Lecture #1 16.31 Feedback Control Copyright c2001 by JJonathandHow D
Lecture #1 16.31 Feedback Control Copyright 2001 by Jonathan How. 1
Fall 2001 16.311-1 Introduction d(t) r(t) e(t) yt K(s Goal: Design a controller K(s so that the system has some desired characteristics. Typical objectives Stabilize the system( Stabilization) Regulate the system about some design point(Regulation Follow a given class of command signals(Tracking) Reduce the response to disturbances(Disturbance Rejection Typically think of closed-loop control > so we would analyze the closed-loop dynamics Open-loop control also possible(called"feedforward")-more prone to modeling errors since inputs not changed as a result of measured error Note that a typical control system includes the sensors, actuators and the control lay The sensors and actuators need not always be physical devices (e. g, economic systems) a good selection of the sensor and actuator can greatly simplify the control design process Course concentrates on the design of the control law given the rest of the system(although we will need to model the system)
Fall 2001 16.31 1—1 Introduction K(s) G(s) - 6 ? — u(t) r(t) e(t) y(t) d(t) • Goal: Design a controller K(s) so that the system has some desired characteristics. Typical objectives: — Stabilize the system (Stabilization) — Regulate the system about some design point (Regulation) — Follow a given class of command signals (Tracking) — Reduce the response to disturbances. (Disturbance Rejection) • Typically think of closed-loop control → so we would analyze the closed-loop dynamics. — Open-loop control also possible (called “feedforward”) — more prone to modeling errors since inputs not changed as a result of measured error. • Note that a typical control system includes the sensors, actuators, and the control law. — The sensors and actuators need not always be physical devices (e.g., economic systems). — A good selection of the sensor and actuator can greatly simplify the control design process. — Course concentrates on the design of the control law given the rest of the system (although we will need to model the system)
Fall 2001 6.311-2 Why Control? Easy question to answer for aerospace because many vehicles(space- craft, aircraft, rockets) and aerospace processes(propulsion ) need to be controlled just to function Example: the F-117 does not even fly without computer control and the x-29 is unstable
Fall 2001 16.31 1—2 Why Control? • Easy question to answer for aerospace because many vehicles (spacecraft, aircraft, rockets) and aerospace processes (propulsion) need to be controlled just to function — Example: the F-117 does not even fly without computer control, and the X-29 is unstable
Fall 2001 16.311-3 Feedback Control Approach Establish control objectives -Qualitative- dont use too much fuel Quantitative- settling time of step response reduce size/complexity>Design model Accuracy? Error model? ● Design controller Select technique(SISO, MIMO),(classical, state-space) Choose parameters(ROT, optimization Analyze closed-loop performance Meet objectives? Analysis, simulation, experimentation Ye→done,No→ iterate
Fall 2001 16.31 1—3 Feedback Control Approach • Establish control objectives — Qualitative — don’t use too much fuel — Quantitative — settling time of step response <3 sec — Typically requires that you understand the process (expected commands and disturbances) and the overall goals (bandwidths). — Often requires that you have a strong understanding of the physical dynamics of the system so that you do not “fight” them in appropriate (i.e., inefficient) ways. • Select sensors & actuators — What aspects of the system are to be sensed and controlled? — Consider sensor noise and linearity as key discriminators. — Cost, reliability, size, . . . • Obtain model — Analytic (FEM) or from measured data (system ID) — Evaluation model → reduce size/complexity → Design model — Accuracy? Error model? • Design controller — Select technique (SISO, MIMO), (classical, state-space) — Choose parameters (ROT, optimization) • Analyze closed-loop performance. Meet objectives? — Analysis, simulation, experimentation, . . . — Yes ⇒ done, No ⇒ iterate . .
