# Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL. Glenn Vinnicombe HANDOUT 5. An Introduction to Feedback Control Systems

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1 Part IB Paper 6: Information Engineering LINEAR SYSTEMS AND CONTROL Glenn Vinnicombe HANDOUT 5 An Introduction to Feedback Control Systems ē(s) ȳ(s) Σ K(s) G(s) z(s) H(s) z(s) = H(s)G(s)K(s) L(s) ē(s)= L(s) ē(s) Return ratio Closed-loop transfer function relating ē(s) and ȳ(s)=g(s)k(s)ē(s)= G(s)K(s) L(s) Closed-loop transfer function relating ȳ(s) and

2 Key Points The Closed-Loop Transfer Functions are the actual transfer functions which determine the behaviour of a feedback system. They relate signals around the loop (such as the plant input and output) to external signals injected into the loop (such as reference signals, disturbances and noise signals). It is possible to infer much about the behaviour of the feedback system from consideration of the Return Ratio alone. The aim of using feedback is for the plant outputy(t) to follow the reference signalr(t) in the presence of uncertainty. A persistent difference between the reference signal and the plant output is called a steady state error. Steady-state errors can be evaluated using the final value theorem. Many simple control problems can be solved using combinations of proportional, derivative and integral action: Proportional action is the basic type of feedback control, but it can be difficult to achieve good damping and small errors simultaneously. Derivative action can often be used to improve damping of the closed-loop system. Integral action can often be used to reduce steady-state errors. 2

3 Contents 5 An Introduction to Feedback Control Systems 5. Open-Loop Control Closed-Loop Control (Feedback Control) Derivation of the closed-loop transfer functions: The Closed-Loop Characteristic Equation What if there are more than two blocks? A note on the Return Ratio Sensitivity and Complementary Sensitivity Summary of notation The Final Value Theorem (revisited) The steady state response summary Some simple controller structures Introduction steady-state errors Proportional Control Proportional Derivative (PD) Control Proportional Integral (PI) Control Proportional Integral Derivative (PID) Control.. 2 3

4 5. Open-Loop Control Demanded Output (Reference) Controller K(s) Plant G(s) Controlled Output ȳ(s) In principle, we could could choose a desired transfer functionf(s) and use K(s) = F(s)/G(s) to obtain In practice, this will not work ȳ(s)=g(s) F(s) G(s) =F(s) because it requires an exact model of the plant and that there be no disturbances (i.e. no uncertainty). Feedback is used to combat the effects of uncertainty For example: Unknown parameters Unknown equations Unknown disturbances 4

5 5.2 Closed-Loop Control (Feedback Control) For Example: Input disturbance Output disturbance Demanded Output (Reference) Error Signal d i (s) Control Signal d o (s) Controlled Output Σ ē(s) Controller K(s) ū(s) Σ Plant G(s) Σ ȳ(s) Figure Derivation of the closed-loop transfer functions: ȳ(s)= d o (s)g(s) [ di (s)k(s)ē(s) ] = ē(s)= ȳ(s) = ȳ(s)= d o (s)g(s) [ )] di (s)k(s) ( ȳ(s) ( ) G(s)K(s) ȳ(s)= d o (s)g(s) d i (s)g(s)k(s) = ȳ(s)= G(s)K(s) d o (s) G(s) G(s)K(s) d i (s) G(s)K(s) G(s)K(s) 5

6 Also: ē(s)= ȳ(s) = G(s)K(s) d G(s) o (s) G(s)K(s) d i (s) ( G(s)K(s) ) G(s)K(s) } {{ } G(s)K(s) The Closed-Loop Characteristic Equation and the Closed-Loop Poles Note: All the Closed-Loop Transfer Functions of the previous section have the same denominator: G(s)K(s) The Closed-Loop Poles (ie the poles of the closed-loop system, or feedback system) are the zeros of this denominator. For the feedback system of Figure 5., the Closed-Loop Poles are the roots of G(s)K(s)=0 Closed-Loop Characteristic Equation (for Fig 5.) The closed-loop poles determine: The stability of the closed-loop system. Characteristics of the closed-loop system s transient response.(e.g. speed of response, presence of any resonances etc) 6

7 5.2.3 What if there are more than two blocks? For Example: Demanded Output (Reference) d i (s) d o (s) Controlled Output Σ ē(s) Controller K(s) Plant Σ G(s) Σ ȳ(s) Sensor H(s) Figure 5.2 We now have ȳ(s)= This time G(s)K(s) H(s)G(s)K(s) H(s)G(s)K(s) d o (s) G(s) H(s)G(s)K(s) d i (s) H(s)G(s)K(s) appears as the denominator of all the closed-loop transfer functions. Let, L(s) = H(s)G(s)K(s) i.e. the product of all the terms around loop, not including the at the summing junction.l(s) is called the Return Ratio of the loop (and is also known as the Loop Transfer Function). The Closed-Loop Characteristic Equation is then L(s)=0 and the Closed-Loop Poles are the roots of this equation. 7

