# AN INTRODUCTION TO THE CONTROL THEORY

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1 Open-Loop controller An Open-Loop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, non-linear dynamics and parameter variations can cause significant errors between the plant output and the reference signal. r(s) Kp u(s) Gp(s) y(s) r(s) is the reference signal u(s) )is the control signal y(s) is the plant output K p is the controller gain G p (s) is the plant transfer function

2 Closed-Loop controller A Closed-Loop (CL) controller improves the plant behaviour by means of a feedback loop; the controller transfer functions are designed so that: y(t) r(t) in a desirable manner. CONTROLLER PLANT r(s) () e(s) () Gc(s) u(s) () Gp(s) y(s) H(s) e(s) = r(s) - y(s) is the error signal G c ()i (s) is the feed-forward df d controller transfer function H(s) is the feedback controller transfer function

3 The CL transfer function (CLTF) is: y ( s ) G ( s ) = r( s) 1+ G( s) H ( s) where G(s)=G c (s)g p (s) The denominator of CLTF is called the closed-loop l characteristic i polynomial and the equation: 1 + G( s) H ( s) = isthe closed-loop l characteristic equation (CLCE). TheCLCEhas n roots inthes-plane.

4 AN INTRODUCTION TO CONTROL THEORY By means of the Laplace transforms the relationship between input r(s) and output y(s) of a dynamic system can be rearranged in the following form: y s G(s) is called the transfer function: = G s r s ( ) ( ) ( ) G ( s ) = y s / r s ( ) ( ) and this is the keystone of control system design. r ( s ) y (s ) G (s )

5 For a linear second order system the transfer function is: G ( s ) λω 2 n = 2 2 s + 2ζω n s + ω n where the parameters λ, ζ, ω n characterise the response of the plant. λ is the low-frequency gain. The response for a unit step input is: 2.5 λ = 2, ζ =.5, ω n = 3 y = λr = y(t) r(t) r = 1.5 t s 4 ζω n 27. s time (s)

6 - The steady-state output is λr - The settling time of the response is t s 4/(ζω n ) - The transient is an exponential curve. The number of overshoots is (approximately) given by: ζ = 1. no overshoot (critical damping) ζ =7.7 1 overshoot (underdamping) ζ =.5 2 overshoots (underdamping)

7 Effects of unmodelled terms - The linear transfer function is an idealised model of the plant dynamics - Unmodelled terms can sometimes have a significant effect. Examples include stiction (i.e. static friction), signal noise, non-linear dynamics, load disturbances and parameter variations with linear stiction with parameter linear variation -.2 (m 2m) with -.6 square-law drag -.7 linear with noise linear time (s) time (s)

8 Open-Loop controller An Open-Loop (OL) controller is characterized by no direct connection between the output of the system and its input; therefore external disturbance, non-linear dynamics and parameter variations can cause significant errors between the plant output and the reference signal. r(s) Kp u(s) Gp(s) y(s) r(s) is the reference signal u(s) )is the control signal y(s) is the plant output K p is the controller gain G p (s) is the plant transfer function

9 Closed-Loop controller A Closed-Loop (CL) controller improves the plant behaviour by means of a feedback loop; the controller transfer functions are designed so that: y(t) r(t) in a desirable manner. CONTROLLER PLANT r(s) () e(s) () Gc(s) u(s) () Gp(s) y(s) H(s) e(s) = r(s) - y(s) is the error signal G c ()i (s) is the feed-forward df d controller transfer function H(s) is the feedback controller transfer function

10 The CL transfer function (CLTF) is: y ( s ) G ( s ) = r( s) 1+ G( s) H ( s) where G(s)=G c (s)g p (s) The denominator of CLTF is called the closed-loop l characteristic i polynomial and the equation: 1 + G( s) H ( s) = isthe closed-loop l characteristic equation (CLCE). TheCLCEhas n roots inthes-plane.