Fall 2001 Example: Blimp control.31 1-4 Control objective Stabilization Red blimp tracks the motion of the green blimp ● Sensors GPS for positioning Compass for heading Gyros/GPS for roll attitude Actuators-electric motors(propellers) are very nonlinear · Dynamics rigid body" with strong apparent mass effect Roll modes · Modeling Analytic models with parameter identification to determine "mass Disturbances- wind
Fall 2001 16.31 1—4 Example: Blimp Control • Control objective — Stabilization — Red blimp tracks the motion of the green blimp • Sensors — GPS for positioning — Compass for heading — Gyros/GPS for roll attitude • Actuators — electric motors (propellers) are very nonlinear. • Dynamics — “rigid body” with strong apparent mass effect. — Roll modes. • Modeling — Analytic models with parameter identification to determine “mass”. • Disturbances — wind
Fall 2001 16.311-5 State-Space Approach Basic questions that we will address about the state-space approach What are state-space models? Why should we use them? How are they related to the transfer functions used in classical control design? How do we develop a state-space model? How do we design a controller using a state-space model? ● Bottom line: 1. What: representation of the dynamics of an n n-order system using n first-order differential equations mq+cq+ kq 0 -k/m-c/mq Ax+ Bu 2. Why: State variable form convenient way to work with complex d namics Matrix format easy to use on computers Transfer functions only deal with input/output behavior, but state-space form provides easy access to the "internal" fea- tures/response of the system Allows us to explore new analysis and synthesis tools Great for multiple-input multiple-output systems MIMO) which are very hard to work with using transfer functions
Fall 2001 16.31 1—5 State-Space Approach • Basic questions that we will address about the state-space approach: — What are state-space models? — Why should we use them? — How are they related to the transfer functions used in classical control design? — How do we develop a state-space model? — How do we design a controller using a state-space model? • Bottom line: 1. What: representation of the dynamics of an nth-order system using n first-order differential equations: mq¨ + cq˙ + kq = u ⇒ q˙ q¨ = 0 1 −k/m −c/m q q˙ + 0 1/m u ⇒ x˙ = Ax + Bu 2. Why: — State variable form convenient way to work with complex dynamics. Matrix format easy to use on computers. — Transfer functions only deal with input/output behavior, but state-space form provides easy access to the “internal” features/response of the system. — Allows us to explore new analysis and synthesis tools. — Great for multiple-input multiple-output systems (MIMO), which are very hard to work with using transfer functions
Fall 2001 16.311-6 3. How: There are a variety of ways to develop these state-space models. We will explore this process in detail Linear systems theor 4. Control design: Split into 3 main parts Full-state feedback-ficticious since requires more information than typically(ever? ) available Observer/ estimator design-process of "estimating"the sys- tem state from the measurements that are available Dymamic output feedback-combines these two parts with provable guarantees on stability (and performance Fortunately there are very simple numerical tools available perform each of these steps Removes much of the" art"and or"magic"required in classi- cal control design - design process more systematic Word of caution:- Linear systems theory involves extensive use of linear algebra Will not focus on the theorems/proofs in class -details will be handed out as necessary, but these are in the textbooks Will focus on using the linear algebra to understand the behav- IOI of the system dynamics so that we can modify them usi control. "Linear algebra in action Even so, this will require more algebra that most math courses that you have taken
Fall 2001 16.31 1—6 3. How: There are a variety of ways to develop these state-space models. We will explore this process in detail. — “Linear systems theory” 4. Control design: Split into 3 main parts — Full-state feedback —ficticious since requires more information than typically (ever?) available — Observer/estimator design —process of “estimating” the system state from the measurements that are available. — Dynamic output feedback —combines these two parts with provable guarantees on stability (and performance). — Fortunately there are very simple numerical tools available to perform each of these steps — Removes much of the “art” and/or “magic” required in classical control design → design process more systematic. • Word of caution: — Linear systems theory involves extensive use of linear algebra. — Will not focus on the theorems/proofs in class — details will be handed out as necessary, but these are in the textbooks. — Will focus on using the linear algebra to understand the behavior of the system dynamics so that we can modify them using control. “Linear algebra in action” — Even so, this will require more algebra that most math courses that you have taken . . .
