16 CONTROL FUNDAMENTALS 16.1 Introduction 16.1.1 Plants, Inputs, and Outputs Controller design is about creating dynamic systems that behave in useful ways. Many target systems are physical; we employ controllers to steer ships, fly jets, position electric motors and hydraulic actuators, and distill alcohol. Controllers are also applied in macro-economics and many other important, non-physical systems. It is the fundamental concept of controller design that a set of input variables acts through a given “plant” to create an output. Feedback control then uses sensed plant outputs to apply corrective inputs: Plant Inputs Outputs Sensors Jet aircraft elevator, rudder, etc. altitude, hdg altimeter, GPS Marine vessel rudder angle heading gyrocompass Hydraulic robot valve position tip position joint angle U.S. economy fed interest rate, etc. prosperity inflation, M1 Nuclear reactor cooling, neutron flux power level temp., pressure (Continued on next page)16.2 Representing Linear Systems 77 16.1.2 The Need for Modeling Effective control system design usually benefits from an accurate model of the plant, although it must be noted that many industrial controllers can be tuned up satisfactorily with no knowledge of the plant. Ziegler and Nichols, for example, developed a general recipe which we detail later. In any event, plant models simply do not match real-world systems exactly; we can only hope to capture the basic components in the form of differential or integro-differential equations. Beyond prediction of plant behavior based on physics, the process of system identification generates a plant model from data. The process is often problematic, however, since the measured response could be corrupted by sensor noise or physical disturbances in the system which cause it to behave in unpredictable ways. At some frequency high enough, most systems exhibit effects that are difficult to model or reproduce, and this is a limit to controller performance. 16.1.3 Nonlinear Control The bulk of this subject is taught using the tools of linear systems analysis. The main reason for this restriction is that nonlinear systems are difficult to model, difficult to design controllers for, and difficult overall! Within the paradigm of linear systems, there are many sets of powerful tools available. The reader interested in nonlinear control is referred to the book by Slotine and Li (1991). 16.2 Representing Linear Systems Except for the most heuristic methods of tuning up simple systems, control system design depends on a model of the plant. The transfer function description of linear systems has already been described in the discussion of the Laplace transform. The state-space form is an entirely equivalent time-domain representation that makes a clean extension to systems with multiple inputs and multiple outputs, and opens the way to standard tools from linear algebra. 16.2.1 Standard State-Space Form We write a linear system in a state-space form as follows x˙ = Ax + Bu + Gw (193) y = Cx + Du + v where • x is a state vector, with as many elements as there are orders in the governing differ-ential equations. • A is a matrix mapping x to its derivative; A captures the natural dynamics of the system without external inputs.78 16 CONTROL FUNDAMENTALS • B is an input gain matrix for the control input u. • G is a gain matrix for unknown disturbance w; w drives the state just like the control u. • y is the observation vector, comprised mainly of a linear combination of states Cx (where C is a matrix). • Du is a direct map from input to output (usually zero for physical systems). • v is an unknown sensor noise which corrupts the measurement. u w B G A 1/s C v D y + + + + + + xx’ 16.2.2 Converting a State-Space Model into a Transfer Function There are a number of canonical state-space forms available, which can create the same transfer function. In the case of no disturbances or noise, the transfer function (or transfer matrix) can be written as y(s)G(s) = = C(sI − A)−1B + D, (194) u(s) where I is the identity matrix with the same size as A. A similar equation holds for y(s)/w(s), and clearly y(s)/v(s) = I. 16.2.3 Converting a Transfer Function into a State-Space Model It may be possible to write the corresponding differential equation along one row of the state vector, and then cascade derivatives. For example, consider the following system: my∗∗(t) + by∗(t) + ky(t) = u∗(t) + u(t) (mass-spring-dashpot) s + 1 G(s) = ms2 + bs + k Setting γx = [y∗, y]T , we obtain the system16.3 PID Controllers 79 ⎬ � ⎬ �dγx −b/m −k/m 1/m= γx + u dt 1 0 0 y = [1 1] γx Note specifically that dx2/dt = x1, leading to an entry of 1 in the off-diagonal of the second row in A. Entries in the C-matrix are easy to write in this case because of linearity; the system response to u∗ is the same as the derivative of the system response to u. 16.3 PID Controllers The most common type of industrial controller is the proportional-integral-derivative (PID) design. If u is the output from the controller, and e is the error signal it receives, this control law has the form � t u(t) = kpe(t) + ki e(φ )dφ + kde∗(t), 0 U (s) kiC(s) = = kp + + kds (195)E(s) s � 1 � = kp 1 + + φds ,φis where the last line is written using the conventions of one overall gain kp, plus a time characteristic to the integral part (φi) and and time characteristic to the derivative part (φd). In words, the proportional part of this control law will create a control action that scales linearly with the error – we often think of this as a spring-like action. The integrator is accumulating the error signal over time, and so the control action from this part will continue to grow as long as an error exists. Finally, the derivative action scales with the derivative of the error. The controller will retard motion toward zero error, which helps to reduce overshoot. The common variations are: P , P D, P I, P ID. 16.4 Example: PID Control Consider the case of a mass (m) sliding on a frictionless table. It has a perfect thruster that generates force u(t), but is also subject to an unknown disturbance d(t). If the linear