SingleInputSingleOutput Systems


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1 TF 502 SingleInputSingleOutput Systems SIST, ShanghaiTech Introduction OpenLoop ControlResponse Proportional Control General PID Control Boris Houska 11
2 Contents Introduction OpenLoop ControlResponse Proportional Control General PID Control SingleInputSingleOutput Systems 12
3 Objectives In this lecture we will learn about SingleInputSingleOutput (SISO) systems Openloop step response Proportional control of SISO systems Integral control PID control Warning: this lecture explains basic tricks to tune a controller based on physical intuition without using mathematical analysis yet (we ll discuss mathematical models later on in the lecture) SingleInputSingleOutput Systems 13
4 SISO Systems Recall from Lecture 1: 1. u(t) R denotes (scalar) input at time t 2. y(t) R denotes (scalar) output at time t SingleInputSingleOutput Systems 14
5 Example 1: oven SISO system: 1. u(t) R voltage at the heating coil 2. y(t) R temperature in the oven SingleInputSingleOutput Systems 15
6 Example 2: pendulum SISO system: 1. u(t) R force of the motor 2. y(t) R excitation angle of the pendulum SingleInputSingleOutput Systems 16
7 Contents Introduction OpenLoop ControlResponse Proportional Control General PID Control SingleInputSingleOutput Systems 17
8 Openloop controlresponse Idea: Test basic system behavior by making openloop experiments Send signal u(t) to actuator and observe y(t) SingleInputSingleOutput Systems 18
9 Parameterization of control functions General affine parameterizations of the control function can be written in the form M u(t) = a i ϕ i (t). i=1 The functions ϕ 1,... ϕ M : R R are given basis functions The scalars a 1, a 2,..., a M R are the control parameterization coefficients SingleInputSingleOutput Systems 19
10 Parameterization of control functions General affine parameterizations of the control function can be written in the form M u(t) = a i ϕ i (t). i=1 The functions ϕ 1,... ϕ M : R R are given basis functions The scalars a 1, a 2,..., a M R are the control parameterization coefficients SingleInputSingleOutput Systems 110
11 Parameterization of control functions General affine parameterizations of the control function can be written in the form M u(t) = a i ϕ i (t). i=1 The functions ϕ 1,... ϕ M : R R are given basis functions The scalars a 1, a 2,..., a M R are the control parameterization coefficients SingleInputSingleOutput Systems 111
12 Openloop step response Openloop experiment: Introduce the basis functions 1 if t < 0 ϕ 1 (t) = 0 otherwise 1 if t 0 and ϕ 2 (t) = 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called step response For a 1 = 0, a 2 = 1: y is called unit step response SingleInputSingleOutput Systems 112
13 Openloop step response Openloop experiment: Introduce the basis functions 1 if t < 0 ϕ 1 (t) = 0 otherwise 1 if t 0 and ϕ 2 (t) = 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called step response For a 1 = 0, a 2 = 1: y is called unit step response SingleInputSingleOutput Systems 113
14 Openloop step response Openloop experiment: Introduce the basis functions 1 if t < 0 ϕ 1 (t) = 0 otherwise 1 if t 0 and ϕ 2 (t) = 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called step response For a 1 = 0, a 2 = 1: y is called unit step response SingleInputSingleOutput Systems 114
15 Openloop step response Openloop experiment: Introduce the basis functions 1 if t < 0 ϕ 1 (t) = 0 otherwise 1 if t 0 and ϕ 2 (t) = 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called step response For a 1 = 0, a 2 = 1: y is called unit step response SingleInputSingleOutput Systems 115
