Book Adaptive control -astrom and witten mark

Topics covered What is adaptive control? (Chap 1) Deterministic self tuning regulators (chap 3) Model reference adaptive systems (chap 5) Properties of adaptive systems (chap 6) Auto tuning (chap 8) Gain scheduling (chap 9) Robust and self oscillating systems (chap 10) Practical issues and implementation (chap 11)

Introduction “to adapt” means to change a behavior to conform to new circumstances. An adaptive controller a controller that can modify its behavior in response to the changes in dynamics of the processes and the disturbances acting on the process.

Contd.. An adaptive controller a controller with adjustable parameters and a mechanism for adjusting the parameters. The parameters are adjusted to compensate for the changes in dynamics of the plant and the disturbances acting on the plant. The controller becomes nonlinear because of the parameter adjustment mechanism

A block diagram of the adaptive controller

Description An adaptive control system can be thought of as having two loops. One loop is a normal feedback with the process and the controller. The other loop is the parameter adjustment loop. The parameter adjustment loop is usually slower than the normal feedback loop.

Circumstances under which adaptive control can be preferred: it is convenient to control a plant with the available conventional PID controllers. Some circumstances under which the adaptive controllers can perform better than the conventional PID controllers are: Change in plant transfer function due to variations in the environment, the size and properties of the raw materials, wear & tear of certain components. Stochastic disturbances (disturbances whose characteristics/behavior are unpredictable )

Contd.. Change in nature of inputs Propagation of disturbances along a chain of unit processes Nonlinear behavior as in case of complex chemical or biochemical reaction Appreciable dead time Unknown parameters, when control system for new process is commissioned.

Effects of process variations The standard approach to control system design is to develop a linear model for the process for some operating condition and to design a controller having constant parameters. A fundamental property is also that feedback systems are intrinsically insensitive to modeling errors and disturbances. The mechanisms causing variation in process dynamics and its effect on the performance of control system is studied in the following section.

Some mechanisms causing variation in process dynamics are: Nonlinear actuators Flow and speed variations Flight control Variation in disturbance characteristics

Nonlinear actuators A very common source of variations is that actuators, like valves have a nonlinear characteristic. Let the static valve characteristics be 0 and let and

Block diagram of a flow control loop with a PI controller and a nonlinear valve

Discussions Linearizing the system around a steady state operating point shows that the incremental gain of the valve is f(u), and hence the loop gain is proportional to f(u). The system can perform well at one operating level and poorly at another

Step responses for PI control of simple flow loop at different operating levels

Discussions The controller is tuned to give a good response at low levels of operating level. For higher values of operating level, the closed loop system even becomes unstable as can be seen in fig.3.

Other examples Flow and speed variations – tank system Flight Control Variations in disturbance characteristics are also discussed for Ship steering control Regulation of quality variable in process control

Adaptive control schemes Gain scheduling Model-Reference Adaptive System (MRAS) Self-Tuning Regulator (STR) Dual Control

Gain Scheduling Gain scheduling is an adaptive control strategy, where the gain of the system is determined and based on its value the controller parameters are changed. This approach is called gain scheduling because the scheme was originally used to measure the gain and then change, that is, schedule the controller to compensate for changes in the process gain.

Block diagram of system with gain scheduling

Description The system can be viewed as having two loops. an inner loop composed of the process and the controller outer loop contains components that adjust the controller parameters on the basis of the operating conditions. regarded as mapping from process parameters to controller parameters. It can be implemented as a function or a table lookup.

Contd.. The concept of gain scheduling originated in connection with the development of flight control systems. In process control, the production rate-a scheduling variable, time constants and time delays are inversely proportional to production rate. Gain scheduling is a very useful technique for reducing the effects of parameter variations.

Advantages and disadvantages Parameters can be changed quickly in response to changes in plant dynamics very easy to apply Drawbacks: It is an open-loop adaptation scheme, with no real learning or intelligence The design required for its implementation is enormous.

Model – Reference Adaptive System (MRAS) Used to solve a problem in which the performance specifications are given in terms of a reference model. This model tells how the process output ideally should respond to the command signal.

Block diagram of MRAS

composed of two loops. The inner loop - the process and an ordinary feedback controller. The outer loop adjusts the controller parameters in such a way that the error, which is the difference between the process output y and model output ym is small. The MRAS was originally introduced for flight control. In this case, the reference model describes the desired response of the aircraft to joystick motions.

The key problem with MRAS is to determine the adjustment mechanism so that a stable system, which brings the error to zero is obtained. parameter adjustment mechanism, called MIT rule was used in original MRAS. e is the error between the plant and model outputs θ is the controller parameter.

The quantity is the sensitivity derivative of error with respect to the parameter θ. The parameter γ is the adaptation rate. It is necessary to make approximation to obtain the sensitivity derivative. The MIT rule can be regarded as a gradient scheme to minimize the squared error e2.

Self Tuning Regulator (STR) The gain scheduling and MRAS are called direct methods, because the adjustment rule tells directly how the controller parameters should be updated. A difference scheme is obtained if the estimates of the process parameters are updated and the controller parameters are obtained from the solution of a design problem using the estimated parameters.

Block Diagram of a STR

composed of two loops. The inner loop - the process and an ordinary feedback controller. The parameters of the controller are adjusted by the outer loop, which is composed of a recursive parameter estimator and a design calculation.

It is sometimes not possible to estimate the process parameters without introducing probing control signals or perturbations. The system may be viewed as an automation of process modeling and design, in which the process model and the control design are updated at each sampling period. A controller of this construction is called a Self Tuning Regulator to emphasize that the controller automatically tunes its parameters to obtain the desired properties of the closed loop system.

Adaptive control problem An adaptive control problem is formulated by defining the following: Description of the process Possible controller structures and Adaptation of controller parameters

Description of the process The process is usually described by linear Single-input Single-output (SISO) system. In the continuous time domain, the process is represented in state space as: Transfer Function form as In discrete time, the process can be described in state space form as: The discrete time system can also be represented in transfer function form as: The process is usually described by linear Single-input Single-output (SISO) system. In the continuous time domain, the process is represented in state space as: Transfer Function form as Where, s is the Laplace Transform variable. In discrete time, the process can be described in state space form as: The discrete time system can also be represented in transfer function form as: Where, z is the z-transform variable. The process is usually described by linear Single-input Single-output (SISO) system. In the continuous time domain, the process is represented in state space as: Transfer Function form as Where, s is the Laplace Transform variable. In discrete time, the process can be described in state space form as: The discrete time system can also be represented in transfer function form as: Where, z is the z-transform variable.

Controller Structures The process is controlled by a controller that has adjustable parameters. Underlying design problem: It is assumed that there exists some kind of design procedure that makes it possible to determine a controller that satisfies some design criteria, if the process and its environment are known.

The adaptive control problem is used to find a method of adjusting the controller when the characteristics of the process and its environment are unknown or changing. In direct adaptive control, the controller parameters are changed directly without the characteristics of the process and its disturbance first being determined. In indirect adaptive methods, the process model and possibly the disturbance characteristics are first determined. The controller parameters are designed on the basis of this information.

Adaptation (adjustment) of controller parameters Various techniques are available like the MIT rule and Lyapunov technique for the MRAS, MDPP LQG for STR. Based on the application and the performance desired; any of the techniques can be chosen.

Construction of an adaptive controller contains the following steps: Characterize the behavior of the closed loop system Determine a suitable control law with adjustable parameters Find a mechanism for adjusting the parameters Implement the control law

Applications of Adaptive control aerospace process control ship steering robotics and automotive biomedical systems.