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Both inverse control and internal model control have been recently used in non-linear control systems. Many o f the control methods using neural/neuro-fiizzy networks are based on the principle o f inverse control. Neural networks have been also used in non­ linear internal model control lately. By studying the control principles of these schemes, it can be seen that the strengths o f internal model control may compensate the disadvantages o f inverse control. The principle o f inverse control is the dynamics cancellation o f the controlled plant. This is a special case of model reference

feedforward control in which the controller is cascaded with the plant. The block combining the controller and the plant is called the reference model. When this reference model is chosen to have no dynamics, the task of the controller is to achieve total cancellation o f the dynamics o f the plant. When they are combined, the two blocks disappear and reduce to an identity transfer function, so that the output from the system is exactly the input to the controller. This is the concept of perfect control. When there is dynamics in that model, the control system can also be viewed as a detuned inverse control system. When implementing a neural network in an inverse control scheme, usually the function approximation ability o f the neural network is used as the controller to perform inverse mapping. Let U denote the input to the process and Y denote the process output. The task o f the neural network is to produce U given Y. However, using Y alone as an input is not sufficient to generate U correctly. The most common remedy is to add other inputs, for example state feedback signals as explained in chapter (4). An inverse control scheme has a few serious problems. It is not possible to obtain an inverse model in some cases and the inverse controller is not robust. A control scheme is called robust when it remains stable under model uncertainty or inaccuracy.

The Internal Model Control (IMC) system was first introduced by [Garcia and Morari, 1985]. They designed an overall structure using a linear single-input single-output (SISO) discrete time process model. Then, they extended the SISO systems to multi­ input multi-output (MIMO) systems. This control structure presents a model predictive process control algorithm. Actually, the name IMC came from the fact that the process model is explicitly an internal part o f the controller. The IMC provides a

straightforward yet effective framework for analysis of control system performance, especially with respect to stability and robustness issues. The design of IMC is also simpler and more transparent than that o f traditional control methods even when the goal is just a conventional PID feedback controller. IMC is composed of an inverse model connected in series with the plant and a forward model connected in parallel with the plant, this structure allows the error feedback to reflect the effect of disturbance and plant mismodelling resulting in a robust control loop. IMC is characterized by its fast smooth response to set-point changes and robustness against parametric changes. Also, if the match between the plant and the plant model is perfect, perfect control is achieved. However, perfect matching between plant and plant model is difficult to obtain and may lead to sensitivity problems. Normally, a pre-filter is introduced before the controller in the control loop forward path to reduce the gain of the feedback system in order to move away from the perfect controller and to introduce desirable robustness to the closed-loop system. Detailed study for IMC robustness and stability issues can be found in [Morari and Zafiriou, 1989]. The IMC structure is shown in figure (5.1). This structure consists o f the plant O p to be controlled, the model of the plant Om, the inverse model o f the plant O c which represents the controller, and R, U, Y, and D the vectors o f the reference inputs to the system, the control inputs to the plant, the system outputs and the external disturbances respectively.

For simplicity, all these quantities are assumed to be o f dimension n. In general the IMC requires that both the plant O p and the controller O c be stable. In the case of an open-loop unstable plant, pre-stabilization for the plant by a conventional feedback

R. Oc

u

Figure (5.1). Standard internal model control structure.

D

R

Controller _Plant

Figure (5.2). Classical feedback control structure.

The particular structure o f IMC shown in Figure (5.1) can be proved to be equivalent to that of the conventional linear feedback control structure illustrated in Figure (5.2) regarding the following transformation:

^ ( Z + C O J - ' C (5.2)

The equivalence o f these two control structures implies that whatever is possible employing a conventional linear control structure can be accomplished with the IMC structure and vice versa. However, it is more straightforward to design O c instead of designing C. Furthermore, the IMC structure allows designers to include robustness as a design objective in a very intuitive manner [Garcia and Morari, 1985]. These can be illustrated by examining the IMC properties.

