Lecture plan: Control Systems II, IDSC, 2017

Transcription

1 Control Systems II MAVT, IDSC, Lecture 8 28/04/2017 G. Ducard

2 Lecture plan: Control Systems II, IDSC, 2017 SISO Control Design Lecture 1 Recalls, Introductory case study Lecture 2 Cascaded Control Lecture 3 Case study: Ball on Wheel (BoW) (Mathworks Guest) Controller Implementation Lecture 4 Control design for time delayed systems (Smith predictor), Robustness criteria Lecture 5 Robustness criteria (continued), Controller implementation, Feedforward action, Anti-reset windup Lecture 6 Bumpless transfer scheme, Digital control: discrete signals, aliasing, emulation MIMO Systems Lecture 7 Discrete control: analysis and synthesis, MIMO systems: Introduction Lecture 8 MIMO systems: analysis Lecture 9 MIMO systems: frequency domain MIMO Controller Synthesis Lecture 10 State feedback (LQR), BoW Lecture 11 Finite horizon LQR, Model Predictive Control (MPC) Lecture 12 State observer Lecture 13 Output feedback (LQG, LTR) G. Ducard 2

3 Lecture 8 Class Content 1. MIMO systems: heat transfer system 2. MIMO systems: analysis 3. MIMO systems: relative gain array G. Ducard 3

4 Lecture 8 1. MIMO Systems: Example: Heat exchanger 4

5 Lecture 8 1. MIMO System: a heat transfer system Assumptions 5

6 Lecture 8 1. MIMO System: a heat transfer system Modeling Definitions Modelling 6

7 Lecture 8 1. MIMO System: a heat transfer system Modeling Using the definitions we can convert the ODEs with physical parameters to ODEs with control-oriented parameters 7

8 Lecture 8 1. MIMO System: a heat transfer system State-space description Equilibria: = e + e 8

9 Lecture 8 1. MIMO System: a heat transfer system System properties Controllability / observability Eigenvalues 9

10 Lecture 8 1. MIMO System: a heat transfer system Transfer function Applying the laplace transform to the plant in state-space description and solving for Y U yields which yields the 2x2 transfer function 10

11 Lecture 8 1. MIMO System: a heat transfer system Static gain Evaluating the transfer function of the plant at = yields the static gain matrix which can be analyzed for different values of σ (importance of heat exchange) 11

12 Lecture 8 1. MIMO System: a heat transfer system Poles and Zeros? The transfer function of the system could have been experimentally obtained by identifying each of the four channels,, individually. Such an approach has some problems because looking at the four SISO transfer functions one might believe that: 1. the system has 8 poles (but it is known to only have two eigenvalues); and 2. the system has 2 minimumphase zeros ζ, = (but it is not clear if these zeros are active in a MIMO τ sense). Physical modelling superior to experimental system identification. Poles and zeros of MIMO systems are complicated. 12

13 Lecture 8 1. MIMO System: a heat transfer system Step response From now on, we use numerical values for the parameters. The controloriented parameters are then given by P 13

14 Lecture 8 1. MIMO System: a heat transfer system Bode diagram P 14

15 Lecture 8 1. MIMO System: a heat transfer system SISO control: one loop First, we would like to control one channel independently using the controller: i =

16 Lecture 7 3. MIMO System: introduction SISO control: two loops Can we also control both channels independently with two identical SISO controllers? Yes, works well! 16

17 Lecture 7 3. MIMO System: introduction Better heat transfer After modifications at the heat exchanger, the heat transfer coefficient could be increased by a factor of 100 (not realistic). k = W m K W m K P 17

18 Lecture 8 1. MIMO System: a heat transfer system SISO control II Clearly, with the stronger cross couplings the SISO approach yield much worse results. When a plant has strong cross couplings, a true MIMO controller is needed. 18

19 Lecture 8 Class Content 1. MIMO systems: heat transfer system 2. MIMO systems: analysis A. System representation B. Observability/ controllability C. Closed-loop stability D. Poles/zeros 3. MIMO systems: Relative Gain Array G. Ducard 19

20 Lecture 8 2. MIMO System: analysis A. State space representation Causes (signals) R MIMO system Effects (signals) : order of the system, size of the state vector : dimension of the input vector p : dimension of the output vector The state-space description of a MIMO system consists of four matrices A,,,, G. Ducard 20

21 Lecture 8 2. MIMO System: analysis A. Representation: transfer functions For SISO systems, the transfer function given by a rational function of the laplace variable. For MIMO systems, the transfer function given by a p matrix of rational functions of the laplace variable where each entry represents a SISO transfer function. 21