8 5.2.4 A note on the Return Ratio 0 0 Σ 0 ē(s) b(s) K(s) Σ G(s) Σ ȳ(s) ā(s) H(s) Figure 5.3 With the switch in the position shown (i.e. open), the loop is open. We then have ā(s)=h(s)g(s)k(s) b(s)= H(s)G(s)K(s) b(s) Formally, the Return Ratio of a loop is defined as times the product of all the terms around the loop. In this case L(s) = H(s)G(s)K(s) = H(s)G(s)K(s) Feedback control systems are often tested in this configuration as a final check before closing the loop (i.e. flicking the switch to the closed position). Note: In general, the block denoted here ash(s) could include filters and other elements of the controller in addition to the sensor dynamics. Furthermore, the block labelledk(s) could include actuator dynamics in addition to the remainder of the designed dynamics of the controller. 8

9 5.2.5 Sensitivity and Complementary Sensitivity The Sensitivity and Complementary Sensitivity are two particularly important closed-loop transfer functions. The following figure depicts just one configuration in which they appear. d o (s) ē(s) ū(s) Σ K(s) G(s) Σ ȳ(s) ( Figure 5.4 L(s) = G(s)K(s) ) ȳ(s)= G(s)K(s) G(s)K(s) G(s)K(s) d o (s) = L(s) L(s) Complementary Sensitivity T(s) L(s) Sensitivity S(s) d o (s) Note: S(s)T(s)= L(s) L(s) L(s) = 9

10 5.3 Summary of notation The system being controlled is often called the plant. The control law is often called the controller ; sometimes it is called the compensator or phase compensator. The demand signal is often called the reference signal or command, or (in the process industries) the set-point. The Return Ratio, the Loop transfer function always refer to the transfer function of the opened loop, that is the product of all the transfer functions appearing in a standard negative feedback loop (ourl(s)). Figure 5. hasl(s)=g(s)k(s), Figure 5.2 has L(s) = H(s)G(s)K(s). The Sensitivity function is the transfer functions(s)= L(s). It characterizes the sensitivity of a control system to disturbances appearing at the output of the plant. The transfer functiont(s)= L(s) is called the L(s) Complementary Sensitivity. The name comes from the fact that S(s)T(s)=. When this appears as the transfer function from the demand to the controlled output, as in Fig 5.4 it is often called simply the Closed-loop transfer function (though this is ambiguous, as there are many closed-loop transfer functions). 0

11 5.4 The Final Value Theorem (revisited) Consider an asymptotically stable system with impulse responseg(t) and transfer functiong(s), i.e. g(t) Impulse response G(s) Transfer Function (assumed asymptotically stable) Lety(t)= t g(τ)dτ denote the step response of this system and 0 note thatȳ(s)= G(s). s We now calculate the final value of this step response: lim y(t)= g(τ)dτ t 0 = exp( 0τ) 0 g(τ)dτ=l ( g(t) ) s=0 =G(0) Hence, Final Value of Step Response Transfer Function evaluated at s= 0 Steady-State Gain or DC gain Note that the same result can be obtained by using the Final Value Theorem: limy(t)= lim sȳ(s) t s 0 = lim s 0 s G(s) s ( for anyy for which both limits exist. =G(0) )

12 5.4. The steady state response summary The term steady-state response means two different things, depending on the input. Given an asymptotically stable system with transfer functiong(s): The steady-state response of the system to a constant inputu is a constant,g(0)u. The steady-state response of the system to a sinusoidal input cos(ωt) is the sinusoid G(jω) cos ( ωt argg(jω) ). These two statements are not entirely unrelated, of course: The steady-state gain of a system,g(0) is the same as the frequency response evaluated atω=0 (i.e. the DC gain). 2

13 5.5 Some simple controller structures 5.5. Introduction steady-state errors r ē ū ȳ Σ K(s) G(s) Return Ratio:L(s)=G(s)K(s). CLTFs:ȳ(s)= L(s) L(s) and ē(s)= L(s) Steady-state error:(for a step demand) Ifr(t)=H(t), then ȳ(s)= L(s) L(s) s and so L(s) limy(t)=s t L(s) s L(0) = s= 0 L(0) and lim e(t)=s t L(s) s s= 0 = L(0) Steady-state error (using the final-value theorem.) Note: These particular formulae only hold for this simple configuration where there is a unit step demand signal and no constant disturbances (although the final value theorem can always be used). 3