11 Closed-Loop performance criteria Statement of the standard control problem: Determine u so that y follows a reference vector, r, ina well-defined and stable manner. Also ensure that the effects of disturbances are rejected in the steady-state. The following figure gives a qualitative idea of a successful control response: Effect of disturbance removed r y Desirable transient t behaviour Zero steady-state error Application of disturbance t

12 Therefore, in order of priority, the main performance criteria for a closed-loop system are: 1) stability 2) relative stability: closed-loop loop (CL) transient behaviour and dominant roots 3) steady-state behaviour 4) disturbance rejection

13 1) Stability The closed-looploop system must be stable under all circumstances. This is governed by the roots of the CLCE, which must be all in the left-half f of s-plane: Stable CL System Unstable CL System Im s-plane Im s-plane Re Re rd : CLCE roots (3 rd -order in this example)

14 2) Relative Stability: Closed-Loop (CL) Transient Behaviour and Dominant Roots The CLCE roots must be placed within specific regions insideid the left-half of s-plane in order to achieve desirable transient performance. Typical measures of transient performance are settling time and number of overshoots: these are largely governed by the distribution of the dominant roots of CLCE. 1 st -Order Dominance 2 nd -Order Underdamped Dominance Im s-plane Im s-planep +jω d -1/T Re -ζω n -jω d Re 2 Damped natural frequency: ω = ω 1 ζ : CLCE roots (4 th -order in this example) d n

15 3) Steady-State (ss) Behaviour This is governed by the characteristics of the reference signal r(s) and the number of integrators in the OLTF (i.e. the number of factors of s in the denominator of the OLTF). This number is called the CL system type, and is usually, 1 or 2. If the CL system has unit feedback (H(s) = 1), then: Type OLTFs give zero ss error when r is an impulse. Type 1 OLTFs give zero ss error when r is an impulse or a step. Type 2 OLTFs give zero ss error when r is an impulse, step or a ramp. In general, the CL steady-statestate error can be calculated from the Final Value Theorem: 1 + G ( s ) H ( s ) G ( s ) e = [ se( s)] = sr s s= ( ) 1+ GsHs () () s=

16 4) Disturbance Rejection Effects of disturbances are analysed in the same manner as steadystate errors. The controller should negate the effect of disturbances, at least at steady-state. Usually, the precise nature of the disturbances is unknown, but their structure can be estimated, for instance disturbances are constant, cyclic, random, etc.

17 Basic controller design The strategy for controller design is: 1) Determine the plant transfer function and its parameters. 2) Determine the required CL performance criteria. 3) Make an engineering judgement on the simplest controller to reach the goal. 4) Determine the controller parameters (gains) that satisfy the CL performance criteria. 5) Simulate the CL performance as design verification (optional, but sometimes essential).

18 Proportional (P) control This is the simplest linear controller: G ( s ) = k ; H ( s ) = 1 The parameter k p is called the proportional p gain: thedesign problem is to find a suitable value for this gain. c p We examine the following transfer function: G p ( s) = 2 s s + 9 With three different values of k p :.2;1;5.

19 Proportional (P) control poles in the s-plane Im Re kp =,2 kp = 1 kp = 5

20 Proportional (P) control reference signal kp =.2 kp = 1 kp = 5

21 Proportional-plus-Integral (PI) control To reduce the steady state error, we introduce a pole in the controller forward dynamics: ki ( s) = k p + ; H ( s) = 1 s The parameter k i isthe integral gain. G c We take k p = 1 and three different values of k i :.5; 1; 2. The plant response has been improved, but the response is too slow and there are too many overshoots.

22 Proportional-plus-Integral (PI) control poles in the s-plane Im Re ki =.5 ki = 1 ki = 2

23 AN INTRODUCTION TO AUTOMATIC CONTROL Proportional-plus-Integral (PI) control reference signal ki =.5 ki = 1 ki = 2

24 AN INTRODUCTION TO AUTOMATIC CONTROL Proportional-Integral-Derivative (PID) control To reduce the settling-timetime and the overshoots, we introduce a zero in the controller forward dynamics: G ki ( s) = k p + + kd s ; H ( s) = 1 s The parameter k d isthe derivative i gain. c We take k p = 5, k i = 2 and three different values of k d :.1;.5; 1. We observe that the plant response is very good for the following combination: k p =5, k i =2, k d =1

25 AN INTRODUCTION TO AUTOMATIC CONTROL Proportional-Integral-Derivative (PID) control poles in the s-plane Im Re kd =.1 kd =.5 kd = 1

26 AN INTRODUCTION TO AUTOMATIC CONTROL Proportional-Integral-Derivative (PID) control reference signal kd =.1 kd =.5 kd = 1

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