6.1 The Mapping Algorithm

Using the state-feedback mapping algorithm proposed by Young and Willems (1972), the feedback matrix K that places the poles of a single-input system at _{1},_{2},...,_n can be found as

where is the controllability matrix

X is the lower Toeplitz matrix

formed from the coefficients of the open-loop system characteristic polynomial

and is a vector whose coefficients are the difference between the coefficients of the desired closed-loop characteristic polynomial

and those of the open-loop characteristic polynomial, that is,

This algorithm tends to be more accurate than Ackermann's, since the construction of the lower Toeplitz matrix X is numerically less demanding than the construction of the characteristic polynomial function (A). While this may not be significant for low-order systems, it can be significant for higher-order systems. The function StateFeedbackGains with the option Method Mapping implements this algorithm.

Method Mapping

compute the feedback matrix using the mapping algorithm

State-feedback design using the mapping algorithm.

Make sure the application is loaded.

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As an example, consider the locally linearized model of an aircraft system described in Section 9.1.1 of Control System Professional.

This is the state-space model of the aircraft.

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Here are the desired closed-loop system poles.

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This finds the state feedback using the mapping algorithm, finding the best combination of the inputs automatically.

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This is the infinity norm of this compensator.

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You can add some importance to the use of a particular system input by specifying weights in the fan-in vector. Here, both inputs are to be used, but input 2 is given a weight of 3 compared with input 1.

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The resulting state-feedback compensator has changed to reflect this weighting, but the closed-loop system poles are as specified.

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The relative weights of the infinity norms of the rows of the resulting compensator are as specified.

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Control System Professional's option choices ControlInput -> n and Method -> Automatic are also available and have the same meaning.

The system is completely controllable through input 2.

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This finds the state feedback using the mapping algorithm, using only the second control input.

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This is the infinity norm of the resulting state-feedback matrix.

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The feedback gains determined using both of the system inputs are considerably less than those obtained using only input 2.

This shows that all algorithms applied to single-input systems give the same result.

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You can also obtain the generic solution to a state-feedback design for a set of closed-loop poles at {p₁, p₂, ..., p_k}.

This is the state-space model of an inverted pendulum system from Section 9.2 of Control System Professional.

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This defines the desired closed-loop poles for the inverted pendulum system.

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These are the generic state-feedback gains for this pole set, using the mapping algorithm

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This is the resulting compensator for a particular set of numerical values of the system parameters and the desired poles.

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This is the inverted pendulum system with the numerical values of the system parameters.

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These are the resulting closed-loop system poles.

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