14 5.5.2 Proportional Control K(s)=k p r Σ ē k p ū G(s) ȳ Typical result of increasing the gaink p, (for control systems where G(s) is itself stable): Increased accuracy of control. Increased control action. Reduced damping. Possible loss of closed-loop stability for largek p.} good bad Example: G(s)= (A critically damped 2 nd order system) (s ) 2 ȳ(s)= k pg(s) k p G(s) = = k p s 2 2s k p k p (s ) 2 k p (s ) 2 So,ω 2 n= k p, 2ζω n = 2 = ω n = k p, ζ= k p Closed-loop poles at s = ± j k p X movement of closed-loop poles for increasingk p root locus diagram 4

15 Steady-state errors using the final value theorem: and ē(s)= So, ifr(t)=h(t), lim y(t)= t ȳ(s)= k p s 2 2s k p k p G(s) = (s ) 2 k p s 2 2s k p s 2 2s k p. s= 0 = final value of step response k p k p and lim e(t)= (s ) 2 t s 2 2s k p s= 0 = k p Steady-state error (Note:L(s)=k p (s ) 2 = L(0)=k p =k p ) Hence, in this example, increasingk p gives smaller steady-state errors, but a larger and more oscillatory transient response. However, by using more complex controllers it is usually possible to remove steady state errors and increase damping at the same time: To increase damping can often use derivative action (or velocity feedback). To remove steady-state errors can often use integral action. 5

16 For reference, the step response: (i.e. response to = ) is given by s k p (2s) k p ȳ(s)= s 2 2s k p k p k p s so y(t)= k p exp( t) k p cos( k p t) k p sin( ( ) kp = kp exp( t) cos( k p t φ) Transient Response whereφ=arctan kp k k p t) p k p k p k p Steady-state response But you don t need to calculate this to draw the conclusions we have made. 6

17 5.5.3 Proportional Derivative (PD) Control K(s)=k p k d s r ū ȳ Σ k p Σ G(s) k d s Typical result of increasing the gaink d, (wheng(s) is itself stable): Increased Damping. Greater sensitivity to noise. (It is usually better to measure the rate of change of the error directly if possible i.e. use velocity feedback) Example: G(s) = (s ) 2, K(s)=k pk d s ȳ(s)= K(s)G(s) (k p k d s) K(s)G(s) = (s) 2 (k p k d s) (s) 2 = k p k d s s 2 (2k d )s k p So,ω 2 n= k p, 2cω n = 2k d = ω n = k p, c= 2k d Xk p= 4,k d= 0 2 k movement of p closed-loop poles for increasingk d X 7

18 5.5.4 Proportional Integral (PI) Control In the absence of disturbances, and for our simple configuration, ē(s)= G(s)K(s) Hence, steady-state error, (for step demand) = G(s)K(s) = s=0 G(0)K(0) To remove the steady-state error, we need to makek(0)= (assuming G(0) 0). e.g K(s)=k p k i s r ū ȳ Σ k p Σ G(s) k i /s Example: G(s)= (s ) 2, K(s)=k pk i /s 8

19 ȳ(s)= K(s)G(s) (k p k i /s) K(s)G(s) = (s) 2 (k p k i /s) (s) 2 = k p sk i s(s ) 2 k p sk i ē(s)= K(s)G(s) = (k p k i /s) (s) 2 = Hence, forr(t)=h(t), s(s ) 2 s(s ) 2 k p sk i lim y(t)= t k p sk i s(s ) 2 k p sk i s=0 = and lim e(t)= s(s ) 2 t s(s ) 2 k p sk i s=0 = 0 = no steady-state error 9

20 PI control General Case In fact, integral action (if stabilizing) always results in zero steady-state error, in the presence of constant disturbances and demands, as we shall now show. Assume that the following system settles down to an equilibrium with lime(t)=a 0, then: t r(t) Σ A e(t) k p k i /s Σ d i (t) u(t) Σ G(s) Σ d o (t) = Contradiction H(s) (as system is not in equilibrium) Hence, with PI control the only equilibrium possible has lim t e(t)=0. That is, lim t e(t)=0 system is provided the closed-loop asymptotically stable. 20

21 5.5.5 Proportional Integral Derivative (PID) Control K(s)=k p k i s k ds k i /s r ū ȳ Σ k p Σ G(s) k d s Characteristic equation: G(s)(k p k d sk i /s)=0 can potentially combine the advantages of both derivative and integral action: but can be difficult to tune. There are many empirical rules for tuning PID controllers (Ziegler-Nichols for example) but to get any further we really need some more theory... 2

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