Control Adaptive Systems Natural and artificial nervous systems are often Inman Harvey thought of as control systems or controllers. Informatics We will start by taking a quick look at Control Engineering because this field has many obvious Lecture 17: parallels, influences, applications and useful Control Engineering concepts with respect to autonomous robotics. WARNING: I expect to run out of time before end of this Lecture – and will complete it next week Spring 2010 Control Overview Control engineering grew out of applied mechanics Linear systems and its development in the mid 20th century was Block diagrams tightly bound up with the Cybernetics movement. Feedforward control (Process regulation, target-seeking devices, etc.) Feedback control Nonlinear control Control engineering is the application of Adaptive control mathematical techniques to the design of Fuzzy control algorithms and devices to control processes or Control theory in neuroscience pieces of machinery. It often uses some kind of model of the thing to be controlled. Spring 2010 Spring 2010 Conant & Ashby (1970) Conant & Ashby (1970) “In this paper a theorem is presented which shows, under very broad conditions, that any regulator that is maximally both successful and simple must be isomorphic with the system being regulated”. “The theorem has the interesting corollary that the living brain, so far as it is to be successful and efficient as a regulator for survival, must proceed, in learning, by the formation of a model (or models) of its environment” Download from: http://www.scribd.com/doc/2947203/Every-Good- (Why not read the paper and see if you agree?) Regulator-of-a-System-Must-Be-a-Model-of-that- System-Conant-Ashby Spring 2010 Spring 2010 Let’s start with linear systems Example If a system can be modelled by a set of linear differential equations there are well understood Forced spring techniques for getting exact analytical solutions, and so for designing controllers so that the output of the system is the required one. Spring 2010 Spring 2010 Real systems … Linearization ... are non-linear in almost every practical application. The most common trick: Approximate the nonlinear system by a linear system in the region of interest. Example: a pendulum Use Taylor series expansion and throw away all non-linear terms. For the pendulum, if θ ≈ 0 then sin(θ) ≈ θ and so: Much of control engineering theory and practice is concerned with linear systems. Control of non- linear systems is a very complex area. Spring 2010 Spring 2010 Block diagrams Block diagrams They have their own algebra; Block diagrams are often used for system analysis and design of controllers as they facilitate the study of input/output relationships: transfer G1(t) G2(t) = G2(t)G1(t) functions. u(t)G(t) y(t) The transfer operator could be anything (a function, a derivative, an integration of the input u(t) = input signal, etc.) It is often convenient to use complex y(t) = output variables applying the Laplace transform which allows a simplified mathematical treatment: G(t) = transfer function or operator: differentiation becomes multiplication, and y(t) = G(t)[u(t)] integration becomes division. This is also known as “working in the frequency domain”, as opposed to “working in the time domain”. Spring 2010 Spring 2010 Open loop (feedforward) control Open loop (feedforward) control reference disturbance disturbance reference disturbance disturbance ++ ++ Controller + Plant + output Controller + Plant + output A reference signal (sometimes called the setpoint) The plant is the control engineer’s name for the gives information about the desired output. This physical system that is being controlled – for could be you setting the knob on your toaster to 4, example, a jet engine. The output is what we want to or 7; it could also be an outside temperature sensor control – it might be thrust, or turbine speed, or in a central heating controller. The controller fuel consumption, or brownness of toast, and it’s generates the control input, which is calculated at often called the “controlled variable”. A disturbance the time or in advance (i.e. at design time) to is any external or internal factor that causes the produce the desired output. output to deviate from what is desired. Spring 2010 Spring 2010 Open loop (feedforward) control Feedforward control of toasters Here’s how the cheapest toasters work. When you reference disturbance disturbance switch the toaster on (usually pushing a lever that also lowers the toast into the toaster) the heating ++ coils are powered for a fixed time, which has been Controller + Plant + output calculated by the manufacturer to give good enough The important thing about the reference signal in a results in most cases. pure feedforward controller is that it doesn’t contain any information about the current output. Technically, this means that the control system cannot compensate for disturbances that occur after the control stage. The controller contains an implicit (or sometimes explicit) model of the plant that will produce