22 Lecture 8 2. MIMO System: analysis A. Transfer function interconnection In MIMO, P and are generally matrices a multiplication (series connection) is not commutative anymore. P P Ys () () s Us () MIMO open-loop gains: 1. we will mostly work with L e s = P s s, where the loop is broken at the signal e. 2. However, when < p, it makes sense to analyze L u s = s P s, where the loop is broken at the signal u. G. Ducard 22

23 Lecture 8 2. MIMO System: analysis A. Transfer function interconnection Return difference: e = I + L e Sensitivity: e = I + L e Complementary sensitivity: e s = I + L e L e s with e s + e s = I 23

24 Lecture 8 2. MIMO System: analysis B. Controllability / observability 24

25 Lecture 8 2. MIMO System: analysis Closed-loop stability However, the operator det destroys important information on the cross couplings between the individual channels, i.e., no information on robustness and performance can be deduced from the Nyquist theorem. 25

26 Lecture 8 2. MIMO System: analysis Closed-loop stability With controller and plant given in the state-space description, the closed-loop stability can be derived from the eigenvalues of the -matrix. Plant: = + u, = R, R, R Controller: = + e, u = R, R, R G. Ducard 26

27 Lecture 8 2. MIMO System: analysis Recap: poles and zeros of SISO systems Poles: P = π e π describes the natural dynamics of the system, i.e., it describes how the system evolves when the input is zero. Zeros: P = ζ = e ζ describes the zero dynamics of the system, i.e., when the system is driven by the input to evolve according the e ζ, the output is zero. 27

28 Lecture 8 2. MIMO System: analysis Matrix minors Matrix minors are the determinants of all square submatrices that can be formed from P. Die Minoren sind die Determinanten aller quadratischen Untermatrizen der Matrix P. P = A maximum minor is a minor that is formed using a submatrix with the largest possible dimension. For square matrices P, obviously, the only maximum minor corresponds to the matrix determinant itself. 28

29 Lecture 8 2. MIMO System: analysis Poles The poles of P are the roots of the least common denominator of all minors of P. Example: 29

30 Lecture 8 2. MIMO System: analysis Zeros The zeros of P ) are the roots of the greatest common divisor of the numerators of the maximum minors of P after normalization to have the pole polynomial of P as denominators. Example continued: 30

31 Lecture 8 2. MIMO System: analysis Zero/pole cancellations In MIMO systems, poles and zeros are associated with directions. Zero/pole cancellations only take place when the frequencies and the directions coincide. 31

32 Lecture 8 Class Content 1. MIMO systems: heat transfer system 2. MIMO systems: analysis A. System representation B. Observability/ controllability C. Closed-loop stability D. Poles/zeros 3. MIMO systems: Relative Gain Array G. Ducard 32

33 Lecture 8 3. MIMO System: Relative Gain Array RGA example 33

34 Lecture 8 3. MIMO System: Relative Gain Array RGA example = P + P P P + P 34

35 Lecture 8 3. MIMO System: Relative Gain Array RGA computation For square matrices, the RGA can be computed by Matlab: RGA = P.* inv(p). For general, non-square matrices, the RGA can be computed using the pseudo-inverse (careful: the function pinv can not be applied to transfer functions, but only to complex matrices) Matlab: RGA = P.* pinv(p). The results is a matrix of transfer functions (or many frequency response matrices), which can be analyzed in a bode diagram. 35

36 Lecture 8 3. MIMO System: Relative Gain Array RGA interpretation 1. An input-output paring such that the diagonal elements of the RGA matrix are close to one is preferable since this paring is related with a diagonal dominant plant. 2. Avoid input-output parings which generate diagonal negative elements on the RGA matrix with =. This condition is closely related to systems with lack of integrity; i.e., a system which cannot maintain stability if one of the diagonal closed loops is open. 3. High positive values in the diagonal elements of the RGA matrix indicate difficulty for designing diagonal controllers. Look at G = SISO control possible iff Diagonal entries positive Bode diagram with G, and G, RGA similar to identity around crossover frequency ω c 36

37 Lecture 8 3. MIMO System: Relative Gain Array RGA example: heat exchanger k = W m K k = W m K G =.... G =

38 Lecture 8 3. MIMO System: Relative Gain Array Additional information The columns and the rows of G always add up to 1. The RGA is invariant with respect to scaling, i.e., for any regular diagonal matrices i the equation G [P ] = G [ P ] holds true. The RGA of a triangular matrix P is the identity matrix. If the diagonal entries are small, but the off-diagonal entries are similar to 1, SISO control may still be used, but the inputs and outputs need to be paired differently. 38