16 Openloop step response Openloop experiment: Introduce the basis functions 1 if t < 0 ϕ 1 (t) = 0 otherwise 1 if t 0 and ϕ 2 (t) = 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called step response For a 1 = 0, a 2 = 1: y is called unit step response SingleInputSingleOutput Systems 116
17 Example 1 SingleInputSingleOutput Systems 117
18 Example 1 SingleInputSingleOutput Systems 118
19 Example 2 SingleInputSingleOutput Systems 119
20 Example 2 SingleInputSingleOutput Systems 120
21 Openloop frequency response Openloop experiment: Choose a test frequency ω > 0. Introduce the basis functions 1 if t < 0 ϕ 1 (t) = and ϕ sin(ωt) if t 0 2(t) = 0 otherwise 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called frequency response SingleInputSingleOutput Systems 121
22 Openloop frequency response Openloop experiment: Choose a test frequency ω > 0. Introduce the basis functions 1 if t < 0 ϕ 1 (t) = and ϕ sin(ωt) if t 0 2(t) = 0 otherwise 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called frequency response SingleInputSingleOutput Systems 122
23 Openloop frequency response Openloop experiment: Choose a test frequency ω > 0. Introduce the basis functions 1 if t < 0 ϕ 1 (t) = and ϕ sin(ωt) if t 0 2(t) = 0 otherwise 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called frequency response SingleInputSingleOutput Systems 123
24 Openloop frequency response Openloop experiment: Choose a test frequency ω > 0. Introduce the basis functions 1 if t < 0 ϕ 1 (t) = and ϕ sin(ωt) if t 0 2(t) = 0 otherwise 0 otherwise Choose control step parameters a 1, a 2 R Send the control input u(t) = a 1 ϕ 1 (t) + a 2 ϕ 2 (t) to actuator Associated output signal y is called frequency response SingleInputSingleOutput Systems 124
25 Example 1 SingleInputSingleOutput Systems 125
26 Example 1 no steady state, but periodic limit cycle SingleInputSingleOutput Systems 126
27 Example 2 (resonance effect) SingleInputSingleOutput Systems 127
28 Example 2 (resonance effect) SingleInputSingleOutput Systems 128
29 BoundedInputBoundedOutput Systems A system is called a boundedinputboundedoutput (BIBO) system if there exists for every contant C 1 < a constant C 2 < such that u L C 1 y L C 2 In words: if the input is bounded, then the output is bounded Warning: the amplification factor y L u L might be large SingleInputSingleOutput Systems 129
30 BoundedInputBoundedOutput Systems A system is called a boundedinputboundedoutput (BIBO) system if there exists for every contant C 1 < a constant C 2 < such that u L C 1 y L C 2 In words: if the input is bounded, then the output is bounded Warning: the amplification factor y L u L might be large SingleInputSingleOutput Systems 130
31 BoundedInputBoundedOutput Systems A system is called a boundedinputboundedoutput (BIBO) system if there exists for every contant C 1 < a constant C 2 < such that u L C 1 y L C 2 In words: if the input is bounded, then the output is bounded Warning: the amplification factor y L u L might be large SingleInputSingleOutput Systems 131
32 Resonance catastrophe Small inputs can lead to large outputs... SingleInputSingleOutput Systems 132
33 System Response Norms The largest possible ratio between input and output norm, sup u y u is called system response norm (largest possible amplification factor). Common choices for are L 2 or L norms SingleInputSingleOutput Systems 133
34 System Response Norms The largest possible ratio between input and output norm, sup u y u is called system response norm (largest possible amplification factor). Common choices for are L 2 or L norms SingleInputSingleOutput Systems 134
35 Contents Introduction OpenLoop ControlResponse Proportional Control General PID Control SingleInputSingleOutput Systems 135