the “expected” output. Spring 2010 Spring 2010 Feedforward control of toasters Biological feedforward control In more expensive toasters (now more or less Saccadic eye movements use standard) you can turn a knob to vary the time feedforward control. They according to how brown you want your toast. show highly stereotyped velocity profiles. In both types, the controller uses no direct information about the current state of the toast. The fastest movement would use ‘bang-bang’ control (either The output is almost always either “on” or “off”. maximum speed or zero), but This is known as “bang-bang” control. It sounds since noise in muscles is crude, but it can be very efficient, and crops up proportional to the control even in very sophisticated controllers. signal, errors would be large Actual profiles are well predicted by optimising for end-point accuracy Spring 2010 Spring 2010 Feedback – negative and positive Feedback – negative and positive Consider a system – physical, biological, economic – If the net effect of the feedback is to oppose, and that is in equilibrium. What will happen if you therefore reduce, the original disturbance, it is introduce a small disturbance somewhere in the called negative feedback. A system with negative system? feedback is therefore in principle capable of maintaining its equilibrium, and so negative feedback The effects of the disturbance will propagate is associated with stability. through the system. If the parts of the system are But if the net effect of the feedback is to sufficiently well interconnected, some of these reinforce, and thereby increase, the original effects will find their way back to the site of the disturbance, it is called positive feedback. If disturbance – this is called feedback. sustained, this can lead to a runaway increase in the size of the disturbance, taking the system far from equilibrium, and so positive feedback is associated with instability. Spring 2010 Spring 2010 Feedback in biology Feedback in biology Both kinds of feedback are seen in biological Negative feedback, exploited for its stabilising systems. For example, positive feedback features in effects, is found at all levels in biology, from some examples of ant behaviour. When a colony of physiology, where its involvement in homeostasis is Pheidole dentata is attacked by large numbers of perhaps best known, to the control of behaviour. fire ants outside the nest, some Pheidole workers It was the convergence between ideas of negative retreat to the nest and begin to run through the feedback in the control of behaviour and of galleries, laying odour trails and stimulating other machines that influenced Norbert Wiener in his workers do to the same. This behaviour spreads original formulation of cybernetics. rapidly through the nest, and when some threshold Classic paper: Arturo Rosenblueth, Norbert Wiener is reached, the workers seize the eggs, larvae, and and Julian Bigelow in: Philosophy of Science, pupae, and flee the nest, returning some time later 10(1943), S. 18–24. when the fire ants have departed. Spring 2010 Spring 2010 Feedback in engineering Feedback in toasters If you have a system that you would like to control If you have a system that you would like to control automatically, then adding a controller with some automatically, then adding a controller with some form of feedback from the output may be a good form of feedback from the output may be a good option. option. For example, some expensive toasters now have sensors that measure the colour of the toast, and stop the toaster when the desired colour is reached. Some others have radioactive sensors that detect the particles released by toasting, and stop the toaster when they reach a certain density. These are feedback controlled toasters… Spring 2010 Spring 2010 Feedback control (often called PID: >90% of real world control closed-loop control Reference/ Proportional control: input is proportional to error. setpoint disturbance disturbance For high gains or large errors, it will tend to overshoot and oscillate. There is always a steady- state error that cannot be corrected. + + +++ Controller Plant output - Integral control: can get rid of steady-state error by integrating it over time The output is sensed and compared with the Derivative control: can reduce settling time by reference. The resulting error signal is fed back giving a better dynamical response. Reduces to the controller. The controller generates a overshoot by damping. control input calculated to bring the output closer to the reference – i.e. to reduce the error. Spring 2010 Spring 2010 PID: >90% of real world control PID control Spring 2010 Spring 2010 PID: >90% of real world control PID: >90% of real world control Q: So how do you find the right parameters? Q: So how do you find the right parameters? A: (i) It depends what you mean by ‘right’ A: (i) It depends what you mean by ‘right’ (ii) They may not exist (ii) They may not exist Ideally, you want your system to be “critically damped” – with zero overshoot and minimum settling time.