From the block diagram shown in figure (5.1), the input output (from R to Y) transfer function Or , and the disturbance transfer function (from D to Y) Od of the IMC system can be derived as:

(5.3) [ 7 + 0 0 - 0 O 1 L p c m e J (5.4) r / - o $ + 0 ) 0 1 L m e p c J

Equations (5.3) and (5.4) can be rewritten as:

(j>D= [<jr'+(dr1-<j>m

The most important property o f Equations (5.3) and (5.4) is that if the IMC controller is designed to be equal to the plant inverse model ( O c= O ”1), perfect reference tracking (Y = R) with asymptotically vanishing control error and disturbance rejection can be achieved despite any model/plant mismatch (i.e. O p ^ O "1). This can be seen from Equations (5.5) and (5.6) as for ( O c = O m[), the input transfer function and the disturbance transfer function become <DR = I and O d = 0 , respectively. If a low-pass pre-filter F is introduced in the control loop, Equation (5.5) can be rewritten as:

+ (5.7)

In the ideal case, i.e. when the plant model is perfect and there is no disturbance, the above equation results in O R = F , which means that a desired closed-loop robust performance of the control system can be easily achieved by a proper design o f the pre­ filter. Furthermore, by choosing the pre-filter dynamics appropriately, the stability of the closed-loop system can be achieved for any degree o f plant/model mismatch. In general, slower filters are required for large model errors. This can be interpreted as (ide-tuning) o f an ideal controller, while the procedure is more straightforward and intuitive than that o f conventional linear controller.

Another important property o f the IMC is that if both of the plant <DP and the IMC controller O c are stable, the stability o f the overall IMC system is achieved subjected to perfect plant modelling ( ^ p — O m ).This can be seen from Figure (5.1) as for(O p= O m), the plant input control signal and plant output Y can be derived as follows:

U = cDc( R - D < D p) (5.8)

Y = (Dp<DcR + ( / - a ) p<Dc)D ® p

(5.9)

From Equations (5.8) and (5.9), the internal stability o f O p and O c determines the stability of U and Y. Therefore, the overall IMC system in Figure (5.1) will be stable for stable <bp and <DC.

It is clear from the literature that the IMC approach has not been widely applied to the control of mechanical systems. The reason for this could be that the IMC scheme, in its original design form is applicable only to asymptotically stable systems, which is not the case for most mechanical systems. IMC is a powerful control strategy for linear systems, however its performance when applied to non-linear processes is not good enough. The development o f a general non-linear extension of IMC faces the difficulty that non-linear systems are usually described by non-linear models while linear IMC is based on transfer function models in addition to the lack in powerful tools for design

controllers have been reported due to recent advances in intelligent modelling techniques. Actually, the key characteristics of the IMC described above also apply in the non-linear case. For example, a number of researchers have suggested Neural Networks to provide the non-linear plant models necessary for IMC from input/output data collected from the plant. Likewise, the application of neural networks to the inverse modelling o f non-linear systems is common in the literature, particularly in the field of robotics control. This due to the fact that Neural Networks parallel processing architecture, adaptation and learning capabilities, and fast processing for large-scale dynamic systems provide solid base to represent the robot forward and inverse model within the IMC controller structure. Li et al. proposed compensations procedure for the robot dynamics, before the standard IMC scheme can be applied. This compensation procedure consists o f two stages, namely pre-linearization using approximate inverse dynamic model and pre-stabilization using a conventional PD feedback loop [Li et al., 1995]. Li et al. proposed an adaptive algorithm based on Neural Networks to construct a joint-based IMC for robot manipulators. In this method, a Neural Network inverse model and a conventional PD feedback were used to pre-linearise and pre-stabilize the plant in a fixed structure IMC controller. The utilized Neural Network consists of an n sub-network structure, each sub-network operates independently based on each link angle, velocity, and acceleration to generate respective link actuating torque.