**2. The optimal tracking problem**

The optimal tracking problem is introduced in (Kwakernaak & Sivan, 1972) (Anderson & Moore, 1989). The nth order system is

$$\begin{aligned} \dot{\mathbf{x}} &= A\mathbf{x} + Bu; \; \mathbf{x}(t\_o) = \mathbf{x}\_o, \\ y &= \mathbf{C} \mathbf{x} \end{aligned} \tag{1}$$

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

The th order reference trajectory generator is

$$\begin{aligned} \dot{\mathbf{x}}\_r &= A\_r \mathbf{x}\_r + B\_r w\_r; \; \mathbf{x}\_r(t\_o) = \mathbf{x}\_{ro}, \\ y\_r &= \mathbf{C}\_r \mathbf{x}\_r \end{aligned} \tag{2}$$

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

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 is augmented by

$$\begin{aligned} e\_x &= x\_r - x \\ \dot{\eta}\_1 &= C\_{c1} [x\_r - x] = C\_{c1} e\_x \\ \dot{\eta}\_2 &= C\_{c2} \eta\_1; \\ \vdots \\ \dot{\eta}\_m &= C\_{cm} \eta\_{m-1} \end{aligned} \tag{3}$$

$$\eta = \begin{bmatrix} \eta\_1 \\ \eta\_2 \\ \vdots \\ \eta\_m \end{bmatrix}; \qquad \eta(t\_o) = \eta\_o, \tag{4}$$

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

$$J = \frac{1}{2}E\begin{bmatrix} \left[y\_r(t\_f) - y(t\_f)\right]^T G\_1 \left[y\_r(t\_f) - y(t\_f)\right] + \eta(t\_f)^T G\_2 \eta(t\_f) \\ + \int\_{t\_0}^{t\_f} \left[\left[y\_r(t) - y(t)\right]^T Q\_1 \left[y\_r(t) - y(t)\right] + \eta(t)^T Q\_2 \eta(t) + u(t)^T R u(t)\right] \end{bmatrix} \tag{5}$$

Family of the PID Controllers 35

As stated in the introduction Architecture deals with 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 of control and feedback organization theory (Rusnak,

This control architecture is directly derived from the Solution of the Optimal Tracking Problem as derived in (Asseo, 1970) and in (12). The parallel controller can be written

*<sup>C</sup>*<sup>1</sup> <sup>2</sup> *<sup>x</sup>*

1

2 3

1

*C s*

( ) ( ) [( ( ) ( )) [( ( ) ( )) ... ( ) ( )

*x sx s x sx s*

2

*C s*

1

*r*

1

1

(15)

( )( ( ) ( ))]...]} ( )

*x s xs*

2 3

*n n*

1

*C s*

(13)

*us C s x s x s C s x s x s* () () () () () () () 1 1 1 22 2 *r r* (14)

<sup>1</sup> *u x*

*r n rn n*

( ) ( ) ( ) ...... ( ) ( ) ( )

*Csx s xs Csx s x s*

*u s = C s x s x s = C s e s*

*n n*

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

*i ri i i i*

Figure 1 presents the block diagram of the parallel controller architecture for a 2nd order

1 1

*C*2

2 3

*C s C s*

*C s C s*

( ) ( ) ( ){ ( ( ) ( )) [( ( ) ( )) ( )

*C s u s =C s x s x s x s x s*

*<sup>n</sup> nr n r n n n n r n r n*

*n n*

1 2

*n n*

*i i*

 

11 1

For 2nd order system the parallel controller architecture takes the form.

**4. Architectures** 

system.

2006, 2008), and are beyond the scope of this chapter.

1. Parallel controller architecture; 2. Cascade controller architecture; 3. One block controller architecture.

**4.1 Parallel controller architecture** 

directly from (12) in Laplace domain as

*r x*2

**4.2 Cascade controller architecture** 

Fig. 1. Parallel controller architecture for 2nd order system.

By elementary block operation (13) can be written as

*r x*1

In this chapter we deal with three specific architectures. These are:

The optimal tracking problem (Kwakernaak & Sivan, 1972) is to find an admissible input *u(t)* such that the tracking objective (5) is minimized subject to the dynamic constraints (1- 4).

All vectors and matrices are of the proper dimension.
