Adaptive control is the control method used by a controller which must adapt to a controlled system with parameters which vary, or are initially uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; a control law is needed that adapts itself to such changing conditions. Adaptive control is different from robust control

Robust control

Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. Robust control methods are designed to function properly so long as uncertain parameters or disturbances are within some set...

 in that it does not need a priori information about the bounds on these uncertain or time-varying parameters; robust control guarantees that if the changes are within given bounds the control law need not be changed, while adaptive control is concerned with control law changes themselves.

Parameter estimation

The foundation of adaptive control is parameter estimation. Common methods of estimation include recursive least squares and gradient descent

Gradient descent

Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient of the function at the current point...

. Both of these methods provide update laws which are used to modify estimates in real time (i.e., as the system operates). Lyapunov stability

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

 is used to derive these update laws and show convergence criterion (typically persistent excitation). Projection (mathematics)

Projection (mathematics)

Generally speaking, in mathematics, a projection is a mapping of a set which is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a left inverse. Bot notions are strongly related, as follows...

 and normalization are commonly used to improve the robustness of estimation algorithms.

Classification of adaptive control techniques

In general one should distinguish between:

1. Feedforward Adaptive Control
2. Feedback Adaptive Control

as well as between

1. Direct Methods and
2. Indirect Methods

Direct methods are ones wherein the estimated parameters are those directly used in the adaptive controller. In contrast, indirect methods are those in which the estimated parameters are used to calculate required controller parameters

There are several broad categories of feedback adaptive control (classification can vary):

• Dual Adaptive Controllers [based on Dual control theory
  Dual control theory
  Dual control theory is a branch of control theory that deals with the control of systems whose characteristics are initially unknown. It is called dual because in controlling such a system the controller's objectives are twofold:...
  
  ]
  • Optimal Dual Controllers [difficult to design]
  • Suboptimal Dual Controllers
• Nondual Adaptive Controllers
  • Adaptive Pole Placement
  • Extremum Seeking Controllers
  • Iterative learning control
    Iterative learning control
    Iterative Learning Control is a method of tracking control for systems that work in a repetitive mode. Examples of systems that operate in a repetitive manner include robot arm manipulators, chemical batch processes and reliability testing rigs. In each of these tasks the system is required to...
    
    
  • Gain scheduling
    Gain scheduling
    In control theory, gain scheduling is an approach to control of non-linear systems that uses a family of linear controllers, each of which provides satisfactory control for a different operating point of the system....
    
    
  • Model Reference Adaptive Controllers (MRACs) [incorporate a reference model defining desired closed loop performance]
    • Gradient Optimization MRACs [use local rule for adjusting params when performance differs from reference. Ex.: "MIT rule".]
    • Stability Optimized MRACs
  • Model Identification Adaptive Controllers (MIACs) [perform System identification
    System identification
    In control engineering, the field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data...
    
     while the system is running]
    • Cautious Adaptive Controllers [use current SI to modify control law, allowing for SI uncertainty]
    • Certainty Equivalent Adaptive Controllers [take current SI to be the true system, assume no uncertainty]
      • Nonparametric Adaptive Controllers
      • Parametric Adaptive Controllers
        • Explicit Parameter Adaptive Controllers
        • Implicit Parameter Adaptive Controllers

Some special topics in adaptive control can be introduced as well:

1. Adaptive Control Based on Discrete-Time Process Identification
2. Adaptive Control Based on the Model Reference Technique
3. Adaptive Control based on Continuous-Time Process Models
4. Adaptive Control of Multivariable Processes
5. Adaptive Control of Nonlinear Processes

Applications

When designing adaptive control systems, special consideration is necessary of convergence and robustness

Robustness (computer science)

In computer science, robustness is the ability of a computer system to cope with errors during execution or the ability of an algorithm to continue to operate despite abnormalities in input, calculations, etc. Formal techniques, such as fuzz testing, are essential to showing robustness since this...

 issues. Lyapunov stability

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

is typically used to derive control adaptation laws and show convergence.

Typical applications of adaptive control are (in general):

• Self-tuning of subsequently fixed linear controllers during the implementation phase for one operating point;
• Self-tuning of subsequently fixed robust controllers during the implementation phase for whole range of operating points;
• Self-tuning of fixed controllers on request if the process behaviour changes due to ageing, drift, wear etc.;
• Adaptive control of linear controllers for nonlinear or time-varying processes;
• Adaptive control or self-tuning control of nonlinear controllers for nonlinear processes;
• Adaptive control or self-tuning control of multivariable controllers for multivariable processes (MIMO systems);

Usually these methods adapt the controllers to both the process statics and dynamics. In special cases the adaptation can be limited to the static behavior alone, leading to adaptive control based on characteristic curves for the steady-states or to extremum value control, optimizing the steady state. Hence, there are several ways to apply adaptive control algorithms.

Further reading

• K. J. Astrom and B. Wittenmark, Adaptive Control, Addison-Wesley, 1989, 2d ed. 1994.

External links

• Shankar Sastry and Marc Bodson, Adaptive Control: Stability, Convergence, and Robustness, Prentice-Hall, 1989-1994 (book)
• K. Sevcik: Tutorial on Model Reference Adaptive Control (Drexel University)