# Closed-loop system 2/1/2016. Generally MIMO case. Two-degrees-of-freedom (2 DOF) control structure. (2 DOF structure) The closed loop equations become

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1 Closed-loop system enerally MIMO case Two-degrees-of-freedom (2 DOF) control structure (2 DOF structure) 2 The closed loop equations become solving for z gives where is the closed loop transfer function (matrix) Control error e = r - z Use the following abbreviations S is the sensitivity function T is the complementary sensitivity function 3 4

2 is the input sensitivity function enerally: (in all frequencies) Fundamental relationship The most important formula in control engineering! Note that the transfer functions (and matrices) are complex-valued. Often in which case (One-degree-of-freedom ( DOF) control configuration) What about the control signal where System with inputs and outputs 5 6 Internal stability But how about transfer function from reference to control? Zero-pole-cancellations are problematic Ex. process controller closed loop system; stable The controller is unstable, and therefore useless. The unstable mode corresponding to s was not observable in the closed loop transfer function

3 Ex. process Sensitivity function controller closed loop All stable; is there no s problem now? Calculate the transfer function from wu to output. Closed loop from r (or w) to control u or unstable! r does not activate this mode 9 But do safe cancellations exist? Internal stability of the closed-loop system Ex. The process is unstable all is nice and beautiful; Feedback can stabilize an unstable process. Reason for a definition: The system is internally stable, if (after all cancellations in the calculation of the transfer functions have been made) the following transfer functions 2 3

4 Robustness are stable and the pre-compensator Fr is also stable. nominal model true system model error Similarly: Is the closed loop stable in spite of model error? (robust stability) Does the system meet performance specifications in spite of model error? ( robust performance) 3 4 Earlier the matrix inversion lemma was presented. nominal model Now consider the push-through -rule, which is often useful in matrix manipulations related to multivariable systems. Let A and B be such matrices that both AB and BA are defined and square matrices. Then it holds Note. A and B need not be square matrices. Assume that the disturbances are zero, but there is model error in real output output predicted by the model 5 6 4

5 The following results can be derived But what about stability: It is seen that the sensitivity function S shows, how the model error maps into the output error. For those frequencies where the sensitivity function is small, the effect of the modeling error in output is also small. Consider the loop gain, set the reference to zero, for simplicity. Idea: study the transfer function from x to 7 8 ; The Small gain theorem guarantees that the closed loop is stable, if subsystems are stable and the gain of their product is smaller than one. (When applying the Small gain theorem, note that the system is linear.) Use the push-through -rule 9 2 5

6 Because and T are both stable transfer functions, the system is stable if (sufficient condition) for which it is necessary that Design specifications for the closed loop system Design the compensator such that the controlled variable follows the reference as close as possible in spite of disturbances, measurement errors and model uncertainties. Use the control signal as little as possible. Sufficient condition for robust stability 2 22 Let It then holds that. small, closed loop tf. close to I. 2. Sensitivity function S small, so that disturbances and model errors would have a minor impact on the output. Now it is easy to list demands for control: 3. Complementary sensitivity function T should be small, so that measurement disturbances would not affect much and the closed loop stability would not be in danger. 4. The tfs. and should not be large. But: always hold, there are inevitable conflicts (fundamental limitations in control performance)

7 Specifications in time domain:..9 y M e Static error corresponding to step input (if T = c) To minimize the static error the sensitivity must be small at low frequancies. M overshoot e static error Tr rise time Ts settling time Tr Ts t That means that the compensator must have high gain (e.g. integration) at low frequencies. Other criteria: design the compensator such that c and S are as desired; or their poles are at desired locations Frequency domain specifications: Minimize where a natural way to formalize the goal as an optimal control problem, in which the control law must be found such that the given criterion is minimized. But: it is not straightforward to calculate a control law, that directly fulfils some time domain criteria. Specifications in frequency domain are easier to deal with in some sense. ain crossover S -3 db from below T -3 db from above There are different definitions for bandwidth (meaning the frequency range where system can follow the sinusoidal input)

8 A typical S-curve T small to compensate measurement disturbances, but also to guarantee robust stability. When we want to limit the use of control signal, we set suppress the disturbances 29 3 Design specifications: Choose the weights Unfortunately this is not always possible. and design the controller such that Instead: minimize SISO: N is a vector; hence H control problem: K is the compensator

9 9 Main topics 33 Closed loop equations Effect of model errors if, Robust stability ) (, ) (, ) ( T F I S z S I z r z I y c

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