Fall 2001 16.311-7 My reasons for the review of classical design State-space techniques are just another to design a controller But it is essential that you understand the basics of the control design process Otherwise these are just a"bunch of numerical tools To truly understand the output of the state-space control design process, I think it is important that you be able ze l from a classical perspective * Try to answer“ why did it do that”? k Not always possible, but always a good goal Feedback: muddy cards and office hours Help me to know whether my assumptions about your back grounds is correct and whether there are any questions about the material Matlab will be required extensively. If you have not used it before then start practicing
Fall 2001 16.31 1—7 • My reasons for the review of classical design: — State-space techniques are just another to design a controller — But it is essential that you understand the basics of the control design process — Otherwise these are just a “bunch of numerical tools” — To truly understand the output of the state-space control design process, I think it is important that you be able to analyze it from a classical perspective. ∗ Try to answer “why did it do that”? ∗ Not always possible, but always a good goal. • Feedback: muddy cards and office hours. — Help me to know whether my assumptions about your backgrounds is correct and whether there are any questions about the material. • Matlab will be required extensively. If you have not used it before, then start practicing
Fall 2001 16.311-8 System Modeling Investigate the model of a simple system to explore the basics of system dynamics Provide insight on the connection between the system response d the pole locations M2 Consider the simple mechanical system(2MSS)-derive the system model 1. Start with a free body diagram 2. Develop the 2 equations of motion m1=k(x2-x1) k( )+F 3. How determine the relationships between 1, c2 and F? Numerical integration- good for simulation, but not analysis Use laplace transform to get transfer functions Fast /easy/lots of tables k Provides lots of information(poles and zeros)
Fall 2001 16.31 1—8 System Modeling • Investigate the model of a simple system to explore the basics of system dynamics. — Provide insight on the connection between the system response and the pole locations. • Consider the simple mechanical system (2MSS) — derive the system model 1. Start with a free body diagram 2. Develop the 2 equations of motion m1x¨1 = k(x2 − x1) m2x¨2 = k(x1 − x2) + F 3. How determine the relationships between x1, x2 and F? — Numerical integration - good for simulation, but not analysis — Use Laplace transform to get transfer functions ∗ Fast/easy/lots of tables ∗ Provides lots of information (poles and zeros)
Fall 2001 16.311-9 Laplace transform Cif(t)=-f(t)e-stat Key point: If Ca(t)=X(s, then Ci(t)=sX(s)assuming that the initial conditions are zero Apply to the model C{m1x1-k(x2-x1}=(m1s2+k)X1(s)-kX(s)=0 C{m2xi2-k(x1-m2)-F}=(m2s2+k)X2(s)-kX1(s)-F(s)=0 m1s+k k X 0 k +k[X2(s)」[F(s Perform some algebra to get m1s+k F(s)m1m294(82+k(1/m1+1/mn=G2(s) G2(s) is the transfer function between the input F and the system response a2
Fall 2001 16.31 1—9 • Laplace transform L{f(t)} ≡ 8 ∞ 0− f(t)e−stdt — Key point: If L{x(t)} = X(s), then L{x˙(t)} = sX(s) assuming that the initial conditions are zero. • Apply to the model L{m1x¨1 − k(x2 − x1)} = (m1s2 + k)X1(s) − kX2(s)=0 L{m2x¨2 − k(x1 − x2) − F} = (m2s2 + k)X2(s) − kX1(s) − F(s)=0 m1s2 + k −k −k m2s2 + k X1(s) X2(s) = 0 F(s) • Perform some algebra to get X2(s) F(s) = m1s2 + k m1m2s2(s2 + k(1/m1 + 1/m2)) ≡ G2(s) • G2(s) is the transfer function between the input F and the system response x2