36 Set points Goal: Design the closedloop system such that y(t ) y ref after a short transient phase T > 0 Notation: The point y ref R is called set point The function e(t) = y(t) y ref is called the tracking error SingleInputSingleOutput Systems 136
37 Set points Goal: Design the closedloop system such that y(t ) y ref after a short transient phase T > 0 Notation: The point y ref R is called set point The function e(t) = y(t) y ref is called the tracking error SingleInputSingleOutput Systems 137
38 Feedback laws Strategy: If we observe y(t), we set u(t) = µ(y(t)) The function µ : R R is called feedback law Secret assumption : Evaluation of µ is fast compared to the system dynamics SingleInputSingleOutput Systems 138
39 Feedback laws Strategy: If we observe y(t), we set u(t) = µ(y(t)) The function µ : R R is called feedback law Secret assumption : Evaluation of µ is fast compared to the system dynamics SingleInputSingleOutput Systems 139
40 Affine feedback (Pcontrol) Proportional control is based on affine feedback laws, µ(y(t)) = u ref + K(y(t) y ref ) = u ref + K e(t) Constant u ref R is a control offset The constant K R is called the proportional gain SingleInputSingleOutput Systems 140
41 Affine feedback (Pcontrol) Proportional control is based on affine feedback laws, µ(y(t)) = u ref + K(y(t) y ref ) = u ref + K e(t) Constant u ref R is a control offset The constant K R is called the proportional gain SingleInputSingleOutput Systems 141
42 Affine feedback (Pcontrol) Proportional control is based on affine feedback laws, µ(y(t)) = u ref + K(y(t) y ref ) = u ref + K e(t) Constant u ref R is a control offset The constant K R is called the proportional gain SingleInputSingleOutput Systems 142
43 Pcontrol tuning: Step 1 If the system is stable: Play around with the open loop system. Adjust u(t) = u ref such that openloop system satisfies y(t ) y ref after (a possibly long) time T. SingleInputSingleOutput Systems 143
44 Pcontrol tuning: Step 1 If the system is stable: Play around with the open loop system. Adjust u(t) = u ref such that openloop system satisfies y(t ) y ref after (a possibly long) time T. SingleInputSingleOutput Systems 144
45 Tuning Step 1 SingleInputSingleOutput Systems 145
46 Tuning Step 1 SingleInputSingleOutput Systems 146
47 Tuning Step 1 SingleInputSingleOutput Systems 147
48 Pcontrol tuning: Step 2 Increase / decrease the control gain K and test the closedloop behavior of the system Finetuning of u ref and K if needed. SingleInputSingleOutput Systems 148
49 Pcontrol tuning: Step 2 Increase / decrease the control gain K and test the closedloop behavior of the system Finetuning of u ref and K if needed. SingleInputSingleOutput Systems 149
50 Tuning Step 2 SingleInputSingleOutput Systems 150
51 Tuning Step 2 K = 0.25 [ ] V C SingleInputSingleOutput Systems 151
52 Tuning Step 2 K = 0.50 [ ] V C SingleInputSingleOutput Systems 152
53 Tuning Step 2 K = 2.00 [ ] V C SingleInputSingleOutput Systems 153
54 Tuning Step 2 K = 1.00 [ ] V C SingleInputSingleOutput Systems 154
55 Pcontrol tuning If the system is unstable at the setpoint: openloop experiments are not possible / difficult prestabilize the system with a suitable K Tune u ref (and K) to reduce output error SingleInputSingleOutput Systems 155
56 Pcontrol tuning If the system is unstable at the setpoint: openloop experiments are not possible / difficult prestabilize the system with a suitable K Tune u ref (and K) to reduce output error SingleInputSingleOutput Systems 156
57 Pcontrol tuning If the system is unstable at the setpoint: openloop experiments are not possible / difficult prestabilize the system with a suitable K Tune u ref (and K) to reduce output error SingleInputSingleOutput Systems 157
