9/11/2014 AS-74.3123 Model based control systems (4 cr. ) Main idea of the course • Classical control theory: SISO-systems, linear or linearized system models • Extension to multivariable (MIMO) systems • Performance and limitations of control • Robust control, IMC-control • Dynamic programming, • Linear quadratic control, LQ-, LQG-controllers • Loop-shaping methods, H-infinity control Kai Zenger, TuAs 3567, first.lastname(at)aalto.fi Lectures on Thursdays at 10.15 – 12.00 AS3 Sergey Samokhin, first.lastname(at)aalto.fi Exercises on Mondays at 12.15 – 14.00 AS3 Requirements for passing the course: Exam and an assignment Textbook : Glad, Ljung: Control Theory, (multivariable and nonlinear methods), Taylor and Francis, 2000. 2 Time-table • • • • • • • • • • Time-table(cont..) L1: Introduction, the control problem L2,3: Linear control systems L4: Disturbances and the Kalman filter L5,6: Fundamental limitations of control L7: (cont..) L8: Control structures and control design L9: Dynamic programming L10: LQ-, LQG-control L11,12: Modern design methods L13: Exam • Exam on Thursday 11.12.2014, 10:00-12:00 AS3. Next one 26.1.2015, 16-19. Note that the first exam is in the last week of the study period and lasts 2 hours. (The problems in all following exams are planned two hours long). • A project work (mandatory) is given during the course. After being accepted it gives 1-5 bonus points to the exam. The bonus points are valid until the course lectures start again (September 2015). • Homework problems are given during the course. Together they constitute one problem in the exam. • Lecture slides and problems with solutions appear on the course pages in the Noppa portal. • Use the web-oodi to register yourself to the course and to the exam. 3 4 1 9/11/2014 Example 1 Basics.... Multivariable system: • Basics in continuous control theory, e.g. Lewis, Yang: ”Basic Control Systems Engineering”, Dorf, Bishop: ”Modern Control Systems”, Wilkie, Johnson, Katebi: ”Control Engineering, an introductory course”. • Basics in discrete time control theory, e.g. Åström, Wittenmark: ”Computer-Controlled Systems, theory and design”, Franklin, Powell, Workman: ”Digital Control of Dynamic Systems”. Two inputs, two outputs. General multivariable system. MIMO = multiple inputs, multiple outputs Interconnections between the two loops ”Pairing problem” • one more book about control: (Glad, Ljung: Control Theory) 5 6 Let PI-controller (control 1 – output 1) Transfer function, when u2=0 Correspondingly (control 2 – output2) Transfer function, when u1=0 The result is as good as expected. 7 8 2 9/11/2014 But we can always change the pairing (control 1 – output 2), (control 2 – output1) This leads to the transfer function (ref. 1 – output1): In order to be stable, the coefficients in the denominator must be positive; K1 or K2 must be negative. Difficult to tune The good response is lost, when both controllers are operating, Reason? Interconnections? Analysis? 9 10 Example 2 Control signal limitations Double integrator in which the control is limited as Control law Open loop transfer function Analysis will later show that the difficulties are because the system has a RHP-zero (zero in the right half plane). Cross-over frequency 3 rad/s, phase margin 55 degrees 11 12 3 9/11/2014 The control problem The output deteriorates clearly because of the control signal limitation ”Given a system S and measurements y. Determine a control u such that the controlled variable z follows the reference (set point) r as close as possible irrespective of process disturbances w and measurement disturbances n.” 13 14 System In order to design the controller (R), the system (S) and disturbances w must be described (modelling, identification). On the other hand, there must be different design methods and approaches for different model classes (continuous/discrete, SISO/MIMO, linear/nonlinear). Modeling, classification of models, and analysis and synthesis methods of a wide application area are needed. causal/non-causal, static/dynamic, continuous/discrete, SISO/MIMO, time-invariant/timevarying, linear/nonlinear 15 16 4 9/11/2014 Signal and system ”sizes” If the signal z is a n-dimensional vector, its ”size” at time t can be defined as the Euclidean vector norm The size of the whole signal can be measured e.g. by -norm, ”infinity”-norm ”The regulator must be such that it compensates measurable disturbances, and the effect of nonmeasurable disturbances is as small as possible.” -norm, 2-norm 17 18 More generally: consider the system S discrete-time 2-norm Gain: The gain of the system is defined as The matrix equation can be understood as a system, in which the matrix A maps the input x to the output y in which u covers all possible signals with a finite 2-norm. The gain can be infinite, too. The operator (or matrix) A norm is defined as the largest possible gain, as x changes For a cascade connection of systems 19 20 5 9/11/2014 Ex. static nonlinear system Ex. Integrator Consider the system and choose the input in which and where the equality holds for some We obtain which has the 2-norm equal to 1. The output is identically 1, when The gain is then smaller or equal to K. But choose so that the gain is K. The 2-norm is infinite and the gain of the integrator is thus infinite. 21 22 Ex. Linear SISO system The gain is smaller or equal to K and in fact this value can be approached arbitrarily close. The gain is then By Parseval’s equation it follows Assume that which is denoted as and that the equality holds for some (H-infinity norm) The system norm is the same as the matrix norm Then it follows 23 24 6 9/11/2014 Stability and the ”Small Gain Theorem” ”Small Gain Theorem”: The closed loop system is BIBO stable, if the product of the system gains is smaller than1. The system is called input-output stable (BIBO-stable), if it has a finite gain. If S1 and S2 are linear, a weaker condition follows ”Proof”: Writing the system equations Inputs: r1, r2 Outputs: e1, e2, y1, y2 25 gives by the triangle inequality 26 Ex. Nonlinear static feedback and The gain from r1 and r2 to the output e1 is finite. Other cases correspondingly. Obs. It does not matter, whether the feedbacks are positive or negative (the norms of S and –S are the same). The result is ”conservative” ? If K < 2.5 the system is stable for sure Slope is K 27 28 7 9/11/2014 Stability of the solution trajectory: Do small changes in the initial conditions of a differential (or difference) equation change the solution essentially from the nominal solution? Lyapunovstability If for all ε there exists a δ such that the trajectory, when deviated originally by δ from the nominal solution, stays within ε, the solution is stable in the sense of 29 Lyapunov. It was discussed.... • Introduction of the course • Example of a multivariable system; analysis is difficult by classical SISO methods • Example of a nonlinear system; analysis difficult • General system models • Signal and system norms • Small Gain Theorem, Lyapunov-stability 30 8