order to reject constant disturbances we introduce integral action, Kwakernaak and Sivan (1972) , Anderson and Moore (1989) , i.e. we consider the integral of the tracking error
PRELIMINARIES OF PROBLEM STATEMENT
In order to achieve robustness we consider the sensitivity functions of the system performance with respect to the system's parameters, . That is, the gradient of the system output with respect to the plant's parameters, Frank (1978) 
PRELIMINARIES OF PROBLEM STATEMENT
and in order to drive the sensitivity to small values (ideally to zero) we consider the integral of the sensitivity function as well, i.e.
PRELIMINARIES OF PROBLEM STATEMENT
All vectors and matrices are of the appropriate dimensions.
PRELIMINARIES OF PROBLEM STATEMENT
The observer canonical form is used for convenience. For n th order system there are at most 2n coefficients-parameters that the state space representation can "capture". From this point of view any canonical could have fit.
THE SENSITIVITY SYSTEM
In section 2, eq. (2.5) the sensitivity functions, Frank (1978) , have been introduced. The sensitivity functions will be minimized in the optimization process to derive robust PID controller. The sensitivity system is, Frank (1978) ,
THE SENSITIVITY SYSTEM
Equations (3.1) can be written The associated transfer functions are
THE SENSITIVITY SYSTEM
Remark: For n th order system there are 2n parameters.
THE ROBUST PID CONTROLLER PROBLEM
The robust PID controller problem is defined. That is, the problem being dealt with here is finding the optimal control u*(t) that minimizes the quadratic criterion:
THE ROBUST PID CONTROLLER PROBLEM
subject to (2.1-6, 3.1-6). That is, the objective is the simultaneous minimization of the squares of the tracking error, e=x-x r , the sensitivity of the output with respect to the plant's parameters,
THE ROBUST PID CONTROLLER PROBLEM
and the input, u(t).
THE AUGMENTED PROBLEM
The problem of the robust PID controller in (4.1), can be written as the minimization of
THE AUGMENTED PROBLEM
where the tracking error is 
THE AUGMENTED PROBLEM
Notice that the solution is independent of the reference trajectory driving input, w' r . For the formal independence w' r is required to be white. It is assumed that for all practical purposes, over the ensemble average of all reference trajectories, this is so.
SOLUTION OF THE ROBUST PID CONTROLLER
PROBLEM By corresponding partition of the gain matrix, K, in eq. (5.7) the optimal input can be written, formally, as
ARCHITECTURES AND STRUCTURES OF THE ROBUST PID CONTROLLER PROBLEM
From section 6 several architectures emerge. Three of them are presented here.
Direct Full State Sensitivity Robust PID Architecture
This architecture and structure are direct implementation of equation (6.4), and is the same architecture as depicted in Frank (1978) , chapter 9, figure 9.4-1, that is This architecture requires the availability-measurement of the full state. 
Indirect State Sensitivity Robust PID Architecture
This architecture and structure do not require the availabilitymeasurement of the full state and the measurement of the input and output is sufficient. The outputs are not used for the sensitivity function computation. This is called the indirect state sensitivity robust PID architecture. The state used for derivation of the sensitivity functions are derive from a plant emulator that is using (2.1) For single input systems we get
Indirect State Sensitivity Robust PID Architecture
The above means that the robust PID controller applies the PID controller, C PID (s) and then in cascade the robustifying controller C Robust (s) . This is a one block controller architecture whose structure is given by
Mixed State Sensitivity Robust PID Architecture
In section 7.1 availability of the full state has been assumed. In this section partial state availability is assumed. The rest of the state is derived by model of the system as in section 7.2 (in stochastic system it will an estimator, issue that is not treated in this paper). For deterministic system for those states that are measured it is the direct state sensitivity and for those states that are not measured it is the indirect state sensitivity. Block diagram of this approach is presented in figure 3 . 
Discussion
The Direct Full State Sensitivity Robust PID Architecture, Figure 1 , section 7.1, is the optimal architecture. However in practice not all states are measurable. The Indirect State Sensitivity Robust PID Architecture, Figure 2 , section 7.2, assumes that no state is available. It is suboptimal but it is the one that is usually implemented. This architecture is the one practiced in design of robust systems. The Mixed State Sensitivity Robust PID Architecture, Figure  3 , section 7.3, a newly proposed architecture, uses the measured states for building the respective sensitivity function and the remaining sensitivity function are derived from a plant's emulator.
ARCHITECTURE AND STRUCTURE OF ROBUST PI CONTROLLERS FOR FIRST ORDER SYSTEM
The case of desensitizing of the performance with respect to the time constant of a first order system is considered. For known systems this approach leads to the P and PI controllers, Rusnak (2011) . The system and the respective transfer function are
ARCHITECTURE AND STRUCTURE OF ROBUST PI CONTROLLERS FOR FIRST ORDER SYSTEM
The objective is (4.1). The sensitivity system is u y
ARCHITECTURE AND STRUCTURE OF ROBUST PI CONTROLLERS FOR FIRST ORDER SYSTEM
, then augmentation of the state and sensitivity function gives
ARCHITECTURE AND STRUCTURE OF ROBUST PI CONTROLLERS FOR FIRST ORDER SYSTEM
In realization of the controller one must remember that the actual time constant, , of the system is unknown. Therefore, the controller is realized with an estimate (the nominal value) of this time constant. The computation of the sensitivity function is realized in the controller. 
Direct Full State Sensitivity

Indirect Full State Sensitivity Robust PID Architecture

P controller
For P controller in the  loop we have 
PI controller
For PI controller in the  loop we have These are the robust P and PI controllers. The main observation is that the robust P/PI controller is a cascade of classical P/PI controller and a second order Lead-Lag. Figure  5 presents a block diagram of this architecture. 
ARCHITECTURE AND STRUCTURE OF ROBUST PID CONTROLLERS FOR SECOND ORDER SYSTEM
The case of desensitizing a second order system with respect to the time constant is dealt with. For known systems this approach leads to the PD and PID controllers, Rusnak (2011) .
ARCHITECTURE AND STRUCTURE OF ROBUST PID CONTROLLERS FOR SECOND ORDER SYSTEM
(Here, for convenience the system is represented in the companion canonical form.) The system and the respective transfer function are
ARCHITECTURE AND STRUCTURE OF ROBUST PID CONTROLLERS FOR SECOND ORDER SYSTEM
The objective is (4.1). Denote 
CONCLUSIONS
The optimal linear quadratic tracking theory and system sensitivity theory have been used to formulate a problem and show solutions that give a family of robust PID controllers. The architecture generating property of the LQT criterion has been exploited to derive architectures and structure of the family of optimal robust PID controllers. This way heuristics were avoided and systematic approach has been presented in derivation of robust PID controllers. Examples of architectures and structures for first and second order systems were presented.

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