58 Inverted pendulum: Pcontrol u ref = 0 K = 0.5 [ ] N rad SingleInputSingleOutput Systems 158
59 Inverted pendulum: Pcontrol u ref = 0 K = 1.0 [ ] N rad SingleInputSingleOutput Systems 159
60 Inverted pendulum: Pcontrol u ref = 0 K = 2.0 [ ] N rad SingleInputSingleOutput Systems 160
61 Inverted pendulum: Pcontrol u ref = 0 K = 4.0 [ ] N rad SingleInputSingleOutput Systems 161
62 Inverted pendulum: Pcontrol u ref = 0 K = 8.0 [ ] N rad SingleInputSingleOutput Systems 162
63 Proportional Control Circuit Replace computer by electrical circuit (e.g. for our oven): (here y(t), y ref and K(y(t) y ref ) are voltages) Use summing amplifier (or modify above circuit) to add offset SingleInputSingleOutput Systems 163
64 Proportional Control Circuit Replace computer by electrical circuit (e.g. for our oven): (here y(t), y ref and K(y(t) y ref ) are voltages) Use summing amplifier (or modify above circuit) to add offset SingleInputSingleOutput Systems 164
65 Contents Introduction OpenLoop ControlResponse Proportional Control General PID Control SingleInputSingleOutput Systems 165
66 Affine feedback (PIcontrol) ProportionalIntegral (PI) control uses feedback laws of the form t u(t) = u ref + K(y(t) y ref ) + K I (y(τ) y ref ) dτ t = u ref + Ke(t) + K I e(τ) dτ The constant K I R is called the integral gain For discretetime systems: replace integral by running sum 0 0 t 0 e(τ) dτ where δ is the samping time t/δ 1 k=0 e(kδ) δ, SingleInputSingleOutput Systems 166
67 Affine feedback (PIcontrol) ProportionalIntegral (PI) control uses feedback laws of the form t u(t) = u ref + K(y(t) y ref ) + K I (y(τ) y ref ) dτ t = u ref + Ke(t) + K I e(τ) dτ The constant K I R is called the integral gain For discretetime systems: replace integral by running sum 0 0 t 0 e(τ) dτ where δ is the samping time t/δ 1 k=0 e(kδ) δ, SingleInputSingleOutput Systems 167
68 Affine feedback (PIcontrol) ProportionalIntegral (PI) control uses feedback laws of the form t u(t) = u ref + K(y(t) y ref ) + K I (y(τ) y ref ) dτ t = u ref + Ke(t) + K I e(τ) dτ The constant K I R is called the integral gain For discretetime systems: replace integral by running sum 0 0 t 0 e(τ) dτ where δ is the samping time t/δ 1 k=0 e(kδ) δ, SingleInputSingleOutput Systems 168
69 Integral Gain Tuning K = 1.0 [ ] V C K I = 0.0 [ ] V C s SingleInputSingleOutput Systems 169
70 Integral Gain Tuning K = 1.0 [ ] V C K I = 1.0 [ V C s ] SingleInputSingleOutput Systems 170
71 Integral Gain Tuning K = 1.0 [ ] V C K I = 0.5 [ V C s ] SingleInputSingleOutput Systems 171
72 Integral Gain Tuning K = 1.0 [ ] V C K I = 0.1 [ V C s ] SingleInputSingleOutput Systems 172
73 Integral Gain Circuit Operational amplifier integrator: Needs to be refined for nonideal opamps. SingleInputSingleOutput Systems 173
74 Integral Gain Circuit Operational amplifier integrator: Needs to be refined for nonideal opamps. SingleInputSingleOutput Systems 174
75 Affine feedback (PDcontrol) ProportionalDifferential (PD) control uses feedback laws of the form u(t) = u ref + Ke(t) + K D ė(τ) The function ė(t) denotes the time derivative of the error function e(t) = y(t) y ref. The constant K D R is called the differential gain Assumes that we can measure ẏ(t) directly ( velocity measurement) SingleInputSingleOutput Systems 175
76 Affine feedback (PDcontrol) ProportionalDifferential (PD) control uses feedback laws of the form u(t) = u ref + Ke(t) + K D ė(τ) The function ė(t) denotes the time derivative of the error function e(t) = y(t) y ref. The constant K D R is called the differential gain Assumes that we can measure ẏ(t) directly ( velocity measurement) SingleInputSingleOutput Systems 176
