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## 22. Exploiting SISO Techniques in MIMO Control

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The previous chapter gave an introduction to MIMO control system synthesis by showing how SISO methods could sometimes be used in MIMO problems. However, some MIMO problems require a fundamentally MIMO approach. This is the topic of the current chapter. We will emphasize methods based on optimal control theory. There are three reasons for this choice:

1
It is relatively easy to understand.
2
It has been used in a myriad of applications (Indeed, the authors have used these methods on approximately 20 industrial applications).
3
It is a valuable precursor to other advanced methods - e.g. model predictive control which is explained in the next chapter.

The analysis presented in this chapter builds on the results in Chapter 18 where state space design methods were briefly described in the SISO context. We recall from that chapter, that the two key elements were

• state estimation by an observer
• state estimate feedback

We will mirror these elements here for the MIMO case.

### Summary

• Full multivariable control incorporates the interaction dynamics rigorously and explicitly.

• The fundamental SISO synthesis result that, under mild conditions, the nominal closed loop poles can be assigned arbitrarily carries over to the MIMO case.

• Equivalence of state feedback and frequency domain pole placement by solving the (multivariable) Diophantine Equation carries over as well.

• Due to the complexities of multivariable systems, criterion based synthesis (briefly alluded to in the SISO case) gains additional motivation; it is also a powerful way to pre-compensate a system which is subsequently trimmed with a MIMO Q-parametrization.

• A popular family of criteria are functionals involving quadratic forms of control error and control effort.

• For a general nonlinear formulation, the optimal solution is characterized by a two-point boundary value problem.

• In the linear case (the so-called linear quadratic regulator, LQR), the general problem reduces to the solution of the continuous time dynamic Riccati equation which can be feasibly solved, leading to time-variable state feedback.

• After initial conditions decay, the optimal time-varying solution converges to a constant state feedback, the so-called steady state LQR solution.

• It is frequently sufficient to neglect the initial transient of the strict LQR and only implement the steady state LQR.

• The steady state LQR is equivalent to either
• a model matching approach, where a desired complementary sensitivity is specified and a controller is computed that matches it as closely as possible according to some selected measure.
• pole placement, where a closed loop polynomial is specified and a controller is computed to achieve it.

• Thus, LQR, model-matching and pole-placement are mathematically equivalent, although they do offer different tuning parameters.

 Equivalent synthesis techniques Tuning parameters LQR Relative penalties on control error versus control effort. Model matching Closed loop complementary sensitivity reference model and weighted penalty on the difference to the control loop Pole placement Closed loop polynomial

• These techniques can be extended to discrete time systems.

• There is a very close connection to the ‘dual’ problem of filtering, i.e., the problem of inferring a state from a related (but not exactly invertible) set of measurements.

• Optimal filter design based on quadratic criteria leads again to a Riccati equation.

• The filters can be synthesized and interpreted equivalently in a