The Theory of Control: a Brief Overview

The Theory of Control: A Brief Overview Robin H. Pearce Abstract Control systems form a vital part of engineering, where anything that needs to be regulated or optimised can be done so with a technique that is encompassed by control theory. Since the 16th century, problems have been modelled as dynamical systems so that a mathematical control strategy could be devised and an equivalent physical strategy invented. Methods for controlling mechanical, analogue and digital systems have been invented and some are widely used. I will briefly introduce and explain many of the relevant topics considered to be part of control theory. \Control Theory" is somewhat an umbrella term, are both linear and time-invariant. These systems used to describe a wide range of analytic techniques are known as LTI systems and are extremely useful and styles which are applied to the control of a math- in control and signal processing. ematical system. Much of this theory is linked to sys- When considering the simplest, stateless LTI SISO tems theory, since many of the problems that can be system, we have that solved can also be modelled as a dynamical system. The methods involved range from classical open or y(t) = (u ? h)(t) closed loop control, up to more modern disturbance rejection and noise filtering, as well as stabilisation where h(t) is the impulse response of the system. By and fine-tuning. taking the Laplace transform of both sides, we end up with Y (s) = U(s)H(s) Systems where Y (s), U(s) and H(s) are the laplace transforms Many of the systems can be modelled as a system of of y(t), u(t) and h(t) respectively. The function H(s) differential equations. Usually these are modelled in is known as the transfer function of the system and continuous time, however discrete time equations can can be written as also be useful. The basic system model in continuous Y (s) time is H(s) = U(s) x_(t) = f(x(t); u(t); t) Closed loop system y(t) = h(x(t); u(t); t) One of the simplest and most useful control systems where x(t) is the state, u(t) is the control input and is a closed-loop feedback controller, similar to the one y(t) is the system output at time t. The easiest sys- seen in Figure 1. In this system, the output is mea- tem to consider is one where x; y and u are all scalar. sured and sent back into the system to help regulate This is known as a SISO system, which stands for sin- the response of the system. Using the same notation gle input single output. Generally, these systems look before, we can consider two controllers, G1(S) and at signal processing, particularly for systems which G2(s), so that we can find the closed-loop transfer 1 The Theory of Control Robin H. Pearce function H~ (s), that is the function which satisfies where N(s) and D(s) are polynomials, then the sta- ~ bility can be determined by the location of all s which Y (s) = R(s)H(s) satisfy D(s) = 0. By the Routh-Hurwitz criterion, where R(s) is the laplace transform of the reference the system will be stable if the real parts of all such signal r(t) and U(s) is the laplace transform of the s are negative. If we are given a particular system signal u(t) before it enters the plant H(s). where H(s) is unstable, then we can design a feedback controller like the one in Figure 1, such that H~ (s) or Hc(s) is stable by choosing G1(s) and G2(s). If NH (s) N1(s) N2(s) H(s) = ;G1(s) = ;G2(s) = ; DH (s) D1(s) D2(s) then we can rewrite the closed loop transfer function as NH N1D2 Hc(s) = : DH D1D2 + NH N1N2 Figure 1: Closed-loop feedback controller This system will now be stable if the solutions to DH D1D2 + NH N1N2 = 0 are all in the left-hand By starting with Y (s) = U(s)H(s) and substitut- plane, that is the real part of all the solutions are negative. The simplest set of controls for the closed ing back for U(s), G1(s), E(s), R(s) and G2(s), we end up with the expression loop system are to take G1(s) = K and G2(s) = 1, so now the system is stable if the zeros of 1 + KH(s) G (s)H(s) Y (s) = R(s) 1 are in the left-hand plane. 1 + G2(s)G1(s)H(s) so we have an expression for H~ (s), where Y (s) = States R(s)H~ (s). Quite often these systems are being con- trolled because some property of stability is desired. So far, the control methods considered are for systems where the state doesn't matter. In many real life examples this is not the case, so we need to con- Stability sider methods which control the state of a system. Most often, systems will be either linear or easily ap- There are two main types of stability in control sys- proximated by a linear system. These linear systems tems: BIBO stability and internal stability. BIBO take the form stands for bounded inupt bounded output, and con- cerns the systems ability to return bounded outputs when given bounded inputs. The second type of sta- x_(t) = Ax(t) + Bu(t) bility is often the most important, and can be determined using the Routh-Hurwitz criterion. y(t) = Cx(t) + Du(t) Internal stability refers to the ability of the system where x(t) 2 n, u(t) 2 m and y(t) 2 p. The to return to a state of equilibrium after an arbitrary R R R matrices are all constant and the system is commonly displacement from equilibrium. This is commonly as- referred to as an (A; B; C; D) system. The discrete sociated with the location of the poles of the transfer time equivalent is just the same, except t is replaced function, H(s). That is, if by n andx _(t) becomes x(n + 1). N(s) For a linear autonomous system,x _(t) = Ax(t) + H(s) = A(t−t0) D(s) g(t), solutions take the form Φ(t; t0) = e , 2 The Theory of Control Robin H. Pearce where eA(t−t0) is the matrix exponential. The laplace we get an augmented system which only depends on transform for eAt is (sI −A)−1 which is known as the x(t) and r(t). Also, if (A; B) is controllable, then resolvant. For the full (A; B; C; D) system, the same it is possible to choose the eigenvalues of the state computations lead to transition matrix to be what we want by making the right choice of F . H(s) = C(sI − A)−1B + D Of course, eAt can often be difficult to calculate for Observability non-diagonal systems, so we can consider a similarity transform P x~ = x, where det(P ) 6= 0. The system is A system is observable if we can determine the initial −1 −1 now a (P AP; P B; CP; D) system, where P AP condition for the state given the inputs and outputs At is diagonal and e is in Jordan normal form. over a finite time interval. The observability of a system can be found from the matrices A and C using a Controllability similar method to controllability, where the observability matrix is given by For any (A; B; C; D) system, we can find all states obs(A; C) = [CT ;AT CT ; (A2)T CT ; :::; (An−1)T CT ]T that we can reach through controlling the system. A state xd is considered reachable or controllable-from- The condition for observability of a system is sim- the-origin if there exists a control input u that takes ilar to that for controllability; that is, the observ- the state x(t) from the zero state to xd in a finite ability matrix has full rank if and only if the system amount of time. is observable. One interesting result is that there is The converse definition is that a state xs is called a duality between controllability and observability. controllable if there exists a control input u that takes When given a standard (A; B; C; D) system, we can the state from xs to the zero state in a finite amount define the dual system as of time. Any set of states can also be called reachable or controllable if every state in the set is reachable or T T controllable. If all states are reachable or control- x_(t) = A x(t) + C u(t) lable, then the system is a reachable or controllable T T system. y(t) = B x(t) + D u(t) To determine if a system is controllable, we can The result is that the dual system is controllable calculate the controllability matrix from the matrices if and only if the original system is observable and A and B, defined to be the original system is controllable if and only if the con(A; B) = [B; AB; A2B; :::; An−1B] dual system is observable. In real life, not all systems are observable, so it is useful to design an observer to where the system is controllable if and only if the con- estimate the state of the system and make corrections trollability matrix has full rank.

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