77 Affine feedback (PDcontrol) ProportionalDifferential (PD) control uses feedback laws of the form u(t) = u ref + Ke(t) + K D ė(τ) The function ė(t) denotes the time derivative of the error function e(t) = y(t) y ref. The constant K D R is called the differential gain Assumes that we can measure ẏ(t) directly ( velocity measurement) SingleInputSingleOutput Systems 177
78 Affine feedback (PDcontrol) ProportionalDifferential (PD) control uses feedback laws of the form u(t) = u ref + Ke(t) + K D ė(τ) The function ė(t) denotes the time derivative of the error function e(t) = y(t) y ref. The constant K D R is called the differential gain Assumes that we can measure ẏ(t) directly ( velocity measurement) SingleInputSingleOutput Systems 178
79 Inverted pendulum: PDcontrol u ref = 0 K = 2.0 [ ] N rad [ K D = 0.1 N rad/s ] SingleInputSingleOutput Systems 179
80 Inverted pendulum: PDcontrol u ref = 0 K = 2.0 [ ] N rad [ K D = 0.5 N rad/s ] SingleInputSingleOutput Systems 180
81 Inverted pendulum: PDcontrol u ref = 0 K = 2.0 [ ] N rad [ K D = 1.0 N rad/s ] SingleInputSingleOutput Systems 181
82 Inverted pendulum: PDcontrol u ref = 0 K = 2.0 [ ] N rad [ K D = 4.0 N rad/s ] SingleInputSingleOutput Systems 182
83 Affine feedback (PIDcontrol) ProportionalIntegralDifferential (PID): t u(t) = u ref + Ke(t) + K I e(τ) dτ + K D ė(τ) 0 Ruleofthethumb based PID tuning use K to prestabilize system use K I to reduce /eliminate offsets use K D to reduce oscillations/overshoot Warning: there are systems which behave differently; we ll analyze this later in more detail SingleInputSingleOutput Systems 183
84 Affine feedback (PIDcontrol) ProportionalIntegralDifferential (PID): t u(t) = u ref + Ke(t) + K I e(τ) dτ + K D ė(τ) 0 Ruleofthethumb based PID tuning use K to prestabilize system use K I to reduce /eliminate offsets use K D to reduce oscillations/overshoot Warning: there are systems which behave differently; we ll analyze this later in more detail SingleInputSingleOutput Systems 184
85 Affine feedback (PIDcontrol) ProportionalIntegralDifferential (PID): t u(t) = u ref + Ke(t) + K I e(τ) dτ + K D ė(τ) 0 Ruleofthethumb based PID tuning use K to prestabilize system use K I to reduce /eliminate offsets use K D to reduce oscillations/overshoot Warning: there are systems which behave differently; we ll analyze this later in more detail SingleInputSingleOutput Systems 185
86 Affine feedback (PIDcontrol) ProportionalIntegralDifferential (PID): t u(t) = u ref + Ke(t) + K I e(τ) dτ + K D ė(τ) 0 Ruleofthethumb based PID tuning use K to prestabilize system use K I to reduce /eliminate offsets use K D to reduce oscillations/overshoot Warning: there are systems which behave differently; we ll analyze this later in more detail SingleInputSingleOutput Systems 186
87 Affine feedback (PIDcontrol) ProportionalIntegralDifferential (PID): t u(t) = u ref + Ke(t) + K I e(τ) dτ + K D ė(τ) 0 Ruleofthethumb based PID tuning use K to prestabilize system use K I to reduce /eliminate offsets use K D to reduce oscillations/overshoot Warning: there are systems which behave differently; we ll analyze this later in more detail SingleInputSingleOutput Systems 187
88 PIDcontrol circuit (many variants exist) SingleInputSingleOutput Systems 188
89 Summary / Keywords SISO systems (without using models) Control parameterization Openloop response, step response, frequency response Steady state / periodic limit cycle BIBO systems, system response norm P, PI, PD, PID control + tuning by hand basic electrical circuits for PID control SingleInputSingleOutput Systems 189
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